Abstract group 158400.f: $C_{15}:Q_8\times \SL(2,11)$

• Label: 158400.f
• Order: \( 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 11 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 5 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{2} \cdot 5 \cdot 11 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $40$
• Trans deg.: $2880$
• Rank: $2$

Group information

Description:	$C_{15}:Q_8\times \SL(2,11)$
Order:	\(158400\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 11 \)
Exponent:	\(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \)
Automorphism group:	$(C_2^5\times S_3).C_2^2.\PSL(2,11).C_2$, of order \(1013760\)\(\medspace = 2^{11} \cdot 3^{2} \cdot 5 \cdot 11 \)
Composition factors:	$C_2$ x 4, $C_3$, $C_5$, $\PSL(2,11)$
Derived length:	$2$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	10	11	12	15	20	22	30	33	44	55	60	66	110	132	165	220	330	660
Elements	1	3	332	1788	1324	996	3972	120	10128	3968	44112	360	11904	240	3360	480	51072	720	1440	960	960	13440	2880	3840	158400
Conjugacy classes	1	3	3	11	14	9	42	2	36	22	104	6	66	2	12	8	184	6	24	8	8	48	24	32	675
Divisions	1	3	3	11	4	9	12	1	22	6	29	3	18	1	6	1	28	3	3	2	1	6	3	2	178
Autjugacy classes	1	3	3	6	5	9	15	1	18	7	14	3	21	1	2	1	22	3	3	1	1	2	3	1	146

Dimension	1	2	4	5	6	8	10	11	12	16	20	22	24	40	44	48	80	88	96	160	176	192
Irr. complex chars.	20	25	0	40	40	0	150	20	130	0	125	25	100	0	0	0	0	0	0	0	0	0	675
Irr. rational chars.	4	3	5	0	0	3	16	4	4	1	16	3	11	25	5	27	16	3	17	9	1	5	178

Minimal presentations

Permutation degree:	$40$
Transitive degree:	$2880$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	not computed	none
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $40$ $\langle(36,37,40,38,39), (25,26)(27,29)(28,30)(31,32)(36,38,37,39,40), (1,2)(3,6) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 3 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 3 \end{array}\right), \left(\begin{array}{rrrr} 8 & 3 & 2 & 3 \\ 10 & 8 & 8 & 0 \\ 3 & 7 & 6 & 2 \\ 1 & 7 & 4 & 9 \end{array}\right), \left(\begin{array}{rrrr} 1 & 6 & 0 & 2 \\ 10 & 7 & 9 & 10 \\ 6 & 2 & 9 & 2 \\ 5 & 8 & 8 & 0 \end{array}\right), \left(\begin{array}{rrrr} 9 & 8 & 1 & 0 \\ 7 & 2 & 0 & 10 \\ 9 & 0 & 2 & 8 \\ 0 & 2 & 7 & 9 \end{array}\right), \left(\begin{array}{rrrr} 9 & 9 & 10 & 7 \\ 10 & 7 & 7 & 0 \\ 7 & 0 & 9 & 5 \\ 2 & 5 & 5 & 0 \end{array}\right), \left(\begin{array}{rrrr} 8 & 8 & 6 & 1 \\ 9 & 0 & 1 & 7 \\ 5 & 10 & 9 & 6 \\ 6 & 3 & 6 & 5 \end{array}\right), \left(\begin{array}{rrrr} 8 & 0 & 0 & 0 \\ 0 & 8 & 0 & 0 \\ 0 & 0 & 8 & 0 \\ 0 & 0 & 0 & 8 \end{array}\right), \left(\begin{array}{rrrr} 4 & 2 & 0 & 2 \\ 0 & 2 & 6 & 8 \\ 3 & 6 & 3 & 0 \\ 4 & 9 & 10 & 0 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{11})$
Direct product:	$C_5$ $\, \times\, $ $(C_3:Q_8)$ $\, \times\, $ $\SL(2,11)$
Semidirect product:	$(C_{15}\times \SL(2,11))$ $\,\rtimes\,$ $Q_8$	$C_{15}$ $\,\rtimes\,$ $(Q_8\times \SL(2,11))$	$C_3$ $\,\rtimes\,$ $(C_5\times Q_8\times \SL(2,11))$	$(C_3\times \SL(2,11))$ $\,\rtimes\,$ $(C_5\times Q_8)$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_{20}\times \SL(2,11))$ . $S_3$	$C_{20}$ . $(S_3\times \SL(2,11))$ (2)	$(C_{10}\times \SL(2,11))$ . $D_6$	$C_{10}$ . $(\SL(2,11):D_6)$	all 36

Elements of the group are displayed as matrices in $\GL_{4}(\F_{11})$.

