Abstract group 3630.f: $C_{11}^2:C_{30}$

• Label: 3630.f
• Order: \( 2 \cdot 3 \cdot 5 \cdot 11^{2} \)
• Exponent: \( 2 \cdot 3 \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3 \cdot 5 \)
• $\card{Z(G)}$: \( 2 \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3 \cdot 5 \cdot 11^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 5 \)
• Perm deg.: $40$
• Trans deg.: $330$
• Rank: $2$

Group information

Description:	$C_{11}^2:C_{30}$
Order:	\(3630\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11^{2} \)
Exponent:	\(330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11 \)
Automorphism group:	$C_{11}^2.C_{60}.C_4.C_2^2$, of order \(116160\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \cdot 11^{2} \)
Composition factors:	$C_2$, $C_3$, $C_5$, $C_{11}$ x 2
Derived length:	$2$

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

Group statistics

Order	1	2	3	5	6	10	11	15	22	30	55	110
Elements	1	1	242	4	242	4	120	968	120	968	480	480	3630
Conjugacy classes	1	1	2	4	2	4	40	8	40	8	160	160	430
Divisions	1	1	1	1	1	1	4	1	4	1	4	4	24
Autjugacy classes	1	1	1	1	1	1	1	1	1	1	1	1	12

Dimension	1	2	3	4	8	30	120
Irr. complex chars.	30	0	400	0	0	0	0	430
Irr. rational chars.	2	2	0	2	2	8	8	24

Minimal presentations

Permutation degree:	$40$
Transitive degree:	$330$
Rank:	$2$
Inequivalent generating pairs:	$72$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	3	6	120
Arbitrary	3	6	34

Constructions

Presentation:	$\langle a, b, c \mid a^{30}=b^{11}=c^{11}=[b,c]=1, b^{a}=b^{9}c^{4}, c^{a}=b^{2}c \rangle$
Permutation group:	Degree $40$ $\langle(1,2,5,13,4,7,15,28,12,18,29)(3,8,17,27,6,11,25,24,14,10,20)(9,21,19,32,31,22,33,23,16,30,26) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 3 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 3 \end{array}\right), \left(\begin{array}{rrrr} 0 & 8 & 9 & 4 \\ 5 & 10 & 0 & 9 \\ 8 & 0 & 4 & 3 \\ 1 & 8 & 6 & 3 \end{array}\right), \left(\begin{array}{rrrr} 0 & 1 & 6 & 7 \\ 1 & 9 & 6 & 6 \\ 2 & 6 & 6 & 10 \\ 10 & 2 & 10 & 4 \end{array}\right), \left(\begin{array}{rrrr} 7 & 1 & 10 & 4 \\ 3 & 1 & 7 & 10 \\ 0 & 10 & 8 & 10 \\ 8 & 0 & 8 & 2 \end{array}\right), \left(\begin{array}{rrrr} 8 & 0 & 0 & 0 \\ 0 & 8 & 0 & 0 \\ 0 & 0 & 8 & 0 \\ 0 & 0 & 0 & 8 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{11})$
Direct product:	$C_2$ $\, \times\, $ $C_5$ $\, \times\, $ $(C_{11}^2:C_3)$
Semidirect product:	$C_{11}^2$ $\,\rtimes\,$ $C_{30}$	$(C_{11}\times C_{55})$ $\,\rtimes\,$ $C_6$	$(C_{11}\times C_{110})$ $\,\rtimes\,$ $C_3$	$(C_{11}\times C_{22})$ $\,\rtimes\,$ $C_{15}$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

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

Homology

Abelianization:	$C_{30} \simeq C_{2} \times C_{3} \times C_{5}$
Schur multiplier:	$C_{11}$
Commutator length:	$1$

