Abstract group 5280.s: $C_{120}:D_{22}$

• Label: 5280.s
• Order: \( 2^{5} \cdot 3 \cdot 5 \cdot 11 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 5 \)
• $\card{Z(G)}$: \( 2 \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3 \cdot 5 \cdot 11 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 5 \)
• Perm deg.: $27$
• Trans deg.: $1320$
• Rank: $3$

Group information

Description:	$C_{120}:D_{22}$
Order:	\(5280\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 11 \)
Exponent:	\(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
Automorphism group:	$C_{66}.C_{10}.C_2^6.C_2$, of order \(84480\)\(\medspace = 2^{9} \cdot 3 \cdot 5 \cdot 11 \)
Composition factors:	$C_2$ x 5, $C_3$, $C_5$, $C_{11}$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.

Group statistics

Order	1	2	3	4	5	6	8	10	11	12	15	20	22	24	30	33	40	44	55	60	66	88	110	120	132	165	220	264	330	440	660	1320
Elements	1	167	2	168	4	46	48	668	10	48	8	672	130	96	184	20	192	140	40	192	20	40	520	384	40	80	560	80	80	160	160	320	5280
Conjugacy classes	1	5	1	4	4	3	4	20	5	4	4	16	10	8	12	5	16	10	20	16	5	10	40	32	10	20	40	20	20	40	40	80	525
Divisions	1	5	1	4	1	3	2	5	1	2	1	4	2	2	3	1	2	2	1	2	1	1	2	2	1	1	2	1	1	1	1	1	60
Autjugacy classes	1	4	1	4	1	2	2	4	1	2	1	4	2	2	2	1	2	2	1	2	1	1	2	2	1	1	2	1	1	1	1	1	56

Dimension	1	2	4	8	10	16	20	32	40	80	160	320
Irr. complex chars.	40	210	275	0	0	0	0	0	0	0	0	0	525
Irr. rational chars.	8	6	12	8	4	4	3	2	6	4	2	1	60

Minimal presentations

Permutation degree:	$27$
Transitive degree:	$1320$
Rank:	$3$
Inequivalent generating triples:	$499968$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	not computed	320
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c \mid a^{10}=b^{2}=c^{264}=1, b^{a}=bc^{72}, c^{a}=c^{131}, c^{b}=c^{241} \rangle$
Permutation group:	Degree $27$ $\langle(2,3)(4,10)(5,9)(6,11)(7,8)(12,13,15,16,14), (12,13,15,16,14), (1,2)(3,8) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 3 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 3 \end{array}\right), \left(\begin{array}{rrrr} 6 & 5 & 2 & 2 \\ 0 & 5 & 4 & 2 \\ 8 & 3 & 6 & 6 \\ 6 & 8 & 0 & 5 \end{array}\right), \left(\begin{array}{rrrr} 5 & 6 & 8 & 10 \\ 5 & 0 & 9 & 8 \\ 7 & 1 & 7 & 5 \\ 3 & 7 & 6 & 2 \end{array}\right), \left(\begin{array}{rrrr} 2 & 6 & 0 & 4 \\ 4 & 8 & 8 & 6 \\ 8 & 10 & 0 & 5 \\ 4 & 2 & 10 & 1 \end{array}\right), \left(\begin{array}{rrrr} 7 & 7 & 7 & 5 \\ 0 & 6 & 10 & 7 \\ 4 & 5 & 1 & 10 \\ 9 & 3 & 10 & 8 \end{array}\right), \left(\begin{array}{rrrr} 8 & 0 & 0 & 0 \\ 0 & 8 & 0 & 0 \\ 0 & 0 & 8 & 0 \\ 0 & 0 & 0 & 8 \end{array}\right), \left(\begin{array}{rrrr} 9 & 7 & 3 & 1 \\ 1 & 8 & 9 & 8 \\ 7 & 7 & 1 & 6 \\ 3 & 9 & 3 & 4 \end{array}\right), \left(\begin{array}{rrrr} 4 & 8 & 7 & 9 \\ 3 & 6 & 1 & 7 \\ 2 & 8 & 5 & 3 \\ 8 & 2 & 8 & 7 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{11})$
Direct product:	$C_5$ $\, \times\, $ $D_{11}$ $\, \times\, $ $(C_{24}:C_2)$
Semidirect product:	$C_{440}$ $\,\rtimes\,$ $D_6$	$C_{120}$ $\,\rtimes\,$ $D_{22}$	$C_{1320}$ $\,\rtimes\,$ $C_2^2$	$C_{55}$ $\,\rtimes\,$ $(C_8:D_6)$	all 40
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{660}$ . $C_2^3$	$C_{20}$ . $(S_3\times D_{22})$	$(C_{15}\times D_{22})$ . $D_4$	$(C_{20}\times D_{11})$ . $D_6$	all 36