Homology

Abelianization:	$C_{2} \times C_{10} \simeq C_{2}^{2} \times C_{5}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 142492 subgroups in 1322 conjugacy classes, 72 normal (40 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2\times C_{10}$	$G/Z \simeq$ $D_6.\PSL(2,11)$
Commutator:	$G' \simeq$ $C_6\times \SL(2,11)$	$G/G' \simeq$ $C_2\times C_{10}$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $(C_{10}\times S_3).\PSL(2,11)$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{60}$	$G/\operatorname{Fit} \simeq$ $C_2\times \PSL(2,11)$
Radical:	$R \simeq$ $C_{30}:Q_8$	$G/R \simeq$ $\PSL(2,11)$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{30}$	$G/\operatorname{soc} \simeq$ $C_2^2\times \PSL(2,11)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_8^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^2$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 158400: $C_{15}:Q_8\times \SL(2,11)$

Order 79200: $C_3:C_{20}\times \SL(2,11)$ x 2, $(C_2\times C_{60}).\PSL(2,11)$

Order 52800: $C_5\times Q_8\times \SL(2,11)$

Order 39600: $C_{30}\times \SL(2,11)$

Order 31680: $C_3:Q_8\times \SL(2,11)$

Order 26400: $C_{20}\times \SL(2,11)$ x 3

Order 19800: $C_{15}\times \SL(2,11)$

Order 15840: $C_3:C_4\times \SL(2,11)$ x 2, $C_{12}\times \SL(2,11)$

Order 14400: $C_{15}:Q_8\times \SL(2,5)$ x 2

Order 13200: $C_{132}.C_{10}^2$, $C_{10}\times \SL(2,11)$

Order 10560: $Q_8\times \SL(2,11)$

Order 7920: $C_6\times \SL(2,11)$

Order 7200: $C_3:C_{20}\times \SL(2,5)$ x 4, $C_{60}\times \SL(2,5)$ x 2

Order 6600: $C_{660}.C_{10}$ x 4, $C_{330}:C_{20}$ x 2, $C_{110}:C_{60}$, $C_5\times \SL(2,11)$

Order 5280: $C_4\times \SL(2,11)$ x 3

Order 4800: $C_5\times Q_8\times \SL(2,5)$ x 2

Order 4400: $C_{44}.C_{10}^2$

Order 3960: $C_3\times \SL(2,11)$

Order 3600: $C_{30}\times \SL(2,5)$ x 2

Order 3300: $C_{165}:C_{20}$ x 4, $C_{55}:C_{60}$ x 2, $C_{110}:C_{30}$

Order 2880: $C_3:Q_8\times \SL(2,5)$ x 2, $C_{15}:Q_8\times \SL(2,3)$, $(C_3\times C_{15}):Q_8^2$

Order 2640: $C_2\times C_{132}.C_{10}$ x 5, $C_2\times \SL(2,11)$, $C_{330}:Q_8$

Order 2400: $C_{20}\times \SL(2,5)$ x 6, $C_{15}:(Q_8\times C_{20})$

Order 2200: $C_{220}.C_{10}$ x 4, $C_{110}:C_{20}$ x 3

Order 1800: $C_{15}\times \SL(2,5)$ x 2

Order 1650: $C_{110}:C_{15}$ x 3

Order 1440: $C_5\times C_6^2.C_2^3$ x 4, $(C_3\times C_{60}):Q_8$ x 4, $C_3^2:(Q_8\times C_{20})$ x 4, $C_3:C_4\times \SL(2,5)$ x 4, $C_{12}^2.C_{10}$ x 2, $C_3:C_{20}\times \SL(2,3)$ x 2, $C_{12}\times \SL(2,5)$ x 2, $C_{60}.D_{12}$, $C_{60}\times \SL(2,3)$

Order 1320: $C_{132}.C_{10}$ x 20, $C_{66}:C_{20}$ x 10, $C_{22}:C_{60}$ x 5, $C_{165}:Q_8$ x 4, $C_6:C_{220}$ x 2, $C_2\times C_{660}$, $\SL(2,11)$

Order 1200: $C_{10}\times \SL(2,5)$ x 2, $C_{30}.D_{20}$ x 2, $C_{10}^2.D_6$ x 2, $C_{12}.C_{10}^2$, $C_{60}:C_{20}$, $C_{20}:C_{60}$

Order 1100: $C_{220}:C_5$ x 6, $C_{110}:C_{10}$

Order 960: $Q_8\times \SL(2,5)$ x 2, $C_{15}:Q_8^2$ x 2, $Q_8^2:C_{15}$

Order 880: $Q_8\times C_{11}:C_{10}$ x 5, $Q_8\times C_{110}$

Order 825: $C_{165}:C_5$

Order 800: $C_{10}^2.C_2^3$

Order 720: $C_{30}.D_{12}$ x 8, $C_3^2:Q_8\times C_{10}$ x 4, $C_{30}.D_{12}$ x 4, $C_6.D_4\times C_{15}$ x 4, $C_{12}:C_{60}$ x 4, $C_5\times C_3^2:C_4^2$ x 4, $C_{60}.D_6$ x 2, $C_{60}.D_6$ x 2, $C_6\times \SL(2,5)$ x 2, $C_{12}:C_{60}$ x 2, $C_{30}\times \SL(2,3)$, $C_{12}\times C_{60}$

Order 660: $C_{33}:C_{20}$ x 20, $C_{11}:C_{60}$ x 10, $C_{22}:C_{30}$ x 5, $C_3:C_{220}$ x 4, $C_{660}$ x 2, $C_2\times C_{330}$

