Abstract group 63888.n: $C_{11}^3:(C_2\times D_{12})$

• Label: 63888.n
• Order: \( 2^{4} \cdot 3 \cdot 11^{3} \)
• Exponent: \( 2^{2} \cdot 3 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3 \cdot 5^{2} \cdot 11^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 5^{2} \)
• Perm deg.: $48$
• Trans deg.: not computed
• Rank: $3$

Group information

Description:	$C_{11}^3:(C_2\times D_{12})$
Order:	\(63888\)\(\medspace = 2^{4} \cdot 3 \cdot 11^{3} \)
Exponent:	\(132\)\(\medspace = 2^{2} \cdot 3 \cdot 11 \)
Automorphism group:	$C_{11}^3.C_6.C_{10}^2.C_2^4$, of order \(12777600\)\(\medspace = 2^{7} \cdot 3 \cdot 5^{2} \cdot 11^{3} \)
Composition factors:	$C_2$ x 4, $C_3$, $C_{11}$ x 3
Derived length:	$3$

This group is nonabelian and solvable. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	6	11	12	22	33	44	66	132
Elements	1	4247	242	264	5566	1330	5808	31690	2420	5060	2420	4840	63888
Conjugacy classes	1	7	1	2	3	95	4	225	5	25	5	10	383
Divisions	1	7	1	2	3	19	2	45	1	4	1	1	87

Minimal presentations

Permutation degree:	$48$
Transitive degree:	not computed
Rank:	$3$
Inequivalent generating triples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e \mid b^{22}=c^{12}=d^{11}=e^{11}=[b,e]=[d,e]=1, a^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $48$ $\langle(1,2,7,9,5,8,15,23,16,17,29,19,13,6,4,14,21,18,12,25,3,10)(11,24,22,33,27,28,20,30,31,32,26) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 5 & 9 & 0 & 4 \\ 4 & 7 & 3 & 0 \\ 4 & 2 & 6 & 10 \\ 7 & 10 & 2 & 4 \end{array}\right), \left(\begin{array}{rrrr} 10 & 0 & 0 & 0 \\ 0 & 10 & 0 & 0 \\ 0 & 0 & 10 & 0 \\ 0 & 0 & 0 & 10 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 4 & 5 & 1 & 0 \\ 6 & 6 & 8 & 0 \\ 6 & 6 & 7 & 1 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 9 & 10 & 5 & 0 \\ 0 & 0 & 1 & 0 \\ 2 & 0 & 2 & 10 \end{array}\right), \left(\begin{array}{rrrr} 4 & 10 & 1 & 7 \\ 6 & 8 & 9 & 1 \\ 8 & 9 & 5 & 1 \\ 0 & 8 & 5 & 9 \end{array}\right), \left(\begin{array}{rrrr} 6 & 10 & 10 & 0 \\ 7 & 4 & 2 & 7 \\ 1 & 5 & 6 & 2 \\ 1 & 2 & 6 & 6 \end{array}\right), \left(\begin{array}{rrrr} 6 & 7 & 3 & 3 \\ 8 & 7 & 2 & 7 \\ 1 & 7 & 9 & 10 \\ 2 & 7 & 4 & 0 \end{array}\right), \left(\begin{array}{rrrr} 0 & 2 & 5 & 1 \\ 2 & 6 & 5 & 5 \\ 10 & 8 & 3 & 9 \\ 0 & 10 & 9 & 9 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{11})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not computed
Trans. wreath product:	not computed
Possibly split product:	$(C_{11}^3:C_6)$ . $D_4$	$(C_{22}.D_{11}^2)$ . $S_3$	$(C_{11}^2:D_{22})$ . $D_6$	$(C_{11}^2:D_6)$ . $D_{22}$	all 30

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	not computed
Commutator length:	$1$

Subgroups

There are 41 normal subgroups (27 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_{11}^3:(C_2^2\times S_3)$
Commutator:	$G' \simeq$ $C_{11}^2:C_{66}$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_{11}^3:(C_2^2\times S_3)$
Fitting:	$\operatorname{Fit} \simeq$ $C_{11}^2\times C_{22}$	$G/\operatorname{Fit} \simeq$ $C_2\times D_6$
Radical:	$R \simeq$ $C_{11}^3:(C_2\times D_{12})$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{11}^2\times C_{22}$	$G/\operatorname{soc} \simeq$ $C_2\times D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}^3$

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 1 (order at least 63888) are shown, as well as all normal subgroups of any index.

