Abstract group 15840.m: $\SL(2,11):D_6$

• Label: 15840.m
• Order: \( 2^{5} \cdot 3^{2} \cdot 5 \cdot 11 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{2} \cdot 5 \cdot 11 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $51$
• Trans deg.: $144$
• Rank: $2$

Group information

Description:	$\SL(2,11):D_6$
Order:	\(15840\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \cdot 11 \)
Exponent:	\(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
Automorphism group:	$(S_3\times D_4).\PSL(2,11).C_2$, of order \(63360\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \cdot 11 \)
Composition factors:	$C_2$ x 3, $C_3$, $\PSL(2,11)$
Derived length:	$2$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	8	10	11	12	15	20	22	24	30	33	44	60	66	132
Elements	1	903	332	112	264	1212	1320	3432	120	1544	528	528	120	2640	528	240	240	1056	240	480	15840
Conjugacy classes	1	4	3	2	2	7	2	6	1	10	2	2	1	4	2	2	2	4	2	4	63
Divisions	1	4	3	2	1	7	2	3	1	6	1	1	1	2	1	1	1	1	1	1	41
Autjugacy classes	1	3	3	2	2	6	1	4	1	8	2	2	1	2	2	1	1	2	1	1	46

Dimension	1	2	10	11	12	20	22	24	40	48	96
Irr. complex chars.	4	2	14	4	14	13	2	10	0	0	0	63
Irr. rational chars.	4	2	6	4	0	9	2	5	4	4	1	41

Minimal presentations

Permutation degree:	$51$
Transitive degree:	$144$
Rank:	$2$
Inequivalent generating pairs:	$876$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	12	40
Arbitrary	12	12	22

Constructions

Permutation group:	Degree $51$ $\langle(1,2,4,7)(3,9,11,21)(5,8,13,19)(6,15,17,30)(10,23,25,36)(12,22,14,28)(16,29,32,44) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 7 & 7 & 7 & 8 \\ 7 & 0 & 0 & 8 \\ 7 & 4 & 1 & 0 \\ 4 & 0 & 3 & 3 \end{array}\right), \left(\begin{array}{rrrr} 1 & 1 & 7 & 6 \\ 4 & 7 & 0 & 7 \\ 8 & 3 & 4 & 10 \\ 8 & 8 & 7 & 10 \end{array}\right), \left(\begin{array}{rrrr} 9 & 3 & 10 & 7 \\ 1 & 5 & 0 & 10 \\ 2 & 9 & 7 & 8 \\ 2 & 2 & 10 & 3 \end{array}\right), \left(\begin{array}{rrrr} 9 & 6 & 9 & 3 \\ 8 & 2 & 10 & 9 \\ 4 & 2 & 3 & 5 \\ 3 & 6 & 6 & 8 \end{array}\right), \left(\begin{array}{rrrr} 10 & 0 & 0 & 0 \\ 0 & 10 & 0 & 0 \\ 0 & 0 & 10 & 0 \\ 0 & 0 & 0 & 10 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{11})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$\SL(2,11)$ $\,\rtimes\,$ $D_6$	$(\SL(2,11):S_3)$ $\,\rtimes\,$ $C_2$ (2)	$(\SL(2,11):C_2)$ $\,\rtimes\,$ $S_3$	$(\SL(2,11):C_6)$ $\,\rtimes\,$ $C_2$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{12}$ . $\PGL(2,11)$	$C_6$ . $(C_2\times \PGL(2,11))$	$C_4$ . $(C_3:\PGL(2,11))$	$C_2$ . $(C_2\times C_3.\PSL(2,11).C_2)$	more information

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

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 46708 subgroups in 273 conjugacy classes, 13 normal (11 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2\times C_3.\PSL(2,11).C_2$
Commutator:	$G' \simeq$ $C_3\times \SL(2,11)$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2\times C_3.\PSL(2,11).C_2$
Fitting:	$\operatorname{Fit} \simeq$ $C_{12}$	$G/\operatorname{Fit} \simeq$ $\PGL(2,11)$
Radical:	$R \simeq$ $C_{12}$	$G/R \simeq$ $\PGL(2,11)$
Socle:	$\operatorname{soc} \simeq$ $C_6$	$G/\operatorname{soc} \simeq$ $C_2\times \PGL(2,11)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_8:C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
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 15840: $\SL(2,11):D_6$

Order 7920: $\SL(2,11):S_3$ x 2, $\SL(2,11):C_6$

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

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

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

Order 1320: $C_{132}:C_{10}$, $\SL(2,11)$

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

Order 660: $C_{66}:C_{10}$ x 2, $C_{11}:C_{60}$

Order 440: $C_{44}:C_{10}$

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

Order 330: $C_{33}:C_{10}$ x 2, $C_{11}:C_{30}$

Order 288: $C_{12}.S_4$, $D_{12}:D_6$

Order 264: $D_{132}$

Order 240: $\SL(2,5):C_2$, $D_5\times D_{12}$

Order 220: $C_2\times F_{11}$ x 2, $C_{11}:C_{20}$

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

Order 144: $C_6.D_{12}$ x 2, $C_3:D_{24}$ x 2, $C_3:\GL(2,3)$ x 2, $C_{12}.C_{12}$, $C_6:D_{12}$, $D_{12}:C_6$, $\SL(2,3):C_6$

