Abstract group 52800.f: $\GL(2,11):C_2^2$

• Label: 52800.f
• Order: \( 2^{6} \cdot 3 \cdot 5^{2} \cdot 11 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 5 \)
• $\card{Z(G)}$: \( 2 \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3 \cdot 5 \cdot 11 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $53$
• Trans deg.: $240$
• Rank: $3$

Group information

Description:	$\GL(2,11):C_2^2$
Order:	\(52800\)\(\medspace = 2^{6} \cdot 3 \cdot 5^{2} \cdot 11 \)
Exponent:	\(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
Automorphism group:	$C_2\times C_4\times \PSL(2,11).C_2\times D_4$, of order \(84480\)\(\medspace = 2^{9} \cdot 3 \cdot 5 \cdot 11 \)
Composition factors:	$C_2$ x 4, $C_5$, $\PSL(2,11)$
Derived length:	$1$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	8	10	11	12	15	20	22	24	30	40	44	55	60	110	120	220
Elements	1	511	110	464	1324	770	880	16564	120	880	440	7136	600	1760	3080	3520	240	480	3520	2400	7040	960	52800
Conjugacy classes	1	7	1	5	14	4	5	98	1	5	4	40	3	10	16	20	1	4	20	12	40	4	315
Divisions	1	7	1	5	4	4	4	27	1	4	1	11	3	4	4	4	1	1	4	3	4	1	99
Autjugacy classes	1	5	1	4	5	3	3	21	1	4	1	12	2	6	3	3	1	1	4	2	6	1	90

Dimension	1	4	10	11	12	20	24	40	44	48	80	96	160	320
Irr. complex chars.	40	0	100	40	80	30	25	0	0	0	0	0	0	0	315
Irr. rational chars.	8	8	12	8	0	4	9	13	8	16	5	6	1	1	99

Minimal presentations

Permutation degree:	$53$
Transitive degree:	$240$
Rank:	$3$
Inequivalent generating triples:	$562776480$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	20	40	96
Arbitrary	20	26	28

Constructions

Permutation group:	Degree $53$ $\langle(49,50,53,51,52), (1,2)(3,8)(4,9)(5,10)(6,17)(7,16)(11,20)(12,22)(13,21) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 3 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 3 \end{array}\right), \left(\begin{array}{rrrr} 8 & 4 & 6 & 6 \\ 5 & 5 & 3 & 6 \\ 9 & 8 & 6 & 7 \\ 3 & 9 & 6 & 3 \end{array}\right), \left(\begin{array}{rrrr} 10 & 5 & 1 & 10 \\ 5 & 0 & 8 & 1 \\ 0 & 10 & 10 & 7 \\ 10 & 0 & 7 & 2 \end{array}\right), \left(\begin{array}{rrrr} 9 & 8 & 1 & 0 \\ 7 & 2 & 0 & 10 \\ 9 & 0 & 2 & 8 \\ 0 & 2 & 7 & 9 \end{array}\right), \left(\begin{array}{rrrr} 3 & 5 & 5 & 0 \\ 2 & 5 & 5 & 0 \\ 9 & 4 & 5 & 2 \\ 4 & 8 & 0 & 9 \end{array}\right), \left(\begin{array}{rrrr} 6 & 5 & 1 & 0 \\ 5 & 5 & 0 & 10 \\ 6 & 0 & 5 & 5 \\ 0 & 5 & 5 & 6 \end{array}\right), \left(\begin{array}{rrrr} 8 & 0 & 0 & 0 \\ 0 & 8 & 0 & 0 \\ 0 & 0 & 8 & 0 \\ 0 & 0 & 0 & 8 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{11})$
Direct product:	$C_5$ $\, \times\, $ $(D_4.\PGL(2,11))$
Semidirect product:	$\GL(2,11)$ $\,\rtimes\,$ $C_2^2$ (3)	$(C_2\times \GL(2,11))$ $\,\rtimes\,$ $C_2$ (2)	$(\GL(2,11):C_2)$ $\,\rtimes\,$ $C_2$ (2)	$(\GL(2,11):C_2)$ $\,\rtimes\,$ $C_2$	all 18
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_5\times D_4)$ . $\PGL(2,11)$	$(C_5\times \SL(2,11))$ . $C_2^3$	$C_{20}$ . $(C_2\times \PGL(2,11))$	$C_4$ . $(C_{10}\times \PGL(2,11))$	all 10

