Abstract group 264000.b: $\GL(2,11).D_{10}$

• Label: 264000.b
• Order: \( 2^{6} \cdot 3 \cdot 5^{3} \cdot 11 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 5 \)
• $\card{Z(G)}$: 10
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3 \cdot 5^{2} \cdot 11 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $58$
• Trans deg.: $240$
• Rank: $3$

Group information

Description:	$\GL(2,11).D_{10}$
Order:	\(264000\)\(\medspace = 2^{6} \cdot 3 \cdot 5^{3} \cdot 11 \)
Exponent:	\(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
Automorphism group:	$C_2\times C_4\times (C_2^2\times F_5).\PSL(2,11).C_2$, of order \(844800\)\(\medspace = 2^{10} \cdot 3 \cdot 5^{2} \cdot 11 \)
Composition factors:	$C_2$ x 4, $C_5$ x 2, $\PSL(2,11)$
Derived length:	$2$

This group is nonabelian and nonsolvable. Whether it is almost simple has not been computed.

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	1487	110	1440	6624	2530	2640	83888	120	2640	2640	36560	1560	5280	16720	19360	1200	2880	19360	13440	38720	4800	264000
Conjugacy classes	1	7	1	5	44	4	5	288	1	5	14	60	3	10	46	60	1	14	60	42	120	4	795
Divisions	1	7	1	5	13	4	4	79	1	4	4	17	3	4	12	10	1	4	10	11	10	1	206
Autjugacy classes	1	7	1	5	11	4	4	55	1	5	3	17	3	8	8	8	1	3	11	7	16	1	180

Dimension	1	2	4	8	10	11	12	20	22	24	40	44	48	80	88	96	160	320
Irr. complex chars.	40	40	0	0	100	40	80	250	40	205	0	0	0	0	0	0	0	0	795
Irr. rational chars.	8	0	12	8	12	8	0	4	0	9	19	12	24	19	8	47	10	6	206

Minimal presentations

Permutation degree:	$58$
Transitive degree:	$240$
Rank:	$3$
Inequivalent generating triples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	20	not computed	96
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $58$ $\langle(1,2,8,28,13)(3,14,31,12,9)(4,10,30,11,35)(5,15,7,26,24)(6,22,38,39,27) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 3 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 3 \end{array}\right), \left(\begin{array}{rrrr} 4 & 0 & 7 & 2 \\ 0 & 4 & 4 & 7 \\ 3 & 7 & 7 & 0 \\ 3 & 3 & 0 & 7 \end{array}\right), \left(\begin{array}{rrrr} 10 & 3 & 8 & 2 \\ 10 & 0 & 2 & 8 \\ 7 & 2 & 9 & 8 \\ 8 & 7 & 1 & 10 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 4 & 9 & 5 & 0 \\ 3 & 6 & 2 & 0 \\ 10 & 3 & 7 & 10 \end{array}\right), \left(\begin{array}{rrrr} 8 & 6 & 2 & 10 \\ 6 & 4 & 5 & 2 \\ 5 & 8 & 8 & 5 \\ 4 & 5 & 5 & 4 \end{array}\right), \left(\begin{array}{rrrr} 4 & 10 & 9 & 5 \\ 7 & 7 & 2 & 3 \\ 10 & 1 & 5 & 5 \\ 6 & 6 & 7 & 2 \end{array}\right), \left(\begin{array}{rrrr} 8 & 0 & 0 & 0 \\ 0 & 8 & 0 & 0 \\ 0 & 0 & 8 & 0 \\ 0 & 0 & 0 & 8 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{11})$
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$\GL(2,11)$ . $D_{10}$ (2)	$(D_5\times \GL(2,11))$ . $C_2$	$(C_2\times \GL(2,11))$ . $D_5$	$(\GL(2,11):C_{10})$ . $C_2$	all 53

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 60 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_{10}$
Commutator:	a subgroup isomorphic to $C_5\times \SL(2,11)$
Frattini:	a subgroup isomorphic to $C_2$
Fitting:	not computed
Radical:	not computed
Socle:	not computed

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

All subgroups of index up to 1056 (order at least 250) are shown, as well as all normal subgroups of any index.