Order 600: $C_{60}.C_{10}$ x 4, $C_5\times \SL(2,5)$ x 2, $C_{30}:C_{20}$ x 2, $C_{30}:C_{20}$ x 2, $C_{10}:C_{60}$ x 2, $C_{10}\times C_{60}$

Order 576: $(C_3:Q_8)^2$, $C_3:Q_8\times \SL(2,3)$

Order 550: $C_{110}:C_5$ x 3

Order 528: $C_{66}:Q_8$

Order 480: $C_{60}:Q_8$ x 8, $C_4\times \SL(2,5)$ x 6, $C_{60}:Q_8$ x 5, $C_3:C_{20}\times Q_8$ x 4, $C_{60}:Q_8$ x 4, $C_{20}\times \SL(2,3)$ x 4, $Q_8\times C_{60}$ x 2, $C_{15}:(C_4\times Q_8)$

Order 440: $C_{44}.C_{10}$ x 20, $C_{22}:C_{20}$ x 15, $Q_8\times C_{55}$ x 4, $C_2\times C_{220}$ x 3

Order 400: $C_{20}:C_{20}$ x 3, $C_{20}:C_{20}$ x 3, $C_4.C_{10}^2$

Order 360: $C_6:C_{60}$ x 8, $C_{12}.C_{30}$ x 8, $C_{30}.D_6$ x 4, $C_{30}.D_6$ x 4, $C_6\times C_{60}$ x 3, $C_3\times \SL(2,5)$ x 2, $C_{60}.S_3$ x 2, $C_{15}\times \SL(2,3)$

Order 330: $C_{11}:C_{30}$ x 15, $C_{330}$ x 3

Order 320: $C_5\times Q_8^2$

Order 300: $C_{15}:C_{20}$ x 4, $C_5:C_{60}$ x 3, $C_5\times C_{60}$ x 2, $C_{15}:C_{20}$ x 2, $C_{10}\times C_{30}$

Order 288: $C_6^2.C_2^3$ x 4, $C_6^2.C_2^3$ x 4, $C_6^2.C_2^3$ x 4, $C_{12}^2.C_2$ x 2, $C_3:C_4\times \SL(2,3)$ x 2, $C_{12}.D_{12}$, $C_{12}\times \SL(2,3)$

Order 275: $C_{55}:C_5$

Order 264: $C_{33}:Q_8$ x 4, $C_6:C_{44}$ x 2, $C_2\times C_{132}$

Order 240: $C_{12}:C_{20}$ x 16, $C_{30}:Q_8$ x 15, $C_{30}.D_4$ x 8, $C_{12}:C_{20}$ x 8, $C_4\times C_{60}$ x 5, $C_4:C_{60}$ x 4, $C_2\times \SL(2,5)$ x 2, $C_6.D_{20}$ x 2, $C_{15}:C_4^2$ x 2, $Q_8\times C_{30}$ x 2, $C_{10}\times \SL(2,3)$ x 2, $C_{60}:C_4$, $C_{20}:C_{12}$

Order 220: $C_{11}:C_{20}$ x 30, $C_{220}$ x 6, $C_{22}:C_{10}$ x 5, $C_2\times C_{110}$

Order 200: $Q_8\times C_5^2$ x 4, $C_{10}:C_{20}$ x 4, $C_{10}\times C_{20}$ x 3

Order 192: $C_3.Q_8^2$ x 2, $Q_8\times \SL(2,3)$

Order 180: $C_3:C_{60}$ x 12, $C_3\times C_{60}$ x 5, $C_3^2:C_{20}$ x 4, $C_6\times C_{30}$

Order 176: $Q_8\times C_{22}$

Order 165: $C_{11}:C_{15}$ x 5, $C_{165}$

Order 160: $Q_8\times C_{20}$ x 4, $C_{20}:Q_8$ x 3, $C_5:C_4\times Q_8$

Order 150: $C_5\times C_{30}$ x 3

Order 144: $C_6.D_{12}$ x 8, $C_6.D_{12}$ x 4, $C_6^2.C_2^2$ x 4, $C_{12}:C_{12}$ x 4, $C_3^2:C_4^2$ x 4, $C_2^2.S_3^2$ x 4, $C_{12}:C_{12}$ x 2, $C_{12}.D_6$ x 2, $C_{12}.D_6$ x 2, $C_6\times \SL(2,3)$, $C_{12}^2$

Order 132: $C_3:C_{44}$ x 4, $C_{132}$ x 2, $C_2\times C_{66}$

Order 120: $C_{15}:Q_8$ x 32, $C_6:C_{20}$ x 22, $C_2\times C_{60}$ x 16, $Q_8\times C_{15}$ x 6, $\SL(2,5)$ x 2, $C_{30}:C_4$ x 2, $C_{10}:C_{12}$ x 2, $C_5\times \SL(2,3)$ x 2