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

Homology

Abelianization:	$C_{2}^{2} \times C_{10} \simeq C_{2}^{3} \times C_{5}$
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 3616 subgroups in 272 conjugacy classes, 88 normal (80 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{10}$	$G/Z \simeq$ $D_{11}\times D_{12}$
Commutator:	$G' \simeq$ $C_{132}$	$G/G' \simeq$ $C_2^2\times C_{10}$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_{30}:D_{22}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{1320}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_{120}:D_{22}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{330}$	$G/\operatorname{soc} \simeq$ $C_2\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times \SD_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

Order 5280: $C_{120}:D_{22}$

Order 2640: $C_{24}:C_{110}$, $C_{1320}:C_2$, $D_{11}\times C_{120}$, $C_{220}:D_6$, $C_{165}:\SD_{16}$, $C_{220}.D_6$, $C_{165}:\SD_{16}$

Order 1760: $C_{40}:D_{22}$

Order 1320: $D_{11}\times C_{60}$, $C_{165}:Q_8$, $C_{165}:D_4$, $C_{30}.D_{22}$, $C_{11}:C_{120}$, $C_{30}:D_{22}$, $C_{1320}$, $C_5\times D_{132}$, $C_{165}:Q_8$, $D_{12}\times C_{55}$, $C_{165}:Q_8$

Order 1056: $C_{24}:D_{22}$

Order 880: $C_{55}:\SD_{16}$, $C_{55}:\SD_{16}$, $C_{440}:C_2$, $D_{11}\times C_{40}$, $C_{20}.D_{22}$, $C_{20}:D_{22}$, $\SD_{16}\times C_{55}$

Order 660: $C_{15}:D_{22}$ x 2, $C_{165}:C_4$, $C_3:C_{220}$, $C_{11}:C_{60}$, $C_5\times D_{66}$, $S_3\times C_{110}$, $C_{15}\times D_{22}$, $C_{660}$

Order 528: $C_{12}.D_{22}$, $C_{264}:C_2$, $C_{24}:C_{22}$, $D_{11}\times C_{24}$, $C_{33}:\SD_{16}$, $C_{33}:\SD_{16}$, $D_{11}\times D_{12}$

Order 480: $C_{40}:D_6$

Order 440: $C_{20}\times D_{11}$ x 2, $C_{55}:Q_8$ x 2, $C_{440}$, $C_{10}\times D_{22}$, $Q_8\times C_{55}$, $D_4\times C_{55}$, $C_{11}:C_{40}$, $C_{55}:D_4$, $C_5\times D_{44}$

Order 352: $\SD_{16}\times D_{11}$

Order 330: $C_{15}\times D_{11}$ x 2, $C_5\times D_{33}$, $S_3\times C_{55}$, $C_{330}$

Order 264: $C_6.D_{22}$, $C_{264}$, $S_3\times D_{22}$, $D_{132}$, $C_{33}:Q_8$, $C_{11}\times D_{12}$, $C_{11}:C_{24}$, $C_{33}:Q_8$, $C_{12}\times D_{11}$, $C_{33}:Q_8$, $C_{11}:D_{12}$

Order 240: $C_{120}:C_2$ x 4, $C_2\times C_{120}$, $C_{10}\times D_{12}$, $C_{30}:Q_8$

Order 220: $C_5\times D_{22}$ x 4, $C_{220}$ x 2, $C_{11}:C_{20}$ x 2, $C_2\times C_{110}$

Order 176: $C_{88}:C_2$, $Q_8\times D_{11}$, $D_4\times D_{11}$, $C_8\times D_{11}$, $C_{11}\times \SD_{16}$, $Q_8:D_{11}$, $C_{11}:\SD_{16}$

Order 165: $C_{165}$

Order 160: $C_{10}\times \SD_{16}$

Order 132: $S_3\times D_{11}$ x 2, $D_{66}$, $S_3\times C_{22}$, $C_3\times D_{22}$, $C_{132}$, $C_{33}:C_4$, $C_{11}:C_{12}$, $C_3:C_{44}$