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

Homology

Abelianization:	$C_{2}^{2} \times C_{10} \simeq C_{2}^{3} \times C_{5}$
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 82612 subgroups in 986 conjugacy classes, 44 normal (24 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{10}$	$G/Z \simeq$ $C_2^2.\PSL(2,11).C_2$
Commutator:	$G' \simeq$ $\SL(2,11)$	$G/G' \simeq$ $C_2^2\times C_{10}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $(C_2\times C_{10}).\PSL(2,11).C_2$
Fitting:	$\operatorname{Fit} \simeq$ $C_5\times D_4$	$G/\operatorname{Fit} \simeq$ $\PGL(2,11)$
Radical:	$R \simeq$ $C_5\times D_4$	$G/R \simeq$ $\PGL(2,11)$
Socle:	$\operatorname{soc} \simeq$ $C_{10}$	$G/\operatorname{soc} \simeq$ $C_2^2.\PSL(2,11).C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_8:C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
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

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Classes of subgroups up to automorphism

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Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 52800: $\GL(2,11):C_2^2$

Order 26400: $\GL(2,11):C_2$ x 2, $C_2\times \GL(2,11)$ x 2, $\GL(2,11):C_2$, $(C_2\times \SL(2,11)):C_{10}$, $\GL(2,11):C_2$

Order 13200: $\GL(2,11)$ x 3, $C_{10}\times \SL(2,11)$ x 2, $\SL(2,11):C_{10}$, $C_{10}.\PGL(2,11)$

Order 10560: $D_4.\PGL(2,11)$

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

Order 5280: $C_2^2.\PGL(2,11)$ x 2, $\SL(2,11):C_2^2$ x 2, $\SL(2,11):C_2^2$, $D_4.\PSL(2,11)$, $C_4.\PGL(2,11)$

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

Order 2640: $\SL(2,11):C_2$ x 3, $C_2\times \SL(2,11)$ x 2, $\SL(2,11):C_2$, $C_2.\PGL(2,11)$

Order 2400: $C_5\times D_4.A_5$

Order 2200: $C_{22}.C_{10}^2$ x 2, $C_{22}:C_{10}^2$ x 2, $C_{220}:C_{10}$, $C_{220}:C_{10}$, $C_{20}\times F_{11}$

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

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

Order 1100: $C_{10}\times F_{11}$ x 7, $C_{110}:C_{10}$ x 2, $C_{220}:C_5$, $C_{55}:C_{20}$

Order 960: $C_5\times \GL(2,3):C_2^2$, $C_{120}.C_2^3$

Order 880: $D_4\times F_{11}$ x 5, $C_{20}:D_{22}$

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

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

Order 550: $C_5\times F_{11}$ x 4, $C_{110}:C_5$ x 3

Order 480: $C_{20}.D_{12}$ x 3, $C_{40}:D_6$ x 3, $D_{24}:C_{10}$ x 3, $C_{40}:D_6$ x 3, $\GL(2,3):C_{10}$ x 2, $C_{10}\times \GL(2,3)$ x 2, $D_4.A_5$, $\OD_{16}:C_{30}$, $C_{60}.C_2^3$, $C_{60}.C_2^3$, $(C_2\times Q_8):C_{30}$, $\GL(2,3):C_{10}$, $\GL(2,3):C_{10}$

Order 440: $C_2^2\times F_{11}$ x 10, $D_{22}:C_{10}$ x 10, $C_{44}:C_{10}$ x 5, $C_4\times F_{11}$ x 5, $C_{44}:C_{10}$ x 5, $C_{10}\times D_{22}$ x 2, $C_{55}:D_4$ x 2, $D_4\times C_{55}$, $C_5\times D_{44}$, $C_{20}\times D_{11}$

Order 400: $C_{10}^2:C_2^2$ x 4, $C_{20}.D_{10}$ x 4, $C_{20}:D_{10}$ x 4, $D_{20}:C_{10}$ x 2, $C_4:C_{10}^2$