Order 110: $C_{11}:C_{10}$ x 15, $C_{110}$ x 3

Order 100: $C_5\times C_{20}$ x 6, $C_5:C_{20}$ x 5, $C_{10}^2$

Order 96: $C_{12}:Q_8$ x 8, $C_{12}:Q_8$ x 5, $C_{12}:Q_8$ x 4, $C_4\times \SL(2,3)$ x 4, $C_3:C_4\times Q_8$ x 4, $Q_8\times C_{12}$ x 2

Order 90: $C_3\times C_{30}$ x 3

Order 88: $Q_8\times C_{11}$ x 4, $C_2\times C_{44}$ x 3

Order 80: $Q_8\times C_{10}$ x 7, $C_4:C_{20}$ x 6, $C_4\times C_{20}$ x 3, $C_{20}:C_4$ x 3, $C_{20}:C_4$ x 3

Order 75: $C_5\times C_{15}$

Order 72: $C_6:C_{12}$ x 8, $C_3^2:Q_8$ x 8, $C_6.D_6$ x 4, $C_3^2:Q_8$ x 4, $C_6\times C_{12}$ x 3, $C_3^2:Q_8$ x 2, $C_3\times \SL(2,3)$

Order 66: $C_{66}$ x 3

Order 64: $Q_8^2$

Order 60: $C_3:C_{20}$ x 32, $C_{60}$ x 28, $C_2\times C_{30}$ x 6, $C_5:C_{12}$ x 3, $C_{15}:C_4$ x 2

Order 55: $C_{11}:C_5$ x 5, $C_{55}$

Order 50: $C_5\times C_{10}$ x 3

Order 48: $C_{12}:C_4$ x 16, $C_6:Q_8$ x 12, $C_6.D_4$ x 8, $C_{12}:C_4$ x 8, $C_4\times C_{12}$ x 5, $C_4:C_{12}$ x 4, $C_6\times Q_8$ x 2, $C_2\times \SL(2,3)$ x 2

Order 45: $C_3\times C_{15}$

Order 44: $C_{44}$ x 6, $C_2\times C_{22}$

Order 40: $C_5\times Q_8$ x 20, $C_2\times C_{20}$ x 16, $C_{10}:C_4$ x 4

Order 36: $C_3:C_{12}$ x 12, $C_3\times C_{12}$ x 5, $C_3^2:C_4$ x 4, $C_6^2$

Order 33: $C_{33}$

Order 32: $C_4\times Q_8$ x 4, $C_4:Q_8$ x 3

Order 30: $C_{30}$ x 18

Order 25: $C_5^2$

Order 24: $C_3:Q_8$ x 20, $C_6:C_4$ x 16, $C_2\times C_{12}$ x 13, $C_3\times Q_8$ x 6, $\SL(2,3)$ x 2

Order 22: $C_{22}$ x 3

Order 20: $C_{20}$ x 29, $C_5:C_4$ x 5, $C_2\times C_{10}$ x 4

Order 18: $C_3\times C_6$ x 3

Order 16: $C_4:C_4$ x 6, $C_2\times Q_8$ x 4, $C_4^2$ x 3

Order 15: $C_{15}$ x 6

Order 12: $C_{12}$ x 22, $C_3:C_4$ x 20, $C_2\times C_6$ x 3

Order 11: $C_{11}$

Order 10: $C_{10}$ x 12

Order 9: $C_3^2$

Order 8: $Q_8$ x 8, $C_2\times C_4$ x 7

Order 6: $C_6$ x 9

Order 5: $C_5$ x 4

Order 4: $C_4$ x 11, $C_2^2$

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 158400: $C_{15}:Q_8\times \SL(2,11)$

Order 79200: $(C_2\times C_{60}).\PSL(2,11)$, $C_3:C_{20}\times \SL(2,11)$

Order 52800: $C_5\times Q_8\times \SL(2,11)$

Order 39600: $C_{30}\times \SL(2,11)$

Order 31680: $C_3:Q_8\times \SL(2,11)$

Order 26400: $C_{20}\times \SL(2,11)$ x 2

Order 19800: $C_{15}\times \SL(2,11)$

Order 15840: $C_3:C_4\times \SL(2,11)$, $C_{12}\times \SL(2,11)$

Order 14400: $C_{15}:Q_8\times \SL(2,5)$

Order 13200: $C_{132}.C_{10}^2$, $C_{10}\times \SL(2,11)$

Order 10560: $Q_8\times \SL(2,11)$

Order 7920: $C_6\times \SL(2,11)$

Order 7200: $C_{60}\times \SL(2,5)$, $C_3:C_{20}\times \SL(2,5)$

Order 6600: $C_{330}:C_{20}$, $C_{110}:C_{60}$, $C_{660}.C_{10}$, $C_5\times \SL(2,11)$

Order 5280: $C_4\times \SL(2,11)$ x 2

Order 4800: $C_5\times Q_8\times \SL(2,5)$

Order 4400: $C_{44}.C_{10}^2$

Order 3960: $C_3\times \SL(2,11)$

Order 3600: $C_{30}\times \SL(2,5)$

Order 3300: $C_{110}:C_{30}$, $C_{165}:C_{20}$, $C_{55}:C_{60}$

Order 2880: $C_{15}:Q_8\times \SL(2,3)$, $C_3:Q_8\times \SL(2,5)$, $(C_3\times C_{15}):Q_8^2$