Order 264000: $\GL(2,11).D_{10}$

Order 132000: $(C_{10}\times \SL(2,11)):C_{10}$, $C_{10}^2.\PGL(2,11)$, $\GL(2,11):C_{10}$, $(C_5\times C_5:C_4).\PSL(2,11).C_2$, $D_5\times \GL(2,11)$, $(C_{10}\times D_5).\PSL(2,11).C_2$, $C_{10}\times \GL(2,11)$

Order 66000: $C_5\times \GL(2,11)$ x 2, $C_{10}^2.\PSL(2,11)$, $C_5\times C_{10}.\PGL(2,11)$, $C_5:\GL(2,11)$, $C_5\times \SL(2,11):D_5$, $C_5\times D_5\times \SL(2,11)$

Order 52800: $\GL(2,11):C_2^2$, $\SL(2,11).(C_2\times D_{10})$

Order 33000: $C_5^2\times \SL(2,11)$

Order 26400: $C_2\times \GL(2,11)$ x 5, $\GL(2,11):C_2$ x 2, $\SL(2,11).D_{10}$, $\GL(2,11):C_2$, $\GL(2,11):C_2$, $\GL(2,11):C_2$, $\GL(2,11):C_2$, $\GL(2,11):C_2$, $(C_2\times \SL(2,11)):C_{10}$, $\SL(2,11).D_{10}$, $\GL(2,11):C_2$

Order 22000: $D_{110}:C_{10}^2$

Order 13200: $\GL(2,11)$ x 9, $C_{10}\times \SL(2,11)$ x 5, $\SL(2,11):D_5$, $D_5\times \SL(2,11)$, $C_{10}.\PGL(2,11)$, $\SL(2,11):D_5$, $\SL(2,11):C_{10}$, $C_{10}.\PGL(2,11)$

Order 12000: $(C_{10}\times \SL(2,5)):C_{10}$

Order 11000: $C_{110}.C_{10}^2$, $C_{10}^2\times F_{11}$, $C_{10}^2:F_{11}$, $C_{110}.C_{10}^2$, $C_{110}.C_{10}^2$, $C_{110}.C_{10}^2$, $C_{110}:C_{10}^2$

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

Order 6600: $C_5\times \SL(2,11)$ x 4

Order 6000: $C_5\times D_5\times \SL(2,5)$, $C_5\times \SL(2,5):D_5$, $C_{10}^2.A_5$

Order 5500: $C_{55}:C_{10}^2$ x 4, $C_{55}:C_{10}^2$ x 2, $C_5\times C_{55}:C_{20}$, $C_{55}:C_{10}^2$, $C_{55}:C_{10}^2$, $C_{55}:C_{10}^2$, $C_5\times C_{55}:C_{20}$

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 4800: $C_5\times \GL(2,3):D_{10}$, $C_5\times D_{120}:C_2^2$

Order 4400: $C_5:D_4\times F_{11}$ x 5, $C_{10}^2:D_{22}$, $C_{44}:C_{10}^2$

Order 4000: $C_{10}^3:C_2^2$

Order 3000: $C_5^2\times \SL(2,5)$

Order 2750: $C_5^2\times F_{11}$ x 3, $C_{110}:C_5^2$ x 2, $C_5\times C_{55}:C_{10}$, $C_5^2:F_{11}$

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: $C_{120}:D_{10}$ x 2, $C_{120}:D_{10}$ x 2, $C_{120}.D_{10}$ x 2, $C_{120}.D_{10}$ x 2, $D_{120}:C_{10}$, $D_{20}.C_{60}$, $C_5\times D_{20}.D_6$, $C_{10}^2.D_{12}$, $C_5\times \GL(2,3):D_5$, $C_{10}^2.D_{12}$, $C_5\times D_4.A_5$, $\SL(2,3):C_{10}^2$, $C_5\times \GL(2,3):D_5$, $C_{24}:C_{10}^2$, $C_5\times \GL(2,3):D_5$, $C_5\times D_{12}:D_{10}$, $(Q_8\times C_{10}):C_{30}$, $\SL(2,5).D_{10}$, $C_5\times D_5\times \GL(2,3)$, $C_{10}^2.S_4$

Order 2200: $C_{22}:C_{10}^2$ x 19, $D_{110}:C_{10}$ x 5, $D_{10}\times F_{11}$ x 5, $D_{10}:F_{11}$ x 5, $(C_2\times C_{110}):C_{10}$ x 5, $C_5:C_4\times F_{11}$ x 5, $D_{110}:C_{10}$ x 5, $C_{22}.C_{10}^2$ x 2, $C_{110}:D_{10}$, $C_{220}:C_{10}$, $D_{11}\times C_{10}^2$, $C_{110}.D_{10}$, $C_{220}:C_{10}$, $D_{110}:C_{10}$, $D_{10}:C_{110}$, $D_{110}:C_{10}$, $C_{110}.D_{10}$, $C_{20}\times F_{11}$