Subgroups

There are 544 subgroups in 32 conjugacy classes, 12 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{10}$	$G/Z \simeq$ $C_{11}^2:C_3$
Commutator:	$G' \simeq$ $C_{11}^2$	$G/G' \simeq$ $C_{30}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_{11}^2:C_{30}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{11}\times C_{110}$	$G/\operatorname{Fit} \simeq$ $C_3$
Radical:	$R \simeq$ $C_{11}^2:C_{30}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{11}\times C_{110}$	$G/\operatorname{soc} \simeq$ $C_3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
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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Classes of subgroups up to conjugation

Order 3630: $C_{11}^2:C_{30}$

Order 1815: $C_{11}^2:C_{15}$

Order 1210: $C_{11}\times C_{110}$

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

Order 605: $C_{11}\times C_{55}$

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

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

Order 121: $C_{11}^2$

Order 110: $C_{110}$ x 4

Order 55: $C_{55}$ x 4

Order 30: $C_{30}$

Order 22: $C_{22}$ x 4

Order 15: $C_{15}$

Order 11: $C_{11}$ x 4

Order 10: $C_{10}$

Order 6: $C_6$

Order 5: $C_5$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 3630: $C_{11}^2:C_{30}$

Order 1815: $C_{11}^2:C_{15}$

Order 1210: $C_{11}\times C_{110}$

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

Order 605: $C_{11}\times C_{55}$

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

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

Order 121: $C_{11}^2$

Order 110: $C_{110}$

Order 55: $C_{55}$

Order 30: $C_{30}$

Order 22: $C_{22}$

Order 15: $C_{15}$

Order 11: $C_{11}$

Order 10: $C_{10}$

Order 6: $C_6$

Order 5: $C_5$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 3630: $C_{11}^2:C_{30}$ ($C_1$)

Order 1815: $C_{11}^2:C_{15}$ ($C_2$)

Order 1210: $C_{11}\times C_{110}$ ($C_3$)

Order 726: $C_{11}^2:C_6$ ($C_5$)

Order 605: $C_{11}\times C_{55}$ ($C_6$)

Order 363: $C_{11}^2:C_3$ ($C_{10}$)

Order 242: $C_{11}\times C_{22}$ ($C_{15}$)

Order 121: $C_{11}^2$ ($C_{30}$)

Order 10: $C_{10}$ ($C_{11}^2:C_3$)

Order 5: $C_5$ ($C_{11}^2:C_6$)

Order 2: $C_2$ ($C_{11}^2:C_{15}$)

Order 1: $C_1$ ($C_{11}^2:C_{30}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 3630: $C_{11}^2:C_{30}$ ($C_1$)

Order 1815: $C_{11}^2:C_{15}$ ($C_2$)

Order 1210: $C_{11}\times C_{110}$ ($C_3$)

Order 726: $C_{11}^2:C_6$ ($C_5$)

Order 605: $C_{11}\times C_{55}$ ($C_6$)

Order 363: $C_{11}^2:C_3$ ($C_{10}$)

Order 242: $C_{11}\times C_{22}$ ($C_{15}$)

Order 121: $C_{11}^2$ ($C_{30}$)

Order 10: $C_{10}$ ($C_{11}^2:C_3$)

Order 5: $C_5$ ($C_{11}^2:C_6$)

Order 2: $C_2$ ($C_{11}^2:C_{15}$)

Order 1: $C_1$ ($C_{11}^2:C_{30}$)

Series

Derived series	$C_{11}^2:C_{30}$	$\rhd$	$C_{11}^2$	$\rhd$	$C_1$
Chief series	$C_{11}^2:C_{30}$	$\rhd$	$C_{11}^2:C_{15}$	$\rhd$	$C_{11}\times C_{55}$	$\rhd$	$C_{11}^2$	$\rhd$	$C_1$
Lower central series	$C_{11}^2:C_{30}$	$\rhd$	$C_{11}^2$
Upper central series	$C_1$	$\lhd$	$C_{10}$

Supergroups

This group is a maximal subgroup of 8 larger groups in the database.

This group is a maximal quotient of 4 larger groups in the database.

Character theory

Complex character table

See the $430 \times 430$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $24 \times 24$ rational character table.