Order 2640: $C_2\times C_{132}.C_{10}$ x 2, $C_2\times \SL(2,11)$, $C_{330}:Q_8$

Order 2400: $C_{20}\times \SL(2,5)$ x 2, $C_{15}:(Q_8\times C_{20})$

Order 2200: $C_{110}:C_{20}$ x 2, $C_{220}.C_{10}$

Order 1800: $C_{15}\times \SL(2,5)$

Order 1650: $C_{110}:C_{15}$ x 3

Order 1440: $C_{12}^2.C_{10}$ x 2, $(C_3\times C_{60}):Q_8$ x 2, $C_3^2:(Q_8\times C_{20})$ x 2, $C_{60}.D_{12}$, $C_{60}\times \SL(2,3)$, $C_5\times C_6^2.C_2^3$, $C_3:C_{20}\times \SL(2,3)$, $C_{12}\times \SL(2,5)$, $C_3:C_4\times \SL(2,5)$

Order 1320: $C_{22}:C_{60}$ x 2, $C_{66}:C_{20}$ x 2, $C_{132}.C_{10}$ x 2, $C_2\times C_{660}$, $\SL(2,11)$, $C_6:C_{220}$, $C_{165}:Q_8$

Order 1200: $C_{12}.C_{10}^2$, $C_{10}\times \SL(2,5)$, $C_{60}:C_{20}$, $C_{20}:C_{60}$, $C_{30}.D_{20}$, $C_{10}^2.D_6$

Order 1100: $C_{220}:C_5$ x 2, $C_{110}:C_{10}$

Order 960: $C_{15}:Q_8^2$ x 2, $Q_8^2:C_{15}$, $Q_8\times \SL(2,5)$

Order 880: $Q_8\times C_{11}:C_{10}$ x 2, $Q_8\times C_{110}$

Order 825: $C_{165}:C_5$

Order 800: $C_{10}^2.C_2^3$

Order 720: $C_{60}.D_6$ x 2, $C_{30}.D_{12}$ x 2, $C_{12}:C_{60}$ x 2, $C_6.D_4\times C_{15}$ x 2, $C_{12}:C_{60}$ x 2, $C_{30}.D_{12}$ x 2, $C_{60}.D_6$, $C_{30}\times \SL(2,3)$, $C_3^2:Q_8\times C_{10}$, $C_6\times \SL(2,5)$, $C_{12}\times C_{60}$, $C_5\times C_3^2:C_4^2$

Order 660: $C_{11}:C_{60}$ x 2, $C_{22}:C_{30}$ x 2, $C_{33}:C_{20}$ x 2, $C_3:C_{220}$, $C_2\times C_{330}$, $C_{660}$

Order 600: $C_{10}:C_{60}$ x 2, $C_5\times \SL(2,5)$, $C_{10}\times C_{60}$, $C_{30}:C_{20}$, $C_{30}:C_{20}$, $C_{60}.C_{10}$

Order 576: $(C_3:Q_8)^2$, $C_3:Q_8\times \SL(2,3)$

Order 550: $C_{110}:C_5$ x 3

Order 528: $C_{66}:Q_8$

Order 480: $C_{60}:Q_8$ x 4, $C_{60}:Q_8$ x 3, $C_{60}:Q_8$ x 3, $C_{20}\times \SL(2,3)$ x 3, $Q_8\times C_{60}$ x 2, $C_3:C_{20}\times Q_8$ x 2, $C_4\times \SL(2,5)$ x 2, $C_{15}:(C_4\times Q_8)$

Order 440: $C_{22}:C_{20}$ x 4, $C_2\times C_{220}$ x 2, $C_{44}.C_{10}$ x 2, $Q_8\times C_{55}$

Order 400: $C_{20}:C_{20}$ x 2, $C_{20}:C_{20}$ x 2, $C_4.C_{10}^2$

Order 360: $C_6:C_{60}$ x 4, $C_{12}.C_{30}$ x 4, $C_6\times C_{60}$ x 3, $C_{15}\times \SL(2,3)$, $C_{30}.D_6$, $C_3\times \SL(2,5)$, $C_{30}.D_6$, $C_{60}.S_3$

Order 330: $C_{11}:C_{30}$ x 6, $C_{330}$ x 3

Order 320: $C_5\times Q_8^2$

Order 300: $C_5:C_{60}$ x 3, $C_{10}\times C_{30}$, $C_5\times C_{60}$, $C_{15}:C_{20}$, $C_{15}:C_{20}$

Order 288: $C_{12}^2.C_2$ x 2, $C_6^2.C_2^3$ x 2, $C_6^2.C_2^3$ x 2, $C_{12}.D_{12}$, $C_{12}\times \SL(2,3)$, $C_6^2.C_2^3$, $C_3:C_4\times \SL(2,3)$