Order 132: $D_{66}$ x 2, $C_{132}$

Order 120: $S_3\times D_{10}$ x 2, $C_5:D_{12}$ x 2, $\SL(2,5)$, $D_{60}$, $C_5\times D_{12}$, $D_5\times C_{12}$

Order 110: $F_{11}$ x 2, $C_{11}:C_{10}$

Order 96: $\GL(2,3):C_2$ x 3, $D_4:D_6$, $C_8:D_6$

Order 88: $D_{44}$

Order 80: $D_4\times D_5$

Order 72: $C_3:D_{12}$ x 3, $C_3:C_{24}$ x 2, $C_6:D_6$, $C_6\times C_{12}$, $C_6\wr C_2$, $C_3\times D_{12}$, $S_3\times C_{12}$, $C_3^2:Q_8$, $C_3\times \SL(2,3)$

Order 66: $D_{33}$ x 2, $C_{66}$

Order 60: $S_3\times D_5$ x 4, $D_{30}$ x 2, $S_3\times C_{10}$ x 2, $C_{60}$, $C_5:C_{12}$, $C_3\times D_{10}$

Order 55: $C_{11}:C_5$

Order 48: $\GL(2,3)$ x 6, $C_2\times D_{12}$ x 3, $\SL(2,3):C_2$ x 3, $D_{24}$ x 2, $C_{24}:C_2$ x 2, $Q_8:S_3$ x 2, $C_3:D_8$ x 2, $D_4:C_6$, $D_{12}:C_2$, $C_3\times \OD_{16}$, $C_3:\OD_{16}$

Order 44: $D_{22}$ x 2, $C_{44}$

Order 40: $C_5:D_4$ x 2, $C_2\times D_{10}$ x 2, $D_{20}$, $C_4\times D_5$, $C_5\times D_4$

Order 36: $C_6:S_3$ x 4, $C_3\times C_{12}$ x 2, $C_3:C_{12}$, $C_6^2$, $C_6\times S_3$

Order 33: $C_{33}$

Order 32: $D_8:C_2$

Order 30: $D_{15}$ x 2, $C_3\times D_5$ x 2, $C_5\times S_3$ x 2, $C_{30}$

Order 24: $D_{12}$ x 11, $C_2\times D_6$ x 4, $C_2\times C_{12}$ x 3, $\SL(2,3)$ x 3, $C_{24}$ x 2, $C_3:C_8$ x 2, $C_3:D_4$, $C_4\times S_3$, $C_3:Q_8$, $C_3\times Q_8$, $C_3\times D_4$

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

Order 20: $D_{10}$ x 7, $C_2\times C_{10}$ x 2, $C_{20}$, $C_5:C_4$

Order 18: $C_3\times C_6$ x 2, $C_3:S_3$ x 2, $C_3\times S_3$

Order 16: $\SD_{16}$ x 2, $D_8$ x 2, $\OD_{16}$, $D_4:C_2$, $C_2\times D_4$

Order 15: $C_{15}$

Order 12: $D_6$ x 17, $C_{12}$ x 6, $C_2\times C_6$ x 3, $C_3:C_4$

Order 11: $C_{11}$

Order 10: $D_5$ x 4, $C_{10}$ x 3

Order 9: $C_3^2$

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

Order 6: $S_3$ x 9, $C_6$ x 7

Order 5: $C_5$

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 15840: $\SL(2,11):D_6$

Order 7920: $\SL(2,11):C_6$, $\SL(2,11):S_3$

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

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

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

Order 1320: $C_{132}:C_{10}$, $\SL(2,11)$

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

Order 660: $C_{11}:C_{60}$, $C_{66}:C_{10}$

Order 440: $C_{44}:C_{10}$

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

Order 330: $C_{11}:C_{30}$, $C_{33}:C_{10}$

Order 288: $C_{12}.S_4$, $D_{12}:D_6$

Order 264: $D_{132}$

Order 240: $\SL(2,5):C_2$, $D_5\times D_{12}$

Order 220: $C_2\times F_{11}$, $C_{11}:C_{20}$

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

Order 144: $C_{12}.C_{12}$, $C_6.D_{12}$, $C_3:D_{24}$, $C_6:D_{12}$, $D_{12}:C_6$, $\SL(2,3):C_6$, $C_3:\GL(2,3)$

Order 132: $D_{66}$, $C_{132}$

Order 120: $\SL(2,5)$, $S_3\times D_{10}$, $D_{60}$, $C_5\times D_{12}$, $D_5\times C_{12}$, $C_5:D_{12}$