Order 120: $C_5\times D_{12}$ x 3, $C_{15}:Q_8$ x 3, $C_{120}$ x 2, $C_{10}\times D_6$, $C_2\times C_{60}$, $C_6:C_{20}$

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

Order 96: $C_8:D_6$

Order 88: $C_4\times D_{11}$ x 2, $C_{11}:Q_8$ x 2, $D_4\times C_{11}$, $C_{11}:D_4$, $D_{44}$, $C_{88}$, $C_2\times D_{22}$, $Q_8\times C_{11}$, $C_{11}:C_8$

Order 80: $C_5\times \SD_{16}$ x 4, $Q_8\times C_{10}$, $D_4\times C_{10}$, $C_2\times C_{40}$

Order 66: $C_3\times D_{11}$ x 2, $C_{66}$, $D_{33}$, $S_3\times C_{11}$

Order 60: $S_3\times C_{10}$ x 4, $C_{60}$ x 2, $C_3:C_{20}$ x 2, $C_2\times C_{30}$

Order 55: $C_{55}$

Order 48: $C_{24}:C_2$ x 4, $C_2\times D_{12}$, $C_6:Q_8$, $C_2\times C_{24}$

Order 44: $D_{22}$ x 4, $C_{44}$ x 2, $C_{11}:C_4$ x 2, $C_2\times C_{22}$

Order 40: $C_5\times Q_8$ x 3, $C_5\times D_4$ x 3, $C_2\times C_{20}$ x 2, $C_{40}$ x 2, $C_2^2\times C_{10}$

Order 33: $C_{33}$

Order 32: $C_2\times \SD_{16}$

Order 30: $C_{30}$ x 3, $C_5\times S_3$ x 2

Order 24: $D_{12}$ x 3, $C_3:Q_8$ x 3, $C_{24}$ x 2, $C_2\times C_{12}$, $C_6:C_4$, $C_2\times D_6$

Order 22: $D_{11}$ x 3, $C_{22}$ x 2

Order 20: $C_2\times C_{10}$ x 5, $C_{20}$ x 4

Order 16: $\SD_{16}$ x 4, $C_2\times C_8$, $C_2\times Q_8$, $C_2\times D_4$

Order 15: $C_{15}$

Order 12: $D_6$ x 4, $C_{12}$ x 2, $C_3:C_4$ x 2, $C_2\times C_6$

Order 11: $C_{11}$

Order 10: $C_{10}$ x 5

Order 8: $Q_8$ x 3, $D_4$ x 3, $C_2\times C_4$ x 2, $C_8$ x 2, $C_2^3$

Order 6: $C_6$ x 3, $S_3$ x 2

Order 5: $C_5$

Order 4: $C_2^2$ x 5, $C_4$ x 4

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 5280: $C_{120}:D_{22}$

Order 2640: $C_{24}:C_{110}$, $C_{1320}:C_2$, $D_{11}\times C_{120}$, $C_{220}:D_6$, $C_{165}:\SD_{16}$, $C_{220}.D_6$, $C_{165}:\SD_{16}$

Order 1760: $C_{40}:D_{22}$

Order 1320: $D_{11}\times C_{60}$, $C_{165}:Q_8$, $C_{165}:D_4$, $C_{30}.D_{22}$, $C_{11}:C_{120}$, $C_{30}:D_{22}$, $C_{1320}$, $C_5\times D_{132}$, $C_{165}:Q_8$, $D_{12}\times C_{55}$, $C_{165}:Q_8$

Order 1056: $C_{24}:D_{22}$

Order 880: $C_{55}:\SD_{16}$, $C_{55}:\SD_{16}$, $C_{440}:C_2$, $D_{11}\times C_{40}$, $C_{20}.D_{22}$, $C_{20}:D_{22}$, $\SD_{16}\times C_{55}$

Order 660: $C_{165}:C_4$, $C_3:C_{220}$, $C_{11}:C_{60}$, $C_5\times D_{66}$, $S_3\times C_{110}$, $C_{15}\times D_{22}$, $C_{15}:D_{22}$, $C_{660}$

Order 528: $C_{12}.D_{22}$, $C_{264}:C_2$, $C_{24}:C_{22}$, $D_{11}\times C_{24}$, $C_{33}:\SD_{16}$, $C_{33}:\SD_{16}$, $D_{11}\times D_{12}$