Order 2000: $C_{10}^2:D_{10}$ x 4, $C_{10}^2.D_{10}$ x 4, $D_{10}:C_{10}^2$ x 2, $C_{10}^2:D_{10}$ x 2, $C_{10}^2.D_{10}$ x 2, $C_{10}^3:C_2$

Order 1600: $C_5\times D_{40}:C_2^2$

Order 1375: $C_{11}:C_5^3$

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

Order 1200: $C_{10}\times \SL(2,5)$ x 5, $C_{120}:C_{10}$ x 4, $D_{60}:C_{10}$ x 2, $C_{10}^2:D_6$ x 2, $C_{10}^2.D_6$ x 2, $C_{60}:D_{10}$ x 2, $D_{60}:C_{10}$ x 2, $C_{60}.D_{10}$ x 2, $C_{60}.D_{10}$ x 2, $C_5^2\times \GL(2,3)$ x 2, $C_{120}:C_{10}$ x 2, $C_{120}:C_{10}$ x 2, $D_5\times C_{120}$ x 2, $C_{60}.D_{10}$ x 2, $C_{60}.D_{10}$ x 2, $C_{60}.D_{10}$ x 2, $D_{60}:C_{10}$ x 2, $C_{12}:C_{10}^2$, $C_{12}.C_{10}^2$, $D_{20}:C_{30}$, $C_{10}^2.A_4$, $C_{60}:D_{10}$, $C_{60}.D_{10}$, $C_{60}.D_{10}$, $C_{60}.D_{10}$, $C_5\times D_5\times \SL(2,3)$, $C_5\times \SL(2,3):D_5$, $C_5^2:\GL(2,3)$, $C_5\times C_{10}.S_4$, $\SL(2,5):C_{10}$, $\SL(2,5):D_5$, $D_5\times \SL(2,5)$, $C_{10}\times C_{120}$, $C_{120}.C_{10}$, $C_5\times D_{120}$, $C_{20}.C_{60}$

Order 1100: $C_{10}\times F_{11}$ x 99, $C_{110}:C_{10}$ x 19, $D_5\times F_{11}$ x 10, $C_{110}:C_{10}$ x 5, $C_{110}:C_{10}$ x 5, $C_{55}:C_{20}$ x 5, $C_{55}:C_{20}$ x 5, $C_5^2\times D_{22}$ x 4, $C_{55}:D_{10}$ x 2, $C_{10}\times C_{110}$, $C_5\times D_{110}$, $D_5\times C_{110}$, $C_{55}:C_{20}$, $C_5:C_{220}$, $C_{220}:C_5$, $C_{55}:C_{20}$

Order 1000: $C_{10}^2:C_{10}$ x 8, $C_{10}^2:C_{10}$ x 4, $C_5^2:D_{20}$ x 4, $C_{10}.D_5^2$ x 4, $D_5\times C_{10}^2$ x 2, $C_{10}\times D_5^2$ x 2, $C_{10}.C_{10}^2$ x 2, $C_5^3:Q_8$ x 2, $C_5^3:D_4$ x 2, $C_{10}.D_5^2$ x 2, $C_{10}^3$, $C_{10}^2:C_{10}$, $C_{10}^2.C_{10}$

Order 960: $D_{120}:C_2^2$, $C_5\times \GL(2,3):C_2^2$, $\GL(2,3):D_{10}$, $C_{120}.C_2^3$

Order 880: $D_4\times F_{11}$ x 5, $C_{20}:D_{22}$, $D_{10}:D_{22}$

Order 800: $C_{40}.D_{10}$ x 2, $C_{40}.D_{10}$ x 2, $C_{40}:D_{10}$ x 2, $C_{40}:D_{10}$ x 2, $C_{10}^2.C_2^3$ x 2, $C_8:C_{10}^2$, $C_{10}^2.D_4$, $C_{10}^2.D_4$, $D_{40}:C_{10}$, $C_{40}.D_{10}$, $C_{10}^2.C_2^3$, $C_2^3:D_5^2$

Order 600: $C_5\times \SL(2,5)$ x 4, $D_{12}\times C_5^2$ x 3, $C_{60}.C_{10}$ x 3, $C_{30}.D_{10}$ x 2, $D_{30}:C_{10}$ x 2, $C_{15}:D_{20}$ x 2, $C_{30}.D_{10}$ x 2, $D_{15}:C_{20}$ x 2, $C_{30}.D_{10}$ x 2, $C_{30}.D_{10}$ x 2, $C_5\times C_{120}$ x 2, $C_5:C_{120}$ x 2, $C_{30}:D_{10}$ x 2, $D_{30}:C_{10}$ x 2, $C_5\times D_{60}$ x 2, $C_{20}\times D_{15}$ x 2, $C_{60}.C_{10}$ x 2, $D_{10}:C_{30}$ x 2, $D_5\times C_{60}$ x 2, $S_3\times C_{10}^2$, $C_{10}\times C_{60}$, $C_{30}:C_{20}$, $C_{15}\times D_{20}$, $C_{20}.C_{30}$, $C_5^2\times \SL(2,3)$