Order 110: $C_{11}:C_{10}$, $F_{11}$

Order 96: $\GL(2,3):C_2$ x 2, $D_4:D_6$, $C_8:D_6$

Order 88: $D_{44}$

Order 80: $D_4\times D_5$

Order 72: $C_3:D_{12}$ x 2, $C_6:D_6$, $C_6\times C_{12}$, $C_6\wr C_2$, $C_3\times D_{12}$, $S_3\times C_{12}$, $C_3^2:Q_8$, $C_3\times \SL(2,3)$, $C_3:C_{24}$

Order 66: $C_{66}$, $D_{33}$

Order 60: $S_3\times D_5$, $C_{60}$, $C_5:C_{12}$, $D_{30}$, $S_3\times C_{10}$, $C_3\times D_{10}$

Order 55: $C_{11}:C_5$

Order 48: $C_2\times D_{12}$ x 3, $\SL(2,3):C_2$ x 2, $\GL(2,3)$ x 2, $D_{24}$, $C_{24}:C_2$, $D_4:C_6$, $D_{12}:C_2$, $C_3\times \OD_{16}$, $Q_8:S_3$, $C_3:D_8$, $C_3:\OD_{16}$

Order 44: $D_{22}$, $C_{44}$

Order 40: $C_5:D_4$, $D_{20}$, $C_4\times D_5$, $C_2\times D_{10}$, $C_5\times D_4$

Order 36: $C_3\times C_{12}$ x 2, $C_6:S_3$ x 2, $C_3:C_{12}$, $C_6^2$, $C_6\times S_3$

Order 33: $C_{33}$

Order 32: $D_8:C_2$

Order 30: $C_{30}$, $D_{15}$, $C_3\times D_5$, $C_5\times S_3$

Order 24: $D_{12}$ x 7, $C_2\times C_{12}$ x 3, $C_2\times D_6$ x 3, $\SL(2,3)$ x 2, $C_3:D_4$, $C_4\times S_3$, $C_3:Q_8$, $C_{24}$, $C_3\times Q_8$, $C_3\times D_4$, $C_3:C_8$

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

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

Order 18: $C_3\times C_6$ x 2, $C_3:S_3$, $C_3\times S_3$

Order 16: $\SD_{16}$, $D_8$, $\OD_{16}$, $D_4:C_2$, $C_2\times D_4$

Order 15: $C_{15}$

Order 12: $D_6$ x 7, $C_{12}$ x 6, $C_2\times C_6$ x 3, $C_3:C_4$

Order 11: $C_{11}$

Order 10: $C_{10}$ x 2, $D_5$ x 2

Order 9: $C_3^2$

Order 8: $D_4$ x 3, $C_2^3$, $Q_8$, $C_2\times C_4$, $C_8$

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

Order 5: $C_5$

Order 4: $C_2^2$ x 3, $C_4$ x 2

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 15840: $\SL(2,11):D_6$ ($C_1$)

Order 7920: $\SL(2,11):S_3$ ($C_2$) x 2, $\SL(2,11):C_6$ ($C_2$)

Order 3960: $C_3\times \SL(2,11)$ ($C_2^2$)

Order 2640: $\SL(2,11):C_2$ ($S_3$)

Order 1320: $\SL(2,11)$ ($D_6$)

Order 12: $C_{12}$ ($\PGL(2,11)$)

Order 6: $C_6$ ($C_2\times \PGL(2,11)$)

Order 4: $C_4$ ($C_3:\PGL(2,11)$)

Order 3: $C_3$ ($\SL(2,11):C_2^2$)

Order 2: $C_2$ ($C_2\times C_3.\PSL(2,11).C_2$)

Order 1: $C_1$ ($\SL(2,11):D_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 15840: $\SL(2,11):D_6$ ($C_1$)

Order 7920: $\SL(2,11):C_6$ ($C_2$), $\SL(2,11):S_3$ ($C_2$)

Order 3960: $C_3\times \SL(2,11)$ ($C_2^2$)

Order 2640: $\SL(2,11):C_2$ ($S_3$)

Order 1320: $\SL(2,11)$ ($D_6$)

Order 12: $C_{12}$ ($\PGL(2,11)$)

Order 6: $C_6$ ($C_2\times \PGL(2,11)$)

Order 4: $C_4$ ($C_3:\PGL(2,11)$)

Order 3: $C_3$ ($\SL(2,11):C_2^2$)

Order 2: $C_2$ ($C_2\times C_3.\PSL(2,11).C_2$)

Order 1: $C_1$ ($\SL(2,11):D_6$)

Series

Derived series

$\SL(2,11):D_6$

$\rhd$

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

$\rhd$

$\SL(2,11)$

Chief series

$\SL(2,11):D_6$

$\rhd$

$\SL(2,11):C_6$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$\SL(2,11):D_6$

$\rhd$

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

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_4$

Supergroups

This group is a maximal subgroup of 4 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 $63 \times 63$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $41 \times 41$ rational character table.