Abstract group 73205.j: $C_{11}^3:(C_{11}:C_5)$

• Label: 73205.j
• Order: \( 5 \cdot 11^{4} \)
• Exponent: \( 5 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 5 \)
• $\card{Z(G)}$: \( 11 \)
• $\card{\Aut(G)}$: \( 2^{2} \cdot 5^{2} \cdot 11^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 5 \)
• Perm deg.: $121$
• Trans deg.: $121$
• Rank: $2$

Group information

Description:	$C_{11}^3:(C_{11}:C_5)$
Order:	\(73205\)\(\medspace = 5 \cdot 11^{4} \)
Exponent:	\(55\)\(\medspace = 5 \cdot 11 \)
Automorphism group:	$\He_{11}:C_{10}^2$, of order \(133100\)\(\medspace = 2^{2} \cdot 5^{2} \cdot 11^{3} \)
Composition factors:	$C_5$, $C_{11}$ x 4
Derived length:	$3$

This group is nonabelian and supersolvable (hence solvable and monomial).

Group statistics

Order	1	5	11	55
Elements	1	5324	14640	53240	73205
Conjugacy classes	1	4	56	40	101
Divisions	1	1	8	1	11
Autjugacy classes	1	4	7	4	16

Dimension	1	4	5	10	11	50	55	110	440	550
Irr. complex chars.	5	0	24	0	50	0	22	0	0	0	101
Irr. rational chars.	1	1	0	2	0	2	0	2	1	2	11

Minimal presentations

Permutation degree:	$121$
Transitive degree:	$121$
Rank:	$2$
Inequivalent generating pairs:	$31944$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	11	22	110
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e \mid b^{11}=c^{11}=d^{11}=e^{11}=[a,e]=[b,e]=[c,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $121$ $\langle(1,2,11,5,36,113,39,106,46,47,7)(3,20,48,73,29,105,117,114,121,69,24)(4,27,100,60,58,80,111,59,92,95,32) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 3 & 0 & 10 & 10 \\ 3 & 0 & 3 & 10 \\ 3 & 8 & 2 & 0 \\ 1 & 3 & 8 & 10 \end{array}\right), \left(\begin{array}{rrrr} 10 & 3 & 2 & 1 \\ 9 & 5 & 5 & 7 \\ 5 & 4 & 1 & 1 \\ 8 & 9 & 1 & 7 \end{array}\right), \left(\begin{array}{rrrr} 0 & 3 & 6 & 0 \\ 10 & 7 & 3 & 6 \\ 4 & 3 & 6 & 8 \\ 8 & 4 & 1 & 2 \end{array}\right), \left(\begin{array}{rrrr} 0 & 4 & 7 & 5 \\ 2 & 8 & 0 & 3 \\ 1 & 8 & 7 & 7 \\ 2 & 2 & 9 & 0 \end{array}\right), \left(\begin{array}{rrrr} 6 & 9 & 1 & 4 \\ 4 & 6 & 3 & 1 \\ 8 & 10 & 7 & 2 \\ 2 & 8 & 7 & 7 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{11})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_{11}^3$ $\,\rtimes\,$ $(C_{11}:C_5)$	$\He_{11}$ $\,\rtimes\,$ $(C_{11}:C_5)$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{11}$ . $(\He_{11}:C_5)$	$C_{11}^2$ . $(C_{11}^2:C_5)$			more information
Aut. group:	$\Aut(C_4\wr C_5.C_2^3)$

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

Homology

Abelianization:	$C_{5} $
Schur multiplier:	$C_1$
Commutator length:	$2$

Subgroups

There are 5154 subgroups in 34 conjugacy classes, 7 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{11}$	$G/Z \simeq$ $\He_{11}:C_5$
Commutator:	$G' \simeq$ $C_{11}^3.C_{11}$	$G/G' \simeq$ $C_5$
Frattini:	$\Phi \simeq$ $C_{11}^2$	$G/\Phi \simeq$ $C_{11}^2:C_5$
Fitting:	$\operatorname{Fit} \simeq$ $C_{11}^3.C_{11}$	$G/\operatorname{Fit} \simeq$ $C_5$
Radical:	$R \simeq$ $C_{11}^3:(C_{11}:C_5)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{11}$	$G/\operatorname{soc} \simeq$ $\He_{11}:C_5$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}^3.C_{11}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 73205: $C_{11}^3:(C_{11}:C_5)$