Order 320: $C_{40}.C_2^3$

Order 275: $C_{55}:C_5$

Order 240: $C_{120}:C_2$ x 10, $D_{12}:C_{10}$ x 6, $C_{15}\times \OD_{16}$ x 3, $C_2\times C_{120}$ x 3, $C_{15}:Q_{16}$ x 3, $C_5\times D_{24}$ x 3, $C_{20}.D_6$ x 3, $C_{20}:D_6$ x 3, $C_{10}\times D_{12}$ x 3, $C_{30}:Q_8$ x 3, $C_5\times \GL(2,3)$ x 3, $C_2\times \SL(2,5)$ x 2, $C_{10}\times \SL(2,3)$ x 2, $\SL(2,5):C_2$, $D_4:C_{30}$, $D_{12}:C_{10}$, $C_{20}.D_6$, $\SL(2,3):C_{10}$, $C_{10}.S_4$

Order 220: $C_2\times F_{11}$ x 35, $C_{22}:C_{10}$ x 10, $C_5\times D_{22}$ x 7, $C_{11}:C_{20}$ x 5, $C_{11}:C_{20}$ x 5, $C_2\times C_{110}$ x 2, $C_{220}$, $C_{11}:C_{20}$

Order 200: $C_{10}\wr C_2$ x 12, $C_{10}\times D_{10}$ x 4, $D_4\times C_5^2$ x 4, $C_{10}:C_{20}$ x 4, $D_5\times C_{20}$ x 4, $C_2\times C_{10}^2$ x 2, $C_5\times D_{20}$ x 2, $C_5^2:Q_8$ x 2, $C_{10}\times C_{20}$

Order 192: $\GL(2,3):C_2^2$, $D_{24}:C_2^2$

Order 176: $D_4\times D_{11}$

Order 160: $Q_{16}:C_{10}$ x 3, $D_8:C_{10}$ x 3, $D_8:C_{10}$ x 3, $C_{10}\times \SD_{16}$ x 3, $C_{10}.C_2^4$, $C_{10}.C_2^4$, $D_4:D_{10}$, $\OD_{16}:C_{10}$

Order 120: $C_5\times D_{12}$ x 7, $C_{15}:Q_8$ x 7, $C_{15}:D_4$ x 6, $S_3\times C_{20}$ x 6, $C_{120}$ x 4, $C_{10}\times D_6$ x 3, $D_4\times C_{15}$ x 3, $C_2\times C_{60}$ x 3, $C_6:C_{20}$ x 3, $\SL(2,5)$, $Q_8\times C_{15}$, $C_5\times \SL(2,3)$

Order 110: $F_{11}$ x 20, $C_{11}:C_{10}$ x 15, $C_5\times D_{11}$ x 4, $C_{110}$ x 3

Order 100: $C_5\times D_{10}$ x 8, $C_{10}^2$ x 7, $C_5:C_{20}$ x 4, $C_5\times C_{20}$ x 2

Order 96: $C_4.D_{12}$ x 3, $C_8:D_6$ x 3, $D_{24}:C_2$ x 3, $C_8:D_6$ x 3, $\GL(2,3):C_2$ x 2, $C_2\times \GL(2,3)$ x 2, $D_4.D_6$, $D_4:D_6$, $D_4.A_4$, $\GL(2,3):C_2$, $\GL(2,3):C_2$, $\OD_{16}:C_6$

Order 88: $C_{11}:D_4$ x 2, $C_2\times D_{22}$ x 2, $D_4\times C_{11}$, $D_{44}$, $C_4\times D_{11}$

Order 80: $C_5\times \SD_{16}$ x 10, $D_4\times C_{10}$ x 9, $D_4:C_{10}$ x 8, $C_{10}:D_4$ x 4, $D_4:D_5$ x 4, $D_4\times D_5$ x 4, $Q_8\times C_{10}$ x 3, $C_5\times Q_{16}$ x 3, $C_5\times D_8$ x 3, $C_5\times \OD_{16}$ x 3, $C_2\times C_{40}$ x 3, $D_{20}:C_2$ x 2

Order 64: $D_8:C_2^2$

Order 60: $S_3\times C_{10}$ x 7, $C_{60}$ x 4, $C_3:C_{20}$ x 4, $C_2\times C_{30}$ x 3

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

Order 50: $C_5\times C_{10}$ x 6, $C_5\times D_5$ x 4

Order 48: $C_{24}:C_2$ x 10, $D_{12}:C_2$ x 6, $C_3:Q_{16}$ x 3, $D_{24}$ x 3, $D_4:S_3$ x 3, $S_3\times D_4$ x 3, $C_2\times D_{12}$ x 3, $C_6:Q_8$ x 3, $\GL(2,3)$ x 3, $C_3\times \OD_{16}$ x 3, $C_2\times C_{24}$ x 3, $C_2\times \SL(2,3)$ x 2, $D_4:C_6$, $D_{12}:C_2$, $S_3\times Q_8$, $\SL(2,3):C_2$, $C_2.S_4$

