Abstract group 2420.r: $C_{22}:F_{11}$

• Label: 2420.r
• Order: \( 2^{2} \cdot 5 \cdot 11^{2} \)
• Exponent: \( 2 \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 5 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 5^{2} \cdot 11^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 5 \)
• Perm deg.: $24$
• Trans deg.: $110$
• Rank: $2$

Group information

Description:	$C_{22}:F_{11}$
Order:	\(2420\)\(\medspace = 2^{2} \cdot 5 \cdot 11^{2} \)
Exponent:	\(110\)\(\medspace = 2 \cdot 5 \cdot 11 \)
Automorphism group:	$F_{11}^2:C_2^2$, of order \(48400\)\(\medspace = 2^{4} \cdot 5^{2} \cdot 11^{2} \)
Composition factors:	$C_2$ x 2, $C_5$, $C_{11}$ x 2
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.

Group statistics

Order	1	2	5	10	11	22
Elements	1	243	484	1452	120	120	2420
Conjugacy classes	1	3	4	12	12	12	44
Divisions	1	3	1	3	4	4	16
Autjugacy classes	1	2	2	4	2	2	13

Dimension	1	4	10	50
Irr. complex chars.	20	0	24	0	44
Irr. rational chars.	4	4	4	4	16

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$110$
Rank:	$2$
Inequivalent generating pairs:	$36$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	10	10	50
Arbitrary	10	10	20

Constructions

Presentation:	$\langle a, b, c \mid a^{10}=b^{11}=c^{22}=[b,c]=1, b^{a}=b^{7}, c^{a}=c^{19} \rangle$
Permutation group:	Degree $24$ $\langle(12,13), (2,11,4,8,7)(3,10,6,5,9)(15,21,17,22,19)(16,23,18,24,20), (14,24,15,20,18,23,21,17,19,16,22) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 6 & 1 & 8 & 9 \\ 10 & 6 & 2 & 8 \\ 9 & 6 & 3 & 10 \\ 5 & 9 & 1 & 3 \end{array}\right), \left(\begin{array}{rrrr} 8 & 6 & 8 & 10 \\ 9 & 8 & 4 & 8 \\ 10 & 3 & 3 & 5 \\ 10 & 10 & 2 & 3 \end{array}\right), \left(\begin{array}{rrrr} 0 & 6 & 3 & 7 \\ 1 & 1 & 9 & 3 \\ 2 & 5 & 3 & 5 \\ 6 & 2 & 10 & 4 \end{array}\right), \left(\begin{array}{rrrr} 6 & 9 & 1 & 4 \\ 4 & 6 & 3 & 1 \\ 8 & 10 & 7 & 2 \\ 2 & 8 & 7 & 7 \end{array}\right), \left(\begin{array}{rrrr} 10 & 0 & 0 & 0 \\ 0 & 10 & 0 & 0 \\ 0 & 0 & 10 & 0 \\ 0 & 0 & 0 & 10 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{11})$
Direct product:	$C_2$ $\, \times\, $ $(C_{11}:F_{11})$
Semidirect product:	$C_{22}$ $\,\rtimes\,$ $F_{11}$ (2)	$(C_{11}:D_{22})$ $\,\rtimes\,$ $C_5$	$C_{11}$ $\,\rtimes\,$ $(C_2\times F_{11})$ (2)	$(C_{11}:D_{11})$ $\,\rtimes\,$ $C_{10}$ (2)	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Aut. group:	$\Aut(C_{4383}.C_9)$

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{2} \times C_{10} \simeq C_{2}^{2} \times C_{5}$
Schur multiplier:	$C_{22}$
Commutator length:	$1$

Subgroups

There are 1510 subgroups in 50 conjugacy classes, 16 normal (8 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_{11}:F_{11}$
Commutator:	$G' \simeq$ $C_{11}^2$	$G/G' \simeq$ $C_2\times C_{10}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_{22}:F_{11}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{11}\times C_{22}$	$G/\operatorname{Fit} \simeq$ $C_{10}$
Radical:	$R \simeq$ $C_{22}:F_{11}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{11}\times C_{22}$	$G/\operatorname{soc} \simeq$ $C_{10}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Subgroup information

Click on a subgroup in the diagram to see information about it.

Classes of subgroups up to conjugation

Order 2420: $C_{22}:F_{11}$

Order 1210: $C_{11}:F_{11}$ x 2, $C_{11}^2:C_{10}$

Order 605: $C_{11}^2:C_5$

Order 484: $C_{11}:D_{22}$

Order 242: $C_{11}:D_{11}$ x 2, $C_{11}\times C_{22}$

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

Order 121: $C_{11}^2$

Order 110: $F_{11}$ x 4, $C_{11}:C_{10}$ x 2

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

Order 44: $D_{22}$ x 4

Order 22: $D_{11}$ x 8, $C_{22}$ x 4

Order 20: $C_2\times C_{10}$

Order 11: $C_{11}$ x 4

Order 10: $C_{10}$ x 3

Order 5: $C_5$

Order 4: $C_2^2$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 2420: $C_{22}:F_{11}$