Order 14641: $C_{11}^3.C_{11}$

Order 6655: $\He_{11}:C_5$, $C_{11}^3:C_5$

Order 1331: $\He_{11}$ x 3, $C_{11}^3$

Order 605: $C_{11}:C_{55}$ x 3, $C_{11}^2:C_5$

Order 121: $C_{11}^2$ x 8

Order 55: $C_{11}:C_5$ x 3, $C_{55}$

Order 11: $C_{11}$ x 8

Order 5: $C_5$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 73205: $C_{11}^3:(C_{11}:C_5)$

Order 14641: $C_{11}^3.C_{11}$

Order 6655: $\He_{11}:C_5$, $C_{11}^3:C_5$

Order 1331: $\He_{11}$ x 2, $C_{11}^3$

Order 605: $C_{11}:C_{55}$ x 3, $C_{11}^2:C_5$

Order 121: $C_{11}^2$ x 7

Order 55: $C_{11}:C_5$ x 3, $C_{55}$

Order 11: $C_{11}$ x 7

Order 5: $C_5$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 73205: $C_{11}^3:(C_{11}:C_5)$ ($C_1$)

Order 14641: $C_{11}^3.C_{11}$ ($C_5$)

Order 1331: $C_{11}^3$ ($C_{11}:C_5$), $\He_{11}$ ($C_{11}:C_5$)

Order 121: $C_{11}^2$ ($C_{11}^2:C_5$)

Order 11: $C_{11}$ ($\He_{11}:C_5$)

Order 1: $C_1$ ($C_{11}^3:(C_{11}:C_5)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 73205: $C_{11}^3:(C_{11}:C_5)$ ($C_1$)

Order 14641: $C_{11}^3.C_{11}$ ($C_5$)

Order 1331: $C_{11}^3$ ($C_{11}:C_5$), $\He_{11}$ ($C_{11}:C_5$)

Order 121: $C_{11}^2$ ($C_{11}^2:C_5$)

Order 11: $C_{11}$ ($\He_{11}:C_5$)

Order 1: $C_1$ ($C_{11}^3:(C_{11}:C_5)$)

Series

Derived series

$C_{11}^3:(C_{11}:C_5)$

$\rhd$

$C_{11}^3.C_{11}$

$\rhd$

$C_{11}^2$

$\rhd$

$C_1$

Chief series

$C_{11}^3:(C_{11}:C_5)$

$\rhd$

$C_{11}^3.C_{11}$

$\rhd$

$C_{11}^3$

$\rhd$

$\He_{11}$

$\rhd$

$C_{11}^2$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

Lower central series

$C_{11}^3:(C_{11}:C_5)$

$\rhd$

$C_{11}^3.C_{11}$

Upper central series

$C_1$

$\lhd$

$C_{11}$

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 $101 \times 101$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	5A	11A	11B	11C	11D	11E	11F	11G	11H	55A
	Size	1	5324	10	110	110	550	550	1210	6050	6050	53240
	5 P	1A	5A	11A	11B	11C	11D	11E	11F	11G	11H	55A
	11 P	1A	1A	11A	11B	11C	11D	11E	11F	11G	11H	11A
73205.j.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
73205.j.1b		$4$	$-1$	$4$	$4$	$4$	$4$	$4$	$4$	$4$	$4$	$-1$
73205.j.5a		$10$	$0$	$10$	$-1$	$-1$	$10$	$10$	$-1$	$-1$	$-1$	$0$
73205.j.5b		$10$	$0$	$10$	$10$	$10$	$10$	$10$	$-1$	$-1$	$-1$	$0$
73205.j.5c		$50$	$0$	$50$	$-5$	$-5$	$50$	$50$	$-5$	$6$	$6$	$0$
73205.j.5d		$50$	$0$	$50$	$-5$	$-5$	$50$	$50$	$6$	$-5$	$-5$	$0$
73205.j.11a		$110$	$10$	$-11$	$11$	$-11$	$0$	$0$	$0$	$0$	$0$	$-1$
73205.j.11b		$440$	$-10$	$-44$	$44$	$-44$	$0$	$0$	$0$	$0$	$0$	$1$
73205.j.55a		$110$	$0$	$110$	$0$	$0$	$-11$	$-11$	$0$	$0$	$0$	$0$
73205.j.55b		$550$	$0$	$-55$	$-11$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
73205.j.55c		$550$	$0$	$-55$	$0$	$11$	$0$	$0$	$0$	$0$	$0$	$0$