Abstract group 14520.ba: $C_5\times C_{11}^2:D_{12}$

• Label: 14520.ba
• Order: \( 2^{3} \cdot 3 \cdot 5 \cdot 11^{2} \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 5 \)
• $\card{Z(G)}$: \( 2 \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3 \cdot 5 \cdot 11^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 5 \)
• Perm deg.: $42$
• Trans deg.: $660$
• Rank: $2$

Group information

Description:	$C_5\times C_{11}^2:D_{12}$
Order:	\(14520\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11^{2} \)
Exponent:	\(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \)
Automorphism group:	$C_{11}^2.C_6.C_{10}.C_4.C_2^3$, of order \(232320\)\(\medspace = 2^{7} \cdot 3 \cdot 5 \cdot 11^{2} \)
Composition factors:	$C_2$ x 3, $C_3$, $C_5$, $C_{11}$ x 2
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	5	6	10	11	12	15	20	22	30	55	60	110
Elements	1	133	242	242	4	242	532	120	484	968	968	1440	968	480	1936	5760	14520
Conjugacy classes	1	3	1	1	4	1	12	15	2	4	4	35	4	60	8	140	295
Divisions	1	3	1	1	1	1	3	3	1	1	1	5	1	3	1	5	32
Autjugacy classes	1	2	1	1	1	1	2	2	1	1	1	3	1	2	1	3	24

Dimension	1	2	4	6	8	12	16	30	60	120	240
Irr. complex chars.	20	25	0	200	0	50	0	0	0	0	0	295
Irr. rational chars.	4	3	5	0	3	0	1	4	4	4	4	32

Minimal presentations

Permutation degree:	$42$
Transitive degree:	$660$
Rank:	$2$
Inequivalent generating pairs:	$216$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	12	240
Arbitrary	6	10	36

Constructions

Presentation:	$\langle a, b, c, d \mid a^{10}=b^{12}=c^{11}=d^{11}=[c,d]=1, b^{a}=b^{11}, c^{a}=d^{10}, d^{a}=c^{10}, c^{b}=d^{10}, d^{b}=cd \rangle$
Permutation group:	Degree $42$ $\langle(1,5,2)(3,19,8)(4,22,10)(6,21,13)(7,11,17)(9,29,16)(12,18,25)(14,23,26) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 3 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 3 \end{array}\right), \left(\begin{array}{rrrr} 4 & 10 & 7 & 8 \\ 5 & 4 & 4 & 7 \\ 10 & 7 & 7 & 1 \\ 9 & 10 & 6 & 7 \end{array}\right), \left(\begin{array}{rrrr} 7 & 1 & 5 & 5 \\ 8 & 3 & 9 & 5 \\ 0 & 6 & 4 & 10 \\ 4 & 0 & 3 & 0 \end{array}\right), \left(\begin{array}{rrrr} 0 & 8 & 9 & 4 \\ 2 & 7 & 2 & 9 \\ 9 & 1 & 7 & 3 \\ 2 & 9 & 9 & 3 \end{array}\right), \left(\begin{array}{rrrr} 9 & 6 & 5 & 2 \\ 9 & 8 & 4 & 5 \\ 3 & 8 & 2 & 5 \\ 7 & 3 & 2 & 1 \end{array}\right), \left(\begin{array}{rrrr} 8 & 0 & 0 & 0 \\ 0 & 8 & 0 & 0 \\ 0 & 0 & 8 & 0 \\ 0 & 0 & 0 & 8 \end{array}\right), \left(\begin{array}{rrrr} 7 & 7 & 2 & 9 \\ 8 & 0 & 7 & 8 \\ 7 & 2 & 3 & 5 \\ 6 & 10 & 10 & 1 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{11})$
	$\left\langle \left(\begin{array}{rr} 60 & 23 \\ 46 & 60 \end{array}\right), \left(\begin{array}{rr} 27 & 0 \\ 0 & 27 \end{array}\right), \left(\begin{array}{rr} 45 & 33 \\ 55 & 78 \end{array}\right), \left(\begin{array}{rr} 0 & 70 \\ 19 & 0 \end{array}\right), \left(\begin{array}{rr} 120 & 0 \\ 0 & 120 \end{array}\right), \left(\begin{array}{rr} 111 & 11 \\ 99 & 12 \end{array}\right), \left(\begin{array}{rr} 120 & 0 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/121\Z)$
Direct product:	$C_5$ $\, \times\, $ $(C_{11}^2:D_{12})$
Semidirect product:	$(C_{11}\times C_{55})$ $\,\rtimes\,$ $D_{12}$	$(C_{11}^2:C_{20})$ $\,\rtimes\,$ $S_3$	$(C_{11}^2:C_{15})$ $\,\rtimes\,$ $D_4$	$(C_{11}^2:D_6)$ $\,\rtimes\,$ $C_{10}$ (2)	all 10
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_{11}\times C_{110})$ . $D_6$	$C_{10}$ . $(C_{11}^2:D_6)$	$(C_{11}^2:C_{30})$ . $C_2^2$	$C_2$ . $(C_5\times C_{11}^2:D_6)$	all 6

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{121}\Z)$.