Order 275: $C_{55}:C_5$

Order 264: $C_2\times C_{132}$, $C_6:C_{44}$, $C_{33}:Q_8$

Order 240: $C_{12}:C_{20}$ x 8, $C_{30}:Q_8$ x 8, $C_4\times C_{60}$ x 4, $C_4:C_{60}$ x 3, $C_{30}.D_4$ x 3, $C_{12}:C_{20}$ x 3, $Q_8\times C_{30}$ x 2, $C_{10}\times \SL(2,3)$ x 2, $C_2\times \SL(2,5)$, $C_{60}:C_4$, $C_{20}:C_{12}$, $C_6.D_{20}$, $C_{15}:C_4^2$

Order 220: $C_{11}:C_{20}$ x 4, $C_{22}:C_{10}$ x 2, $C_{220}$ x 2, $C_2\times C_{110}$

Order 200: $C_{10}:C_{20}$ x 3, $C_{10}\times C_{20}$ x 2, $Q_8\times C_5^2$

Order 192: $C_3.Q_8^2$ x 2, $Q_8\times \SL(2,3)$

Order 180: $C_3:C_{60}$ x 5, $C_3\times C_{60}$ x 4, $C_6\times C_{30}$, $C_3^2:C_{20}$

Order 176: $Q_8\times C_{22}$

Order 165: $C_{11}:C_{15}$ x 2, $C_{165}$

Order 160: $Q_8\times C_{20}$ x 3, $C_{20}:Q_8$ x 2, $C_5:C_4\times Q_8$

Order 150: $C_5\times C_{30}$ x 3

Order 144: $C_6.D_{12}$ x 2, $C_{12}:C_{12}$ x 2, $C_6^2.C_2^2$ x 2, $C_{12}:C_{12}$ x 2, $C_6.D_{12}$ x 2, $C_{12}.D_6$ x 2, $C_3^2:C_4^2$, $C_{12}.D_6$, $C_6\times \SL(2,3)$, $C_2^2.S_3^2$, $C_{12}^2$

Order 132: $C_{132}$, $C_2\times C_{66}$, $C_3:C_{44}$

Order 120: $C_2\times C_{60}$ x 13, $C_{15}:Q_8$ x 10, $C_6:C_{20}$ x 8, $Q_8\times C_{15}$ x 3, $C_{10}:C_{12}$ x 2, $C_5\times \SL(2,3)$ x 2, $\SL(2,5)$, $C_{30}:C_4$

Order 110: $C_{11}:C_{10}$ x 6, $C_{110}$ x 3

Order 100: $C_5:C_{20}$ x 4, $C_5\times C_{20}$ x 2, $C_{10}^2$

Order 96: $C_{12}:Q_8$ x 4, $C_{12}:Q_8$ x 3, $C_{12}:Q_8$ x 3, $C_4\times \SL(2,3)$ x 3, $Q_8\times C_{12}$ x 2, $C_3:C_4\times Q_8$ x 2

Order 90: $C_3\times C_{30}$ x 3

Order 88: $C_2\times C_{44}$ x 2, $Q_8\times C_{11}$

Order 80: $Q_8\times C_{10}$ x 5, $C_4:C_{20}$ x 4, $C_4\times C_{20}$ x 2, $C_{20}:C_4$ x 2, $C_{20}:C_4$ x 2

Order 75: $C_5\times C_{15}$

Order 72: $C_6:C_{12}$ x 4, $C_3^2:Q_8$ x 4, $C_6\times C_{12}$ x 3, $C_6.D_6$, $C_3^2:Q_8$, $C_3\times \SL(2,3)$, $C_3^2:Q_8$

Order 66: $C_{66}$ x 3

Order 64: $Q_8^2$

Order 60: $C_{60}$ x 16, $C_3:C_{20}$ x 9, $C_2\times C_{30}$ x 5, $C_5:C_{12}$ x 3, $C_{15}:C_4$

Order 55: $C_{11}:C_5$ x 2, $C_{55}$

Order 50: $C_5\times C_{10}$ x 3

Order 48: $C_{12}:C_4$ x 8, $C_6:Q_8$ x 6, $C_4\times C_{12}$ x 4, $C_4:C_{12}$ x 3, $C_6.D_4$ x 3, $C_{12}:C_4$ x 3, $C_6\times Q_8$ x 2, $C_2\times \SL(2,3)$ x 2

Order 45: $C_3\times C_{15}$

Order 44: $C_{44}$ x 2, $C_2\times C_{22}$

Order 40: $C_2\times C_{20}$ x 9, $C_5\times Q_8$ x 6, $C_{10}:C_4$ x 3

Order 36: $C_3:C_{12}$ x 5, $C_3\times C_{12}$ x 4, $C_3^2:C_4$, $C_6^2$

Order 33: $C_{33}$

Order 32: $C_4\times Q_8$ x 3, $C_4:Q_8$ x 2

Order 30: $C_{30}$ x 15

Order 25: $C_5^2$

Order 24: $C_2\times C_{12}$ x 11, $C_3:Q_8$ x 8, $C_6:C_4$ x 6, $C_3\times Q_8$ x 3, $\SL(2,3)$ x 2