Order 63888: $C_{11}^3:(C_2\times D_{12})$

Order 31944: $C_{11}^3:(C_2\times D_6)$ x 2, $C_{11}^3:D_{12}$ x 2, $C_{11}^2:(C_{12}\times D_{11})$, $C_{11}^2:D_{132}$, $C_{11}^3:D_{12}$

Order 21296: $D_{22}:D_{11}^2$

Order 15972: $C_{11}^2:D_{66}$ x 2, $C_{11}^3:D_6$ x 2, $C_{11}^3:C_{12}$, $C_2\times C_{11}^3:C_6$, $C_{11}^2:C_{132}$

Order 10648: $C_{22}.D_{11}^2$

Order 7986: $C_{11}^3:C_6$ x 2, $C_{11}^2:C_{66}$

Order 5808: $(C_{11}\times C_{44}):D_6$

Order 5324: $C_{11}^2:D_{22}$, $C_{11}^2:C_{44}$, $C_{11}^2:C_{44}$

Order 3993: $C_{11}\wr C_3$

Order 2904: $C_{11}^2:D_{12}$

Order 2662: $C_{11}^3:C_2$ x 2, $C_{11}^2\times C_{22}$

Order 1452: $C_{11}^2:D_6$ x 2, $C_{11}^2:C_{12}$

Order 1331: $C_{11}^3$

Order 726: $C_{11}^2:C_6$

Order 528: $D_{11}\times D_{12}$

Order 484: $C_{11}^2:C_4$

Order 363: $C_{11}^2:C_3$

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

Order 121: $C_{11}^2$

Order 44: $C_{11}:C_4$

Order 22: $C_{22}$

Order 16: $C_2\times D_4$

Order 11: $C_{11}$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 1 (order at least 63888) are shown, as well as all normal subgroups of any index.

Order 63888: $C_{11}^3:(C_2\times D_{12})$

Order 31944: $C_{11}^3:(C_2\times D_6)$, $C_{11}^2:(C_{12}\times D_{11})$, $C_{11}^3:D_{12}$, $C_{11}^2:D_{132}$, $C_{11}^3:D_{12}$

Order 21296: $D_{22}:D_{11}^2$

Order 15972: $C_{11}^3:C_{12}$, $C_2\times C_{11}^3:C_6$, $C_{11}^2:D_{66}$, $C_{11}^3:D_6$, $C_{11}^2:C_{132}$

Order 10648: $C_{22}.D_{11}^2$

Order 7986: $C_{11}^3:C_6$, $C_{11}^2:C_{66}$

Order 5808: $(C_{11}\times C_{44}):D_6$

Order 5324: $C_{11}^2:D_{22}$, $C_{11}^2:C_{44}$, $C_{11}^2:C_{44}$

Order 3993: $C_{11}\wr C_3$

Order 2904: $C_{11}^2:D_{12}$

Order 2662: $C_{11}^2\times C_{22}$, $C_{11}^3:C_2$

Order 1452: $C_{11}^2:D_6$, $C_{11}^2:C_{12}$

Order 1331: $C_{11}^3$

Order 726: $C_{11}^2:C_6$

Order 528: $D_{11}\times D_{12}$

Order 484: $C_{11}^2:C_4$

Order 363: $C_{11}^2:C_3$

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

Order 121: $C_{11}^2$

Order 44: $C_{11}:C_4$

Order 22: $C_{22}$

Order 16: $C_2\times D_4$

Order 11: $C_{11}$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 63888: $C_{11}^3:(C_2\times D_{12})$ ($C_1$)

Order 31944: $C_{11}^3:(C_2\times D_6)$ ($C_2$) x 2, $C_{11}^3:D_{12}$ ($C_2$) x 2, $C_{11}^2:(C_{12}\times D_{11})$ ($C_2$), $C_{11}^2:D_{132}$ ($C_2$), $C_{11}^3:D_{12}$ ($C_2$)

Order 15972: $C_{11}^2:D_{66}$ ($C_2^2$) x 2, $C_{11}^3:D_6$ ($C_2^2$) x 2, $C_{11}^3:C_{12}$ ($C_2^2$), $C_2\times C_{11}^3:C_6$ ($C_2^2$), $C_{11}^2:C_{132}$ ($C_2^2$)

Order 10648: $C_{22}.D_{11}^2$ ($S_3$)

Order 7986: $C_{11}^3:C_6$ ($D_4$) x 2, $C_{11}^2:C_{66}$ ($C_2^3$)

Order 5324: $C_{11}^2:D_{22}$ ($D_6$), $C_{11}^2:C_{44}$ ($D_6$), $C_{11}^2:C_{44}$ ($D_6$)

Order 3993: $C_{11}\wr C_3$ ($C_2\times D_4$)

Order 2904: $C_{11}^2:D_{12}$ ($D_{11}$)

Order 2662: $C_{11}^3:C_2$ ($D_{12}$) x 2, $C_{11}^2\times C_{22}$ ($C_2\times D_6$)