Homology

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

Subgroups

There are 4732 subgroups in 102 conjugacy classes, 22 normal (18 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{10}$	$G/Z \simeq$ $C_{11}^2:D_6$
Commutator:	$G' \simeq$ $C_{11}^2:C_6$	$G/G' \simeq$ $C_2\times C_{10}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_5\times C_{11}^2:D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_{11}\times C_{110}$	$G/\operatorname{Fit} \simeq$ $D_6$
Radical:	$R \simeq$ $C_5\times C_{11}^2:D_{12}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{11}\times C_{110}$	$G/\operatorname{soc} \simeq$ $D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4$
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

There are too many subgroups to show. See a full page version of the diagram.

Classes of subgroups up to automorphism

There are too many subgroups to show. See a full page version of the diagram.

Normal subgroups

Normal subgroups up to automorphism

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

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

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Classes of subgroups up to conjugation

Order 14520: $C_5\times C_{11}^2:D_{12}$

Order 7260: $C_{10}\times C_{11}^2:S_3$ x 2, $C_{11}^2:C_{60}$

Order 4840: $C_{110}.D_{22}$

Order 3630: $C_5\times C_{11}^2:S_3$ x 2, $C_{11}^2:C_{30}$

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

Order 2420: $D_{11}\times C_{110}$ x 2, $C_{11}^2:C_{20}$

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

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

Order 1210: $D_{11}\times C_{55}$ x 2, $C_{11}\times C_{110}$

Order 968: $D_{22}:D_{11}$

Order 726: $C_{11}^2:S_3$ x 2, $C_{11}^2:C_6$

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

Order 484: $C_{11}\times D_{22}$ x 2, $C_{11}^2:C_4$

Order 440: $C_{55}:D_4$ x 2

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

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

Order 220: $C_{11}:C_{20}$ x 3, $C_2\times C_{110}$ x 2, $C_5\times D_{22}$ x 2

Order 121: $C_{11}^2$

Order 120: $C_5\times D_{12}$

Order 110: $C_{110}$ x 5, $C_5\times D_{11}$ x 2

Order 88: $C_{11}:D_4$ x 2

Order 60: $S_3\times C_{10}$ x 2, $C_{60}$

Order 55: $C_{55}$ x 3

Order 44: $C_{11}:C_4$ x 3, $C_2\times C_{22}$ x 2, $D_{22}$ x 2

Order 40: $C_5\times D_4$

Order 30: $C_5\times S_3$ x 2, $C_{30}$

Order 24: $D_{12}$

Order 22: $C_{22}$ x 5, $D_{11}$ x 2

Order 20: $C_2\times C_{10}$ x 2, $C_{20}$

Order 15: $C_{15}$

Order 12: $D_6$ x 2, $C_{12}$

Order 11: $C_{11}$ x 3

Order 10: $C_{10}$ x 3

Order 8: $D_4$

Order 6: $S_3$ x 2, $C_6$

Order 5: $C_5$

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 14520: $C_5\times C_{11}^2:D_{12}$

Order 7260: $C_{10}\times C_{11}^2:S_3$, $C_{11}^2:C_{60}$

Order 4840: $C_{110}.D_{22}$

Order 3630: $C_5\times C_{11}^2:S_3$, $C_{11}^2:C_{30}$

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

Order 2420: $D_{11}\times C_{110}$, $C_{11}^2:C_{20}$

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

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

Order 1210: $C_{11}\times C_{110}$, $D_{11}\times C_{55}$

Order 968: $D_{22}:D_{11}$

Order 726: $C_{11}^2:C_6$, $C_{11}^2:S_3$

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

Order 484: $C_{11}^2:C_4$, $C_{11}\times D_{22}$

Order 440: $C_{55}:D_4$

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

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

Order 220: $C_{11}:C_{20}$ x 2, $C_2\times C_{110}$, $C_5\times D_{22}$

Order 121: $C_{11}^2$

Order 120: $C_5\times D_{12}$

Order 110: $C_{110}$ x 3, $C_5\times D_{11}$

Order 88: $C_{11}:D_4$

Order 60: $C_{60}$, $S_3\times C_{10}$

Order 55: $C_{55}$ x 2

Order 44: $C_{11}:C_4$ x 2, $C_2\times C_{22}$, $D_{22}$

Order 40: $C_5\times D_4$

Order 30: $C_{30}$, $C_5\times S_3$

Order 24: $D_{12}$

Order 22: $C_{22}$ x 3, $D_{11}$

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

Order 15: $C_{15}$

Order 12: $D_6$, $C_{12}$

Order 11: $C_{11}$ x 2

Order 10: $C_{10}$ x 2

Order 8: $D_4$

Order 6: $C_6$, $S_3$

Order 5: $C_5$

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

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 14520: $C_5\times C_{11}^2:D_{12}$ ($C_1$)