Order 550: $C_5\times F_{11}$ x 67, $C_{110}:C_5$ x 49, $C_{55}:C_{10}$ x 5, $C_{55}:C_{10}$ x 5, $D_{11}\times C_5^2$ x 3, $C_5\times C_{110}$ x 2, $C_5\times D_{55}$, $D_5\times C_{55}$

Order 500: $D_{10}\times C_5^2$ x 6, $C_5\times C_{10}^2$ x 4, $C_5^2:C_{20}$ x 4, $C_5^2:D_{10}$ x 3, $C_5\times D_5^2$ x 2, $C_5^2:C_{20}$ x 2

Order 480: $C_{40}:D_6$ x 6, $C_{10}\times \GL(2,3)$ x 5, $C_{20}.D_{12}$ x 3, $C_{40}:D_6$ x 3, $D_{24}:C_{10}$ x 3, $C_{24}.D_{10}$ x 2, $C_{24}.D_{10}$ x 2, $C_{24}:D_{10}$ x 2, $C_{24}:D_{10}$ x 2, $\GL(2,3):C_{10}$ x 2, $D_5\times \GL(2,3)$, $\GL(2,3):D_5$, $\GL(2,3):D_5$, $\GL(2,3):D_5$, $D_4.A_5$, $\OD_{16}:C_{30}$, $D_{120}:C_2$, $C_{24}.D_{10}$, $C_{20}.D_{12}$, $D_{60}:C_2^2$, $C_{60}.C_2^3$, $C_{60}.C_2^3$, $(C_2\times Q_8):C_{30}$, $D_{12}:D_{10}$, $D_{20}.D_6$, $\SL(2,3).D_{10}$, $Q_8.D_{30}$, $\GL(2,3):C_{10}$, $\GL(2,3):C_{10}$

Order 440: $C_2^2\times F_{11}$ x 20, $D_{22}:C_{10}$ x 10, $C_{44}:C_{10}$ x 5, $C_4\times F_{11}$ x 5, $C_{10}\times D_{22}$ x 5, $C_{44}:C_{10}$ x 5, $C_{55}:D_4$ x 2, $D_5\times D_{22}$, $D_4\times C_{55}$, $C_{55}:D_4$, $D_{10}:C_{22}$, $C_5\times D_{44}$, $C_{20}\times D_{11}$, $C_5:D_{44}$, $C_{55}:D_4$, $C_{10}.D_{22}$

Order 400: $C_{10}^2:C_2^2$ x 14, $C_{20}.D_{10}$ x 7, $C_{20}:D_{10}$ x 7, $D_{20}:C_{10}$ x 5, $D_{10}:D_{10}$ x 4, $D_{10}.D_{10}$ x 4, $\SD_{16}\times C_5^2$ x 4, $C_5^2:Q_{16}$ x 2, $C_{20}.D_{10}$ x 2, $C_{20}.D_{10}$ x 2, $D_{20}:C_{10}$ x 2, $C_{40}:C_{10}$ x 2, $C_{40}:C_{10}$ x 2, $D_5\times C_{40}$ x 2, $C_4:C_{10}^2$ x 2, $D_{20}:C_{10}$ x 2, $C_{20}.D_{10}$ x 2, $D_{10}:D_{10}$ x 2, $D_{10}.D_{10}$ x 2, $C_{20}.C_{20}$, $C_5^2:Q_{16}$, $C_5\times D_{40}$, $C_4.C_{10}^2$, $C_{10}^2:C_2^2$, $C_{10}\times C_{40}$

Order 320: $C_{40}.C_2^3$, $D_{40}:C_2^2$

Order 300: $D_6\times C_5^2$ x 3, $C_5\times D_{30}$ x 2, $C_{15}\times D_{10}$ x 2, $C_{15}:D_{10}$ x 2, $C_5\times C_{60}$ x 2, $C_{15}:C_{20}$ x 2, $C_{15}:C_{20}$ x 2, $C_5:C_{60}$ x 2, $C_{10}\times C_{30}$