Order 22: $C_{22}$ x 3

Order 20: $C_{20}$ x 10, $C_5:C_4$ x 4, $C_2\times C_{10}$ x 3

Order 18: $C_3\times C_6$ x 3

Order 16: $C_4:C_4$ x 4, $C_2\times Q_8$ x 3, $C_4^2$ x 2

Order 15: $C_{15}$ x 5

Order 12: $C_{12}$ x 14, $C_3:C_4$ x 7, $C_2\times C_6$ x 3

Order 11: $C_{11}$

Order 10: $C_{10}$ x 9

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 5, $Q_8$ x 4

Order 6: $C_6$ x 9

Order 5: $C_5$ x 3

Order 4: $C_4$ x 6, $C_2^2$

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 158400: $C_{15}:Q_8\times \SL(2,11)$ ($C_1$)

Order 79200: $C_3:C_{20}\times \SL(2,11)$ ($C_2$) x 2, $(C_2\times C_{60}).\PSL(2,11)$ ($C_2$)

Order 39600: $C_{30}\times \SL(2,11)$ ($C_2^2$)

Order 31680: $C_3:Q_8\times \SL(2,11)$ ($C_5$)

Order 26400: $C_{20}\times \SL(2,11)$ ($S_3$)

Order 19800: $C_{15}\times \SL(2,11)$ ($Q_8$)

Order 15840: $C_3:C_4\times \SL(2,11)$ ($C_{10}$) x 2, $C_{12}\times \SL(2,11)$ ($C_{10}$)

Order 13200: $C_{10}\times \SL(2,11)$ ($D_6$)

Order 7920: $C_6\times \SL(2,11)$ ($C_2\times C_{10}$)

Order 6600: $C_5\times \SL(2,11)$ ($C_3:Q_8$)

Order 5280: $C_4\times \SL(2,11)$ ($C_5\times S_3$)

Order 3960: $C_3\times \SL(2,11)$ ($C_5\times Q_8$)

Order 2640: $C_2\times \SL(2,11)$ ($S_3\times C_{10}$)

Order 1320: $\SL(2,11)$ ($C_{15}:Q_8$)

Order 240: $C_{30}:Q_8$ ($\PSL(2,11)$)

Order 120: $C_{15}:Q_8$ ($\SL(2,11)$) x 4, $C_6:C_{20}$ ($C_2\times \PSL(2,11)$) x 2, $C_2\times C_{60}$ ($C_2\times \PSL(2,11)$)

Order 60: $C_3:C_{20}$ ($C_2\times \SL(2,11)$) x 4, $C_{60}$ ($C_2\times \SL(2,11)$) x 2, $C_2\times C_{30}$ ($C_2^2\times \PSL(2,11)$)

Order 48: $C_6:Q_8$ ($C_5\times \PSL(2,11)$)

Order 40: $C_2\times C_{20}$ ($S_3\times \PSL(2,11)$)

Order 30: $C_{30}$ ($C_2^3.\PSL(2,11)$), $C_{30}$ ($Q_8.\PSL(2,11)$), $C_{30}$ ($\SL(2,11):C_2^2$)

Order 24: $C_3:Q_8$ ($C_5\times \SL(2,11)$) x 4, $C_6:C_4$ ($C_{10}.\PSL(2,11)$) x 2, $C_2\times C_{12}$ ($C_{10}.\PSL(2,11)$)

Order 20: $C_{20}$ ($S_3\times \SL(2,11)$) x 2, $C_2\times C_{10}$ ($D_6.\PSL(2,11)$)

Order 15: $C_{15}$ ($Q_8\times \SL(2,11)$)

Order 12: $C_3:C_4$ ($C_{10}\times \SL(2,11)$) x 4, $C_{12}$ ($C_{10}\times \SL(2,11)$) x 2, $C_2\times C_6$ ($(C_2\times C_{10}).\PSL(2,11)$)

Order 10: $C_{10}$ ($(C_2^2\times S_3).\PSL(2,11)$), $C_{10}$ ($C_3:Q_8.\PSL(2,11)$), $C_{10}$ ($\SL(2,11):D_6$)

Order 8: $C_2\times C_4$ ($(C_5\times S_3).\PSL(2,11)$)

Order 6: $C_6$ ($(C_5\times Q_8).\PSL(2,11)$), $C_6$ ($(C_2^2\times C_{10}).\PSL(2,11)$), $C_6$ ($C_5\times \SL(2,11):C_2^2$)

Order 5: $C_5$ ($C_3:Q_8\times \SL(2,11)$)

Order 4: $C_4$ ($C_5\times S_3\times \SL(2,11)$) x 2, $C_2^2$ ($(C_{10}\times S_3).\PSL(2,11)$)

Order 3: $C_3$ ($C_5\times Q_8\times \SL(2,11)$)

Order 2: $C_2$ ($(C_5\times C_3:Q_8).\PSL(2,11)$), $C_2$ ($(C_2\times C_{10}\times S_3).\PSL(2,11)$), $C_2$ ($C_5\times \SL(2,11):D_6$)

Order 1: $C_1$ ($C_{15}:Q_8\times \SL(2,11)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 158400: $C_{15}:Q_8\times \SL(2,11)$ ($C_1$)