Order 1452: $C_{11}^2:D_6$ ($D_{22}$) x 2, $C_{11}^2:C_{12}$ ($D_{22}$)

Order 1331: $C_{11}^3$ ($C_2\times D_{12}$)

Order 726: $C_{11}^2:C_6$ ($C_2\times D_{22}$)

Order 484: $C_{11}^2:C_4$ ($S_3\times D_{11}$)

Order 363: $C_{11}^2:C_3$ ($D_4\times D_{11}$)

Order 242: $C_{11}\times C_{22}$ ($S_3\times D_{22}$)

Order 121: $C_{11}^2$ ($D_{11}\times D_{12}$)

Order 44: $C_{11}:C_4$ ($C_{11}^2:D_6$)

Order 22: $C_{22}$ ($C_2\times C_{11}^2:D_6$)

Order 11: $C_{11}$ ($(C_{11}\times C_{44}):D_6$)

Order 2: $C_2$ ($C_{11}^3:(C_2^2\times S_3)$)

Order 1: $C_1$ ($C_{11}^3:(C_2\times D_{12})$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 63888: $C_{11}^3:(C_2\times D_{12})$ ($C_1$)

Order 31944: $C_{11}^3:(C_2\times D_6)$ ($C_2$), $C_{11}^2:(C_{12}\times D_{11})$ ($C_2$), $C_{11}^3:D_{12}$ ($C_2$), $C_{11}^2:D_{132}$ ($C_2$), $C_{11}^3:D_{12}$ ($C_2$)

Order 15972: $C_{11}^3:C_{12}$ ($C_2^2$), $C_2\times C_{11}^3:C_6$ ($C_2^2$), $C_{11}^2:D_{66}$ ($C_2^2$), $C_{11}^3:D_6$ ($C_2^2$), $C_{11}^2:C_{132}$ ($C_2^2$)

Order 10648: $C_{22}.D_{11}^2$ ($S_3$)

Order 7986: $C_{11}^3:C_6$ ($D_4$), $C_{11}^2:C_{66}$ ($C_2^3$)

Order 5324: $C_{11}^2:D_{22}$ ($D_6$), $C_{11}^2:C_{44}$ ($D_6$), $C_{11}^2:C_{44}$ ($D_6$)

Order 3993: $C_{11}\wr C_3$ ($C_2\times D_4$)

Order 2904: $C_{11}^2:D_{12}$ ($D_{11}$)

Order 2662: $C_{11}^2\times C_{22}$ ($C_2\times D_6$), $C_{11}^3:C_2$ ($D_{12}$)

Order 1452: $C_{11}^2:D_6$ ($D_{22}$), $C_{11}^2:C_{12}$ ($D_{22}$)

Order 1331: $C_{11}^3$ ($C_2\times D_{12}$)

Order 726: $C_{11}^2:C_6$ ($C_2\times D_{22}$)

Order 484: $C_{11}^2:C_4$ ($S_3\times D_{11}$)

Order 363: $C_{11}^2:C_3$ ($D_4\times D_{11}$)

Order 242: $C_{11}\times C_{22}$ ($S_3\times D_{22}$)

Order 121: $C_{11}^2$ ($D_{11}\times D_{12}$)

Order 44: $C_{11}:C_4$ ($C_{11}^2:D_6$)

Order 22: $C_{22}$ ($C_2\times C_{11}^2:D_6$)

Order 11: $C_{11}$ ($(C_{11}\times C_{44}):D_6$)

Order 2: $C_2$ ($C_{11}^3:(C_2^2\times S_3)$)

Order 1: $C_1$ ($C_{11}^3:(C_2\times D_{12})$)

Series

Derived series

$C_{11}^3:(C_2\times D_{12})$

$\rhd$

$C_{11}^2:C_{66}$

$\rhd$

$C_{11}^2$

$\rhd$

$C_1$

Chief series

$C_{11}^3:(C_2\times D_{12})$

$\rhd$

$C_{11}^3:(C_2\times D_6)$

$\rhd$

$C_2\times C_{11}^3:C_6$

$\rhd$

$C_{11}^2:C_{66}$

$\rhd$

$C_{11}\wr C_3$

$\rhd$

$C_{11}^3$

$\rhd$

$C_{11}^2$

$\rhd$

$C_1$

Lower central series

$C_{11}^3:(C_2\times D_{12})$

$\rhd$

$C_{11}^2:C_{66}$

$\rhd$

$C_{11}\wr C_3$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

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

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

Character theory

Complex character table

The $383 \times 383$ character table is not available for this group.

Rational character table

The $87 \times 87$ rational character table is not available for this group.