Abstract group 1742400.b: $\SL(2,11)^2$

• Label: 1742400.b
• Order: \( 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 11^{2} \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 1 \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{2} \cdot 5^{2} \cdot 11^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $48$
• Trans deg.: $576$
• Rank: $2$

Group information

Description:	$\SL(2,11)^2$
Order:	\(1742400\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 11^{2} \)
Exponent:	\(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \)
Automorphism group:	$\PGL(2,11)\wr C_2$, of order \(3484800\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5^{2} \cdot 11^{2} \)
Composition factors:	$C_2$ x 2, $\PSL(2,11)$ x 2
Derived length:	$0$

This group is nonabelian and perfect (hence nonsolvable).

Group statistics

Order	1	2	3	4	5	6	10	11	12	15	20	22	30	33	44	55	60	66	110	132
Elements	1	3	12320	12540	70224	36960	210672	14640	242880	58080	116160	43920	174240	26400	52800	63360	232320	79200	190080	105600	1742400
Conjugacy classes	1	3	3	5	8	9	24	8	28	4	8	24	12	4	8	8	16	12	24	16	225
Divisions	1	3	3	5	4	9	12	4	16	2	4	12	6	2	4	2	4	6	6	4	109
Autjugacy classes	1	2	2	3	5	5	13	2	15	2	4	5	6	1	2	2	8	3	6	4	91

Dimension	1	5	6	10	11	12	20	24	25	30	36	50	55	60	66	72	100	110	120	121	132	144	200	220	240	264	288	480
Irr. complex chars.	1	4	4	10	2	8	0	0	4	8	4	20	4	36	4	16	25	10	40	1	8	16	0	0	0	0	0	0	225
Irr. rational chars.	1	0	0	8	2	2	2	4	0	0	0	2	0	4	0	2	15	8	6	1	2	0	10	2	18	4	12	4	109

Minimal presentations

Permutation degree:	$48$
Transitive degree:	$576$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

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

Constructions

Groups of Lie type:	$\SpinPlus(4,11)$
Permutation group:	Degree $48$ $\langle(26,28,32,29,33,38,35,31,36,42,47)(27,30,34,39,37,43,48,41,45,44,40), (25,26,27,29,32,36,41,34,30,33,37) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 4 & 6 & 9 & 0 \\ 8 & 7 & 0 & 2 \\ 4 & 0 & 7 & 6 \\ 0 & 7 & 8 & 4 \end{array}\right), \left(\begin{array}{rrrr} 4 & 5 & 7 & 1 \\ 10 & 9 & 4 & 5 \\ 7 & 4 & 7 & 0 \\ 2 & 0 & 4 & 4 \end{array}\right), \left(\begin{array}{rrrr} 8 & 8 & 0 & 10 \\ 0 & 2 & 9 & 2 \\ 1 & 1 & 6 & 9 \\ 5 & 8 & 8 & 8 \end{array}\right), \left(\begin{array}{rrrr} 4 & 9 & 3 & 10 \\ 5 & 9 & 1 & 6 \\ 6 & 5 & 1 & 0 \\ 9 & 0 & 7 & 7 \end{array}\right), \left(\begin{array}{rrrr} 9 & 8 & 7 & 6 \\ 4 & 2 & 4 & 8 \\ 10 & 1 & 3 & 6 \\ 7 & 8 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrr} 10 & 0 & 0 & 0 \\ 0 & 10 & 0 & 0 \\ 0 & 0 & 10 & 0 \\ 0 & 0 & 0 & 10 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{11})$
Direct product:	$\SL(2,11)$ ${}^2$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_2^2$ . $\PSL(2,11)^2$	$\SL(2,11)$ . $\SL(2,11)$ (2)	$C_2$ . $\OmegaPlus(4,11)$	$(C_2\times \SL(2,11))$ . $\PSL(2,11)$ (2)	all 5

Elements of the group are displayed as matrices in $\SpinPlus(4,11)$.

Homology

Abelianization:	$C_1 $
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 3050582 subgroups in 829 conjugacy classes, 10 normal (4 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $\PSL(2,11)^2$
Commutator:	$G' \simeq$ $\SL(2,11)^2$	$G/G' \simeq$ $C_1$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $\PSL(2,11)^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2$	$G/\operatorname{Fit} \simeq$ $\PSL(2,11)^2$
Radical:	$R \simeq$ $C_2^2$	$G/R \simeq$ $\PSL(2,11)^2$
Socle:	$\operatorname{soc} \simeq$ $C_2^2$	$G/\operatorname{soc} \simeq$ $\PSL(2,11)^2$
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:	$P_{ 11 } \simeq$ $C_{11}^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