Order 79200: $(C_2\times C_{60}).\PSL(2,11)$ ($C_2$), $C_3:C_{20}\times \SL(2,11)$ ($C_2$)

Order 39600: $C_{30}\times \SL(2,11)$ ($C_2^2$)

Order 31680: $C_3:Q_8\times \SL(2,11)$ ($C_5$)

Order 26400: $C_{20}\times \SL(2,11)$ ($S_3$)

Order 19800: $C_{15}\times \SL(2,11)$ ($Q_8$)

Order 15840: $C_3:C_4\times \SL(2,11)$ ($C_{10}$), $C_{12}\times \SL(2,11)$ ($C_{10}$)

Order 13200: $C_{10}\times \SL(2,11)$ ($D_6$)

Order 7920: $C_6\times \SL(2,11)$ ($C_2\times C_{10}$)

Order 6600: $C_5\times \SL(2,11)$ ($C_3:Q_8$)

Order 5280: $C_4\times \SL(2,11)$ ($C_5\times S_3$)

Order 3960: $C_3\times \SL(2,11)$ ($C_5\times Q_8$)

Order 2640: $C_2\times \SL(2,11)$ ($S_3\times C_{10}$)

Order 1320: $\SL(2,11)$ ($C_{15}:Q_8$)

Order 240: $C_{30}:Q_8$ ($\PSL(2,11)$)

Order 120: $C_2\times C_{60}$ ($C_2\times \PSL(2,11)$), $C_6:C_{20}$ ($C_2\times \PSL(2,11)$), $C_{15}:Q_8$ ($\SL(2,11)$)

Order 60: $C_{60}$ ($C_2\times \SL(2,11)$), $C_2\times C_{30}$ ($C_2^2\times \PSL(2,11)$), $C_3:C_{20}$ ($C_2\times \SL(2,11)$)

Order 48: $C_6:Q_8$ ($C_5\times \PSL(2,11)$)

Order 40: $C_2\times C_{20}$ ($S_3\times \PSL(2,11)$)

Order 30: $C_{30}$ ($C_2^3.\PSL(2,11)$), $C_{30}$ ($Q_8.\PSL(2,11)$), $C_{30}$ ($\SL(2,11):C_2^2$)

Order 24: $C_2\times C_{12}$ ($C_{10}.\PSL(2,11)$), $C_6:C_4$ ($C_{10}.\PSL(2,11)$), $C_3:Q_8$ ($C_5\times \SL(2,11)$)

Order 20: $C_2\times C_{10}$ ($D_6.\PSL(2,11)$), $C_{20}$ ($S_3\times \SL(2,11)$)

Order 15: $C_{15}$ ($Q_8\times \SL(2,11)$)

Order 12: $C_2\times C_6$ ($(C_2\times C_{10}).\PSL(2,11)$), $C_{12}$ ($C_{10}\times \SL(2,11)$), $C_3:C_4$ ($C_{10}\times \SL(2,11)$)

Order 10: $C_{10}$ ($(C_2^2\times S_3).\PSL(2,11)$), $C_{10}$ ($C_3:Q_8.\PSL(2,11)$), $C_{10}$ ($\SL(2,11):D_6$)

Order 8: $C_2\times C_4$ ($(C_5\times S_3).\PSL(2,11)$)

Order 6: $C_6$ ($(C_5\times Q_8).\PSL(2,11)$), $C_6$ ($(C_2^2\times C_{10}).\PSL(2,11)$), $C_6$ ($C_5\times \SL(2,11):C_2^2$)

Order 5: $C_5$ ($C_3:Q_8\times \SL(2,11)$)

Order 4: $C_2^2$ ($(C_{10}\times S_3).\PSL(2,11)$), $C_4$ ($C_5\times S_3\times \SL(2,11)$)

Order 3: $C_3$ ($C_5\times Q_8\times \SL(2,11)$)

Order 2: $C_2$ ($(C_5\times C_3:Q_8).\PSL(2,11)$), $C_2$ ($(C_2\times C_{10}\times S_3).\PSL(2,11)$), $C_2$ ($C_5\times \SL(2,11):D_6$)

Order 1: $C_1$ ($C_{15}:Q_8\times \SL(2,11)$)

Series

Derived series

$C_{15}:Q_8\times \SL(2,11)$

$\rhd$

$C_6\times \SL(2,11)$

$\rhd$

$\SL(2,11)$

Chief series

$C_{15}:Q_8\times \SL(2,11)$

$\rhd$

$C_{30}:Q_8$

$\rhd$

$C_6:C_{20}$

$\rhd$

$C_2\times C_{30}$

$\rhd$

$C_{30}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_{15}:Q_8\times \SL(2,11)$

$\rhd$

$C_6\times \SL(2,11)$

$\rhd$

$C_3\times \SL(2,11)$

Upper central series

$C_1$

$\lhd$

$C_2\times C_{10}$

$\lhd$

$C_2\times C_{20}$

Supergroups

This group is a maximal subgroup of 2 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

The $675 \times 675$ character table is not available for this group.

Rational character table

See the $178 \times 178$ rational character table (warning: may be slow to load).