Order 44: $D_{22}$ x 7, $C_2\times C_{22}$ x 2, $C_{44}$, $C_{11}:C_4$

Order 40: $C_5\times D_4$ x 25, $C_5:D_4$ x 12, $C_2\times C_{20}$ x 10, $C_2^2\times C_{10}$ x 9, $C_5\times Q_8$ x 5, $C_{10}:C_4$ x 4, $C_4\times D_5$ x 4, $C_{40}$ x 4, $C_2\times D_{10}$ x 4, $D_{20}$ x 2, $C_5:Q_8$ x 2

Order 32: $Q_{16}:C_2$ x 3, $D_8:C_2$ x 3, $D_8:C_2$ x 3, $C_2\times \SD_{16}$ x 3, $C_4.C_2^3$, $D_4:C_2^2$, $\OD_{16}:C_2$

Order 30: $C_{30}$ x 4, $C_5\times S_3$ x 4

Order 25: $C_5^2$

Order 24: $D_{12}$ x 7, $C_3:Q_8$ x 7, $C_3:D_4$ x 6, $C_4\times S_3$ x 6, $C_{24}$ x 4, $C_2\times C_{12}$ x 3, $C_6:C_4$ x 3, $C_2\times D_6$ x 3, $C_3\times D_4$ x 3, $\SL(2,3)$, $C_3\times Q_8$

Order 22: $D_{11}$ x 4, $C_{22}$ x 3

Order 20: $C_2\times C_{10}$ x 34, $C_{20}$ x 11, $D_{10}$ x 8, $C_5:C_4$ x 4

Order 16: $\SD_{16}$ x 10, $D_4:C_2$ x 8, $C_2\times D_4$ x 6, $Q_{16}$ x 3, $D_8$ x 3, $\OD_{16}$ x 3, $C_2\times C_8$ x 3, $C_2\times Q_8$ x 3

Order 15: $C_{15}$

Order 12: $D_6$ x 7, $C_{12}$ x 4, $C_3:C_4$ x 4, $C_2\times C_6$ x 3

Order 11: $C_{11}$

Order 10: $C_{10}$ x 27, $D_5$ x 4

Order 8: $D_4$ x 13, $C_2\times C_4$ x 7, $Q_8$ x 5, $C_8$ x 4, $C_2^3$ x 3

Order 6: $C_6$ x 4, $S_3$ x 4

Order 5: $C_5$ x 4

Order 4: $C_2^2$ x 9, $C_4$ x 5

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 52800: $\GL(2,11):C_2^2$

Order 26400: $\GL(2,11):C_2$, $\GL(2,11):C_2$, $(C_2\times \SL(2,11)):C_{10}$, $\GL(2,11):C_2$, $C_2\times \GL(2,11)$

Order 13200: $\GL(2,11)$ x 2, $\SL(2,11):C_{10}$, $C_{10}.\PGL(2,11)$, $C_{10}\times \SL(2,11)$

Order 10560: $D_4.\PGL(2,11)$

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

Order 5280: $\SL(2,11):C_2^2$, $D_4.\PSL(2,11)$, $C_2^2.\PGL(2,11)$, $C_4.\PGL(2,11)$, $\SL(2,11):C_2^2$

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

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

Order 2400: $C_5\times D_4.A_5$

Order 2200: $C_{220}:C_{10}$, $C_{22}.C_{10}^2$, $C_{220}:C_{10}$, $C_{20}\times F_{11}$, $C_{22}:C_{10}^2$

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

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

Order 1100: $C_{10}\times F_{11}$ x 3, $C_{110}:C_{10}$, $C_{220}:C_5$, $C_{55}:C_{20}$