Order 1742400: $\SL(2,11)^2$

Order 158400: $\SL(2,11).\SL(2,5)$ x 4

Order 145200: $C_{11}:C_{10}\times \SL(2,11)$ x 2

Order 72600: $C_{11}:C_5\times \SL(2,11)$ x 2

Order 31680: $\SL(2,11)\times \SL(2,3)$ x 2, $C_3:Q_8\times \SL(2,11)$ x 2

Order 29040: $C_{22}\times \SL(2,11)$ x 2

Order 26400: $C_5:C_4\times \SL(2,11)$ x 2

Order 15840: $C_3:C_4\times \SL(2,11)$ x 4, $C_{12}\times \SL(2,11)$ x 2

Order 14520: $C_{11}\times \SL(2,11)$ x 2

Order 14400: $\SL(2,5)^2$ x 4

Order 13200: $C_{11}:C_{10}\times \SL(2,5)$ x 4, $C_{10}\times \SL(2,11)$ x 2

Order 12100: $C_{11}^2:C_{10}^2$

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

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

Order 6600: $C_{11}:C_5\times \SL(2,5)$ x 4, $C_5\times \SL(2,11)$ x 2

Order 6050: $C_2\times C_{11}^2:C_5^2$ x 3

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

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

Order 3025: $C_{11}^2:C_5^2$

Order 2880: $\SL(2,3)\times \SL(2,5)$ x 4, $C_3:Q_8\times \SL(2,5)$ x 4

Order 2640: $C_2\times \SL(2,11)$ x 4, $C_{22}\times \SL(2,5)$ x 4, $C_2\times C_{132}.C_{10}$ x 2, $C_{11}:C_{10}\times \SL(2,3)$ x 2

Order 2420: $C_{22}^2:C_5$ x 2, $C_{22}:C_{110}$ x 2, $C_{22}^2:C_5$, $C_{22}^2:C_5$

Order 2400: $C_5:C_4\times \SL(2,5)$ x 4

Order 2200: $C_{110}:C_{20}$ x 2

Order 1440: $C_3:C_4\times \SL(2,5)$ x 8, $C_{12}\times \SL(2,5)$ x 4

Order 1320: $C_{132}.C_{10}$ x 8, $C_{66}:C_{20}$ x 4, $C_{11}\times \SL(2,5)$ x 4, $\SL(2,11)$ x 4, $C_{22}:C_{60}$ x 2, $C_{11}:C_5\times \SL(2,3)$ x 2

Order 1210: $C_{11}^2:C_{10}$ x 6, $C_{11}:C_{110}$ x 6, $C_{11}^2:C_{10}$ x 3, $C_{11}^2:C_{10}$ x 3

Order 1200: $C_{10}\times \SL(2,5)$ x 4

Order 1100: $C_{55}:C_{20}$ x 4, $C_{110}:C_{10}$ x 2

Order 960: $Q_8\times \SL(2,5)$ x 4

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

Order 720: $C_6\times \SL(2,5)$ x 4

Order 660: $C_{33}:C_{20}$ x 8, $C_{11}:C_{60}$ x 4, $C_{22}:C_{30}$ x 2

Order 605: $C_{11}^2:C_5$ x 2, $C_{11}:C_{55}$ x 2, $C_{11}^2:C_5$, $C_{11}^2:C_5$

Order 600: $C_5\times \SL(2,5)$ x 4

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

Order 550: $C_{110}:C_5$ x 6

Order 528: $C_{66}:Q_8$ x 2, $C_{22}\times \SL(2,3)$ x 2

Order 484: $C_{22}^2$

Order 480: $C_4\times \SL(2,5)$ x 4, $C_{15}:(C_4\times Q_8)$ x 2, $C_5:C_4\times \SL(2,3)$ x 2

Order 440: $C_{44}.C_{10}$ x 4, $C_{10}:C_{44}$ x 2, $C_{22}:C_{20}$ x 2

Order 400: $C_5^2:C_4^2$

Order 360: $C_3\times \SL(2,5)$ x 4

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

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

Order 275: $C_{55}:C_5$ x 2

Order 264: $C_{33}:Q_8$ x 8, $C_6:C_{44}$ x 4, $C_2\times C_{132}$ x 2, $C_{11}\times \SL(2,3)$ x 2

Order 242: $C_{11}\times C_{22}$ x 3

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

Order 220: $C_{22}:C_{10}$ x 8, $C_5:C_{44}$ x 4, $C_{11}:C_{20}$ x 4, $C_2\times C_{110}$ x 2