Order 7260: $C_{10}\times C_{11}^2:S_3$ ($C_2$) x 2, $C_{11}^2:C_{60}$ ($C_2$)

Order 3630: $C_{11}^2:C_{30}$ ($C_2^2$)

Order 2904: $C_{11}^2:D_{12}$ ($C_5$)

Order 2420: $C_{11}^2:C_{20}$ ($S_3$)

Order 1815: $C_{11}^2:C_{15}$ ($D_4$)

Order 1452: $C_{11}^2:D_6$ ($C_{10}$) x 2, $C_{11}^2:C_{12}$ ($C_{10}$)

Order 1210: $C_{11}\times C_{110}$ ($D_6$)

Order 726: $C_{11}^2:C_6$ ($C_2\times C_{10}$)

Order 605: $C_{11}\times C_{55}$ ($D_{12}$)

Order 484: $C_{11}^2:C_4$ ($C_5\times S_3$)

Order 363: $C_{11}^2:C_3$ ($C_5\times D_4$)

Order 242: $C_{11}\times C_{22}$ ($S_3\times C_{10}$)

Order 121: $C_{11}^2$ ($C_5\times D_{12}$)

Order 10: $C_{10}$ ($C_{11}^2:D_6$)

Order 5: $C_5$ ($C_{11}^2:D_{12}$)

Order 2: $C_2$ ($C_5\times C_{11}^2:D_6$)

Order 1: $C_1$ ($C_5\times C_{11}^2:D_{12}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 14520: $C_5\times C_{11}^2:D_{12}$ ($C_1$)

Order 7260: $C_{10}\times C_{11}^2:S_3$ ($C_2$), $C_{11}^2:C_{60}$ ($C_2$)

Order 3630: $C_{11}^2:C_{30}$ ($C_2^2$)

Order 2904: $C_{11}^2:D_{12}$ ($C_5$)

Order 2420: $C_{11}^2:C_{20}$ ($S_3$)

Order 1815: $C_{11}^2:C_{15}$ ($D_4$)

Order 1452: $C_{11}^2:D_6$ ($C_{10}$), $C_{11}^2:C_{12}$ ($C_{10}$)

Order 1210: $C_{11}\times C_{110}$ ($D_6$)

Order 726: $C_{11}^2:C_6$ ($C_2\times C_{10}$)

Order 605: $C_{11}\times C_{55}$ ($D_{12}$)

Order 484: $C_{11}^2:C_4$ ($C_5\times S_3$)

Order 363: $C_{11}^2:C_3$ ($C_5\times D_4$)

Order 242: $C_{11}\times C_{22}$ ($S_3\times C_{10}$)

Order 121: $C_{11}^2$ ($C_5\times D_{12}$)

Order 10: $C_{10}$ ($C_{11}^2:D_6$)

Order 5: $C_5$ ($C_{11}^2:D_{12}$)

Order 2: $C_2$ ($C_5\times C_{11}^2:D_6$)

Order 1: $C_1$ ($C_5\times C_{11}^2:D_{12}$)

Series

Derived series

$C_5\times C_{11}^2:D_{12}$

$\rhd$

$C_{11}^2:C_6$

$\rhd$

$C_{11}^2$

$\rhd$

$C_1$

Chief series

$C_5\times C_{11}^2:D_{12}$

$\rhd$

$C_{11}^2:C_{60}$

$\rhd$

$C_{11}^2:C_{12}$

$\rhd$

$C_{11}^2:C_6$

$\rhd$

$C_{11}^2:C_3$

$\rhd$

$C_{11}^2$

$\rhd$

$C_1$

Lower central series

$C_5\times C_{11}^2:D_{12}$

$\rhd$

$C_{11}^2:C_6$

$\rhd$

$C_{11}^2:C_3$

Upper central series

$C_1$

$\lhd$

$C_{10}$

Supergroups

This group is a maximal subgroup of 5 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 $295 \times 295$ 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 $32 \times 32$ rational character table.