Order 275: $C_{55}:C_5$ x 18, $C_5\times C_{55}$

Order 250: $C_5^2\times C_{10}$ x 4, $D_5\times C_5^2$ x 4, $C_5^2:C_{10}$ x 2

Order 200: $C_{10}\wr C_2$

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

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

Order 40: $C_5:D_4$

Order 25: $C_5^2$

Order 20: $C_2\times C_{10}$ x 2, $D_{10}$, $C_5:C_4$

Order 10: $C_{10}$ x 2

Order 5: $C_5$ x 2

Order 4: $C_2^2$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 264000: $\GL(2,11).D_{10}$ ($C_1$)

Order 132000: $(C_{10}\times \SL(2,11)):C_{10}$ ($C_2$), $C_{10}^2.\PGL(2,11)$ ($C_2$), $\GL(2,11):C_{10}$ ($C_2$), $(C_5\times C_5:C_4).\PSL(2,11).C_2$ ($C_2$), $D_5\times \GL(2,11)$ ($C_2$), $(C_{10}\times D_5).\PSL(2,11).C_2$ ($C_2$), $C_{10}\times \GL(2,11)$ ($C_2$)

Order 66000: $C_5\times \GL(2,11)$ ($C_2^2$) x 2, $C_{10}^2.\PSL(2,11)$ ($C_2^2$), $C_5\times C_{10}.\PGL(2,11)$ ($C_2^2$), $C_5:\GL(2,11)$ ($C_2^2$), $C_5\times \SL(2,11):D_5$ ($C_2^2$), $C_5\times D_5\times \SL(2,11)$ ($C_2^2$)

Order 52800: $\SL(2,11).(C_2\times D_{10})$ ($C_5$)

Order 33000: $C_5^2\times \SL(2,11)$ ($C_2^3$)

Order 26400: $\SL(2,11).D_{10}$ ($C_{10}$), $\GL(2,11):C_2$ ($C_{10}$), $\GL(2,11):C_2$ ($C_{10}$), $\GL(2,11):C_2$ ($C_{10}$), $\GL(2,11):C_2$ ($C_{10}$), $\SL(2,11).D_{10}$ ($C_{10}$), $C_2\times \GL(2,11)$ ($C_{10}$), $C_2\times \GL(2,11)$ ($D_5$)

Order 13200: $\GL(2,11)$ ($C_2\times C_{10}$) x 2, $\GL(2,11)$ ($D_{10}$) x 2, $\SL(2,11):D_5$ ($C_2\times C_{10}$), $D_5\times \SL(2,11)$ ($C_2\times C_{10}$), $C_{10}.\PGL(2,11)$ ($C_2\times C_{10}$), $\SL(2,11):D_5$ ($C_2\times C_{10}$), $C_{10}\times \SL(2,11)$ ($C_2\times C_{10}$), $C_{10}\times \SL(2,11)$ ($D_{10}$)

Order 6600: $C_5\times \SL(2,11)$ ($C_2^2\times C_{10}$), $C_5\times \SL(2,11)$ ($C_2\times D_{10}$)

Order 5280: $\SL(2,11):C_2^2$ ($C_5\times D_5$)

Order 2640: $\SL(2,11):C_2$ ($C_5\times D_{10}$) x 2, $C_2\times \SL(2,11)$ ($C_5\times D_{10}$)

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

Order 200: $C_{10}\wr C_2$ ($\PGL(2,11)$)

Order 100: $C_5:C_{20}$ ($C_2\times \PGL(2,11)$), $C_{10}^2$ ($C_2\times \PGL(2,11)$), $C_5\times D_{10}$ ($C_2\times \PGL(2,11)$)

Order 50: $C_5\times C_{10}$ ($C_2^2.\PSL(2,11).C_2$)

Order 40: $C_5:D_4$ ($C_5.\PSL(2,11).C_2$)

Order 25: $C_5^2$ ($D_4.\PGL(2,11)$)

Order 20: $C_2\times C_{10}$ ($D_5.\PSL(2,11).C_2$), $C_2\times C_{10}$ ($C_{10}\times \PGL(2,11)$), $D_{10}$ ($C_{10}\times \PGL(2,11)$), $C_5:C_4$ ($C_{10}\times \PGL(2,11)$)

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

Order 5: $C_5$ ($\GL(2,11):C_2^2$), $C_5$ ($\SL(2,11).(C_2\times D_{10})$)

Order 4: $C_2^2$ ($(C_5\times D_5).\PSL(2,11).C_2$)

Order 2: $C_2$ ($(C_{10}\times D_5).\PSL(2,11).C_2$)

Order 1: $C_1$ ($\GL(2,11).D_{10}$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

Character theory

Complex character table

The $795 \times 795$ character table is not available for this group.

Rational character table

See the $206 \times 206$ rational character table (warning: may be slow to load).