Order 200: $C_{10}:C_{20}$ x 2, $C_{10}.D_{10}$

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

Order 176: $Q_8\times C_{22}$ x 2

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

Order 160: $C_5:C_4\times Q_8$ x 2

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, $C_6\times \SL(2,3)$ x 2, $C_{12}^2$

Order 132: $C_3:C_{44}$ x 8, $C_{132}$ x 4, $C_2\times C_{66}$ x 2

Order 121: $C_{11}^2$

Order 120: $\SL(2,5)$ x 12, $C_{15}:Q_8$ x 8, $C_{30}:C_4$ x 4, $C_6:C_{20}$ x 4, $C_{10}:C_{12}$ x 4, $C_2\times C_{60}$ x 2, $C_5\times \SL(2,3)$ x 2

Order 110: $C_{11}:C_{10}$ x 24, $C_{110}$ x 6

Order 100: $C_5:C_{20}$ x 4, $C_5^2:C_4$ x 2, $C_{10}^2$

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

Order 88: $Q_8\times C_{11}$ x 4, $C_2\times C_{44}$ x 2

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

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, $C_3\times \SL(2,3)$ x 2

Order 66: $C_{66}$ x 6

Order 64: $Q_8^2$

Order 60: $C_3:C_{20}$ x 8, $C_5:C_{12}$ x 6, $C_{60}$ x 4, $C_{15}:C_4$ x 4, $C_2\times C_{30}$ x 2

Order 55: $C_{11}:C_5$ x 8, $C_{55}$ x 2

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

Order 48: $C_{12}:C_4$ x 10, $C_6:Q_8$ x 8, $C_2\times \SL(2,3)$ x 6, $C_6.D_4$ x 4, $C_{12}:C_4$ x 4, $C_4\times C_{12}$ x 3, $C_6\times Q_8$ x 2, $C_4:C_{12}$ x 2

Order 44: $C_2\times C_{22}$ x 4, $C_{44}$ x 4

Order 40: $C_{10}:C_4$ x 6, $C_5\times Q_8$ x 4, $C_2\times C_{20}$ x 2

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: $C_{33}$ x 2

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

Order 30: $C_{30}$ x 6

Order 25: $C_5^2$

Order 24: $C_3:Q_8$ x 16, $C_6:C_4$ x 12, $C_2\times C_{12}$ x 9, $\SL(2,3)$ x 6, $C_3\times Q_8$ x 4

Order 22: $C_{22}$ x 12

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

Order 18: $C_3\times C_6$ x 3

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

Order 15: $C_{15}$ x 2

Order 12: $C_{12}$ x 16, $C_3:C_4$ x 16, $C_2\times C_6$ x 3

Order 11: $C_{11}$ x 4

Order 10: $C_{10}$ x 12

Order 9: $C_3^2$

Order 8: $Q_8$ x 6, $C_2\times C_4$ x 3

Order 6: $C_6$ x 9

Order 5: $C_5$ x 4

Order 4: $C_4$ x 5, $C_2^2$

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1742400: $\SL(2,11)^2$

Order 158400: $\SL(2,11).\SL(2,5)$

Order 12100: $C_{11}^2:C_{10}^2$

Order 3025: $C_{11}^2:C_5^2$

Order 484: $C_{22}^2$

Order 400: $C_5^2:C_4^2$

Order 121: $C_{11}^2$

Order 64: $Q_8^2$

Order 25: $C_5^2$

Order 9: $C_3^2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1742400: $\SL(2,11)^2$ ($C_1$)

Order 2640: $C_2\times \SL(2,11)$ ($\PSL(2,11)$) x 2

Order 1320: $\SL(2,11)$ ($\SL(2,11)$) x 2

Order 4: $C_2^2$ ($\PSL(2,11)^2$)

Order 2: $C_2$ ($\PSL(2,11)\times C_2.\PSL(2,11)$), $C_2$ ($C_2.\PSL(2,11)\times \PSL(2,11)$), $C_2$ ($\OmegaPlus(4,11)$)

Order 1: $C_1$ ($\SL(2,11)^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1742400: $\SL(2,11)^2$ ($C_1$)

Order 2640: $C_2\times \SL(2,11)$ ($\PSL(2,11)$)

Order 1: $C_1$ ($\SL(2,11)^2$)

Series

Derived series	$\SL(2,11)^2$
Chief series	$\SL(2,11)^2$	$\rhd$	$C_2\times \SL(2,11)$	$\rhd$	$C_2^2$	$\rhd$	$C_2$	$\rhd$	$C_1$
Lower central series	$\SL(2,11)^2$
Upper central series	$C_1$	$\lhd$	$C_2^2$

Supergroups

This group is a maximal subgroup of 1 larger groups in the database.

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

See the $225 \times 225$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $109 \times 109$ rational character table.