Order 960: $C_5\times \GL(2,3):C_2^2$, $C_{120}.C_2^3$

Order 880: $D_4\times F_{11}$ x 2, $C_{20}:D_{22}$

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

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

Order 550: $C_5\times F_{11}$ x 2, $C_{110}:C_5$ x 2

Order 480: $C_{20}.D_{12}$ x 2, $C_{40}:D_6$ x 2, $D_{24}:C_{10}$ x 2, $C_{40}:D_6$ x 2, $D_4.A_5$, $\OD_{16}:C_{30}$, $C_{60}.C_2^3$, $C_{60}.C_2^3$, $(C_2\times Q_8):C_{30}$, $\GL(2,3):C_{10}$, $\GL(2,3):C_{10}$, $\GL(2,3):C_{10}$, $C_{10}\times \GL(2,3)$

Order 440: $C_{44}:C_{10}$ x 2, $C_4\times F_{11}$ x 2, $C_2^2\times F_{11}$ x 2, $C_{44}:C_{10}$ x 2, $D_{22}:C_{10}$ x 2, $C_{10}\times D_{22}$, $D_4\times C_{55}$, $C_{55}:D_4$, $C_5\times D_{44}$, $C_{20}\times D_{11}$

Order 400: $C_{20}.D_{10}$ x 3, $C_{20}:D_{10}$ x 3, $C_{10}^2:C_2^2$ x 2, $D_{20}:C_{10}$ x 2, $C_4:C_{10}^2$

Order 320: $C_{40}.C_2^3$

Order 275: $C_{55}:C_5$

Order 240: $C_{120}:C_2$ x 6, $D_{12}:C_{10}$ x 4, $C_{15}\times \OD_{16}$ x 2, $C_2\times C_{120}$ x 2, $C_{15}:Q_{16}$ x 2, $C_5\times D_{24}$ x 2, $C_{20}.D_6$ x 2, $C_{20}:D_6$ x 2, $C_{10}\times D_{12}$ x 2, $C_{30}:Q_8$ x 2, $C_5\times \GL(2,3)$ x 2, $C_2\times \SL(2,5)$, $\SL(2,5):C_2$, $D_4:C_{30}$, $D_{12}:C_{10}$, $C_{20}.D_6$, $\SL(2,3):C_{10}$, $C_{10}\times \SL(2,3)$, $C_{10}.S_4$

Order 220: $C_2\times F_{11}$ x 6, $C_5\times D_{22}$ x 3, $C_{22}:C_{10}$ x 2, $C_{11}:C_{20}$ x 2, $C_{11}:C_{20}$ x 2, $C_{220}$, $C_{11}:C_{20}$, $C_2\times C_{110}$

Order 200: $C_{10}\wr C_2$ x 7, $D_4\times C_5^2$ x 3, $D_5\times C_{20}$ x 3, $C_{10}\times D_{10}$ x 2, $C_{10}:C_{20}$ x 2, $C_5\times D_{20}$ x 2, $C_5^2:Q_8$ x 2, $C_2\times C_{10}^2$, $C_{10}\times C_{20}$

Order 192: $\GL(2,3):C_2^2$, $D_{24}:C_2^2$

Order 176: $D_4\times D_{11}$

Order 160: $Q_{16}:C_{10}$ x 2, $D_8:C_{10}$ x 2, $D_8:C_{10}$ x 2, $C_{10}\times \SD_{16}$ x 2, $C_{10}.C_2^4$, $C_{10}.C_2^4$, $D_4:D_{10}$, $\OD_{16}:C_{10}$

Order 120: $C_5\times D_{12}$ x 5, $C_{15}:Q_8$ x 5, $C_{15}:D_4$ x 4, $S_3\times C_{20}$ x 4, $C_{120}$ x 3, $C_{10}\times D_6$ x 2, $D_4\times C_{15}$ x 2, $C_2\times C_{60}$ x 2, $C_6:C_{20}$ x 2, $\SL(2,5)$, $Q_8\times C_{15}$, $C_5\times \SL(2,3)$

Order 110: $C_{11}:C_{10}$ x 4, $F_{11}$ x 4, $C_{110}$ x 2, $C_5\times D_{11}$ x 2

Order 100: $C_5\times D_{10}$ x 5, $C_{10}^2$ x 4, $C_5:C_{20}$ x 3, $C_5\times C_{20}$ x 2

Order 96: $C_4.D_{12}$ x 2, $C_8:D_6$ x 2, $D_{24}:C_2$ x 2, $C_8:D_6$ x 2, $D_4.D_6$, $D_4:D_6$, $D_4.A_4$, $\GL(2,3):C_2$, $\GL(2,3):C_2$, $\GL(2,3):C_2$, $C_2\times \GL(2,3)$, $\OD_{16}:C_6$