Order 1210: $C_{11}^2:C_{10}$, $C_{11}:F_{11}$

Order 605: $C_{11}^2:C_5$

Order 484: $C_{11}:D_{22}$

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

Order 220: $C_2\times F_{11}$

Order 121: $C_{11}^2$

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

Order 55: $C_{11}:C_5$

Order 44: $D_{22}$ x 2

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

Order 20: $C_2\times C_{10}$

Order 11: $C_{11}$ x 2

Order 10: $C_{10}$ x 2

Order 5: $C_5$

Order 4: $C_2^2$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 2420: $C_{22}:F_{11}$ ($C_1$)

Order 1210: $C_{11}:F_{11}$ ($C_2$) x 2, $C_{11}^2:C_{10}$ ($C_2$)

Order 605: $C_{11}^2:C_5$ ($C_2^2$)

Order 484: $C_{11}:D_{22}$ ($C_5$)

Order 242: $C_{11}:D_{11}$ ($C_{10}$) x 2, $C_{11}\times C_{22}$ ($C_{10}$)

Order 121: $C_{11}^2$ ($C_2\times C_{10}$)

Order 22: $C_{22}$ ($F_{11}$) x 2

Order 11: $C_{11}$ ($C_2\times F_{11}$) x 2

Order 2: $C_2$ ($C_{11}:F_{11}$)

Order 1: $C_1$ ($C_{22}:F_{11}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2420: $C_{22}:F_{11}$ ($C_1$)

Order 1210: $C_{11}^2:C_{10}$ ($C_2$), $C_{11}:F_{11}$ ($C_2$)

Order 605: $C_{11}^2:C_5$ ($C_2^2$)

Order 484: $C_{11}:D_{22}$ ($C_5$)

Order 242: $C_{11}\times C_{22}$ ($C_{10}$), $C_{11}:D_{11}$ ($C_{10}$)

Order 121: $C_{11}^2$ ($C_2\times C_{10}$)

Order 22: $C_{22}$ ($F_{11}$)

Order 11: $C_{11}$ ($C_2\times F_{11}$)

Order 2: $C_2$ ($C_{11}:F_{11}$)

Order 1: $C_1$ ($C_{22}:F_{11}$)

Series

Derived series

$C_{22}:F_{11}$

$\rhd$

$C_{11}^2$

$\rhd$

$C_1$

Chief series

$C_{22}:F_{11}$

$\rhd$

$C_{11}^2:C_{10}$

$\rhd$

$C_{11}\times C_{22}$

$\rhd$

$C_{22}$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

Lower central series

$C_{22}:F_{11}$

$\rhd$

$C_{11}^2$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 13 larger groups in the database.

This group is a maximal quotient of 7 larger groups in the database.

Character theory

Complex character table

See the $44 \times 44$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	2B	2C	5A	10A	10B	10C	11A	11B	11C	11D	22A	22B	22C	22D
	Size	1	1	121	121	484	484	484	484	10	10	50	50	10	10	50	50
	2 P	1A	1A	1A	1A	5A	5A	5A	5A	11A	11B	11C	11D	11A	11B	11C	11D
	5 P	1A	2A	2B	2C	1A	2B	2A	2C	11A	11B	11C	11D	22A	22B	22C	22D
	11 P	1A	2A	2B	2C	5A	10A	10B	10C	1A	1A	1A	1A	2A	2A	2A	2A
2420.r.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
2420.r.1b		$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
2420.r.1c		$1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
2420.r.1d		$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
2420.r.1e		$4$	$4$	$4$	$4$	$-1$	$-1$	$-1$	$-1$	$4$	$4$	$4$	$4$	$4$	$4$	$4$	$4$
2420.r.1f		$4$	$-4$	$-4$	$4$	$-1$	$1$	$-1$	$1$	$4$	$4$	$4$	$4$	$-4$	$-4$	$-4$	$-4$
2420.r.1g		$4$	$-4$	$4$	$-4$	$-1$	$1$	$1$	$1$	$4$	$4$	$4$	$4$	$-4$	$-4$	$-4$	$-4$
2420.r.1h		$4$	$4$	$-4$	$-4$	$-1$	$-1$	$1$	$-1$	$4$	$4$	$4$	$4$	$4$	$4$	$4$	$4$
2420.r.10a		$10$	$10$	$0$	$0$	$0$	$0$	$0$	$0$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$
2420.r.10b		$10$	$10$	$0$	$0$	$0$	$0$	$0$	$0$	$-1$	$-1$	$10$	$-1$	$-1$	$-1$	$10$	$-1$
2420.r.10c		$10$	$-10$	$0$	$0$	$0$	$0$	$0$	$0$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$1$
2420.r.10d		$10$	$-10$	$0$	$0$	$0$	$0$	$0$	$0$	$-1$	$-1$	$10$	$-1$	$1$	$1$	$-10$	$1$
2420.r.10e		$50$	$50$	$0$	$0$	$0$	$0$	$0$	$0$	$-5$	$6$	$-5$	$6$	$-5$	$6$	$-5$	$6$
2420.r.10f		$50$	$50$	$0$	$0$	$0$	$0$	$0$	$0$	$6$	$-5$	$-5$	$-5$	$6$	$-5$	$-5$	$-5$
2420.r.10g		$50$	$-50$	$0$	$0$	$0$	$0$	$0$	$0$	$-5$	$6$	$-5$	$6$	$5$	$-6$	$5$	$-6$
2420.r.10h		$50$	$-50$	$0$	$0$	$0$	$0$	$0$	$0$	$6$	$-5$	$-5$	$-5$	$-6$	$5$	$5$	$5$