Order 88: $D_4\times C_{11}$, $C_{11}:D_4$, $D_{44}$, $C_4\times D_{11}$, $C_2\times D_{22}$

Order 80: $D_4:C_{10}$ x 6, $D_4\times C_{10}$ x 6, $C_5\times \SD_{16}$ x 6, $D_4:D_5$ x 3, $D_4\times D_5$ x 3, $Q_8\times C_{10}$ x 2, $C_{10}:D_4$ x 2, $D_{20}:C_2$ x 2, $C_5\times Q_{16}$ x 2, $C_5\times D_8$ x 2, $C_5\times \OD_{16}$ x 2, $C_2\times C_{40}$ x 2

Order 64: $D_8:C_2^2$

Order 60: $S_3\times C_{10}$ x 5, $C_{60}$ x 3, $C_3:C_{20}$ x 3, $C_2\times C_{30}$ x 2

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

Order 50: $C_5\times C_{10}$ x 4, $C_5\times D_5$ x 3

Order 48: $C_{24}:C_2$ x 6, $D_{12}:C_2$ x 4, $C_3:Q_{16}$ x 2, $D_{24}$ x 2, $D_4:S_3$ x 2, $S_3\times D_4$ x 2, $C_2\times D_{12}$ x 2, $C_6:Q_8$ x 2, $\GL(2,3)$ x 2, $C_3\times \OD_{16}$ x 2, $C_2\times C_{24}$ x 2, $D_4:C_6$, $D_{12}:C_2$, $S_3\times Q_8$, $\SL(2,3):C_2$, $C_2\times \SL(2,3)$, $C_2.S_4$

Order 44: $D_{22}$ x 3, $C_2\times C_{22}$, $C_{44}$, $C_{11}:C_4$

Order 40: $C_5\times D_4$ x 15, $C_2\times C_{20}$ x 7, $C_5:D_4$ x 7, $C_2^2\times C_{10}$ x 4, $C_5\times Q_8$ x 4, $C_4\times D_5$ x 3, $C_{40}$ x 3, $C_{10}:C_4$ x 2, $D_{20}$ x 2, $C_5:Q_8$ x 2, $C_2\times D_{10}$ x 2

Order 32: $Q_{16}:C_2$ x 2, $D_8:C_2$ x 2, $D_8:C_2$ x 2, $C_2\times \SD_{16}$ x 2, $C_4.C_2^3$, $D_4:C_2^2$, $\OD_{16}:C_2$

Order 30: $C_{30}$ x 3, $C_5\times S_3$ x 3

Order 25: $C_5^2$

Order 24: $D_{12}$ x 5, $C_3:Q_8$ x 5, $C_3:D_4$ x 4, $C_4\times S_3$ x 4, $C_{24}$ x 3, $C_2\times C_{12}$ x 2, $C_6:C_4$ x 2, $C_2\times D_6$ x 2, $C_3\times D_4$ x 2, $\SL(2,3)$, $C_3\times Q_8$

Order 22: $C_{22}$ x 2, $D_{11}$ x 2

Order 20: $C_2\times C_{10}$ x 14, $C_{20}$ x 8, $D_{10}$ x 5, $C_5:C_4$ x 3

Order 16: $\SD_{16}$ x 6, $D_4:C_2$ x 6, $C_2\times D_4$ x 4, $Q_{16}$ x 2, $D_8$ x 2, $\OD_{16}$ x 2, $C_2\times C_8$ x 2, $C_2\times Q_8$ x 2

Order 15: $C_{15}$

Order 12: $D_6$ x 5, $C_{12}$ x 3, $C_3:C_4$ x 3, $C_2\times C_6$ x 2

Order 11: $C_{11}$

Order 10: $C_{10}$ x 13, $D_5$ x 3

Order 8: $D_4$ x 9, $C_2\times C_4$ x 5, $Q_8$ x 4, $C_8$ x 3, $C_2^3$ x 2

Order 6: $C_6$ x 3, $S_3$ x 3

Order 5: $C_5$ x 3

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

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 52800: $\GL(2,11):C_2^2$ ($C_1$)

Order 26400: $\GL(2,11):C_2$ ($C_2$) x 2, $C_2\times \GL(2,11)$ ($C_2$) x 2, $\GL(2,11):C_2$ ($C_2$), $(C_2\times \SL(2,11)):C_{10}$ ($C_2$), $\GL(2,11):C_2$ ($C_2$)

Order 13200: $\GL(2,11)$ ($C_2^2$) x 3, $C_{10}\times \SL(2,11)$ ($C_2^2$) x 2, $\SL(2,11):C_{10}$ ($C_2^2$), $C_{10}.\PGL(2,11)$ ($C_2^2$)

Order 10560: $D_4.\PGL(2,11)$ ($C_5$)

Order 6600: $C_5\times \SL(2,11)$ ($C_2^3$)

Order 5280: $C_2^2.\PGL(2,11)$ ($C_{10}$) x 2, $\SL(2,11):C_2^2$ ($C_{10}$) x 2, $\SL(2,11):C_2^2$ ($C_{10}$), $D_4.\PSL(2,11)$ ($C_{10}$), $C_4.\PGL(2,11)$ ($C_{10}$)

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

Order 1320: $\SL(2,11)$ ($C_2^2\times C_{10}$)

Order 40: $C_5\times D_4$ ($\PGL(2,11)$)

Order 20: $C_2\times C_{10}$ ($C_2\times \PGL(2,11)$) x 2, $C_{20}$ ($C_2\times \PGL(2,11)$)

Order 10: $C_{10}$ ($C_2^2.\PSL(2,11).C_2$)

Order 8: $D_4$ ($C_5.\PSL(2,11).C_2$)

Order 5: $C_5$ ($D_4.\PGL(2,11)$)

Order 4: $C_2^2$ ($C_{10}\times \PGL(2,11)$) x 2, $C_4$ ($C_{10}\times \PGL(2,11)$)

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

Order 1: $C_1$ ($\GL(2,11):C_2^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 52800: $\GL(2,11):C_2^2$ ($C_1$)

Order 26400: $\GL(2,11):C_2$ ($C_2$), $\GL(2,11):C_2$ ($C_2$), $(C_2\times \SL(2,11)):C_{10}$ ($C_2$), $\GL(2,11):C_2$ ($C_2$), $C_2\times \GL(2,11)$ ($C_2$)

Order 13200: $\GL(2,11)$ ($C_2^2$) x 2, $\SL(2,11):C_{10}$ ($C_2^2$), $C_{10}.\PGL(2,11)$ ($C_2^2$), $C_{10}\times \SL(2,11)$ ($C_2^2$)

Order 10560: $D_4.\PGL(2,11)$ ($C_5$)

Order 6600: $C_5\times \SL(2,11)$ ($C_2^3$)

Order 5280: $\SL(2,11):C_2^2$ ($C_{10}$), $D_4.\PSL(2,11)$ ($C_{10}$), $C_2^2.\PGL(2,11)$ ($C_{10}$), $C_4.\PGL(2,11)$ ($C_{10}$), $\SL(2,11):C_2^2$ ($C_{10}$)

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

Order 1320: $\SL(2,11)$ ($C_2^2\times C_{10}$)

Order 40: $C_5\times D_4$ ($\PGL(2,11)$)

Order 20: $C_2\times C_{10}$ ($C_2\times \PGL(2,11)$), $C_{20}$ ($C_2\times \PGL(2,11)$)

Order 10: $C_{10}$ ($C_2^2.\PSL(2,11).C_2$)

Order 8: $D_4$ ($C_5.\PSL(2,11).C_2$)

Order 5: $C_5$ ($D_4.\PGL(2,11)$)

Order 4: $C_2^2$ ($C_{10}\times \PGL(2,11)$), $C_4$ ($C_{10}\times \PGL(2,11)$)

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

Order 1: $C_1$ ($\GL(2,11):C_2^2$)

Series

Derived series

$\GL(2,11):C_2^2$

$\rhd$

$\SL(2,11)$

Chief series

$\GL(2,11):C_2^2$

$\rhd$

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

$\rhd$

$C_5\times D_4$

$\rhd$

$C_2\times C_{10}$

$\rhd$

$C_{10}$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$\GL(2,11):C_2^2$

$\rhd$

$\SL(2,11)$

Upper central series

$C_1$

$\lhd$

$C_{10}$

$\lhd$

$C_5\times D_4$

Supergroups

This group is a maximal subgroup of 3 larger groups in the database.

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

Character theory

Complex character table

See the $315 \times 315$ 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 $99 \times 99$ rational character table.