Order 480: $C_{40}:D_6$

Order 440: $C_{20}\times D_{11}$ x 2, $C_{55}:Q_8$ x 2, $C_{440}$, $C_{10}\times D_{22}$, $Q_8\times C_{55}$, $D_4\times C_{55}$, $C_{11}:C_{40}$, $C_{55}:D_4$, $C_5\times D_{44}$

Order 352: $\SD_{16}\times D_{11}$

Order 330: $C_5\times D_{33}$, $S_3\times C_{55}$, $C_{15}\times D_{11}$, $C_{330}$

Order 264: $C_6.D_{22}$, $C_{264}$, $S_3\times D_{22}$, $D_{132}$, $C_{33}:Q_8$, $C_{11}\times D_{12}$, $C_{11}:C_{24}$, $C_{33}:Q_8$, $C_{12}\times D_{11}$, $C_{33}:Q_8$, $C_{11}:D_{12}$

Order 240: $C_{120}:C_2$ x 4, $C_2\times C_{120}$, $C_{10}\times D_{12}$, $C_{30}:Q_8$

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

Order 176: $C_{88}:C_2$, $Q_8\times D_{11}$, $D_4\times D_{11}$, $C_8\times D_{11}$, $C_{11}\times \SD_{16}$, $Q_8:D_{11}$, $C_{11}:\SD_{16}$

Order 165: $C_{165}$

Order 160: $C_{10}\times \SD_{16}$

Order 132: $D_{66}$, $S_3\times C_{22}$, $C_3\times D_{22}$, $S_3\times D_{11}$, $C_{132}$, $C_{33}:C_4$, $C_{11}:C_{12}$, $C_3:C_{44}$

Order 120: $C_5\times D_{12}$ x 3, $C_{15}:Q_8$ x 3, $C_{120}$ x 2, $C_{10}\times D_6$, $C_2\times C_{60}$, $C_6:C_{20}$

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

Order 96: $C_8:D_6$

Order 88: $C_4\times D_{11}$ x 2, $C_{11}:Q_8$ x 2, $D_4\times C_{11}$, $C_{11}:D_4$, $D_{44}$, $C_{88}$, $C_2\times D_{22}$, $Q_8\times C_{11}$, $C_{11}:C_8$

Order 80: $C_5\times \SD_{16}$ x 4, $Q_8\times C_{10}$, $D_4\times C_{10}$, $C_2\times C_{40}$

Order 66: $C_{66}$, $D_{33}$, $C_3\times D_{11}$, $S_3\times C_{11}$

Order 60: $S_3\times C_{10}$ x 3, $C_{60}$ x 2, $C_3:C_{20}$ x 2, $C_2\times C_{30}$

Order 55: $C_{55}$

Order 48: $C_{24}:C_2$ x 4, $C_2\times D_{12}$, $C_6:Q_8$, $C_2\times C_{24}$

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

Order 40: $C_5\times Q_8$ x 3, $C_5\times D_4$ x 3, $C_2\times C_{20}$ x 2, $C_{40}$ x 2, $C_2^2\times C_{10}$

Order 33: $C_{33}$

Order 32: $C_2\times \SD_{16}$

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

Order 24: $D_{12}$ x 3, $C_3:Q_8$ x 3, $C_{24}$ x 2, $C_2\times C_{12}$, $C_6:C_4$, $C_2\times D_6$

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

Order 20: $C_2\times C_{10}$ x 4, $C_{20}$ x 4

Order 16: $\SD_{16}$ x 4, $C_2\times C_8$, $C_2\times Q_8$, $C_2\times D_4$

Order 15: $C_{15}$

Order 12: $D_6$ x 3, $C_{12}$ x 2, $C_3:C_4$ x 2, $C_2\times C_6$

Order 11: $C_{11}$

Order 10: $C_{10}$ x 4

Order 8: $Q_8$ x 3, $D_4$ x 3, $C_2\times C_4$ x 2, $C_8$ x 2, $C_2^3$

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

Order 5: $C_5$

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

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 5280: $C_{120}:D_{22}$ ($C_1$)

Order 2640: $C_{24}:C_{110}$ ($C_2$), $C_{1320}:C_2$ ($C_2$), $D_{11}\times C_{120}$ ($C_2$), $C_{220}:D_6$ ($C_2$), $C_{165}:\SD_{16}$ ($C_2$), $C_{220}.D_6$ ($C_2$), $C_{165}:\SD_{16}$ ($C_2$)

Order 1320: $D_{11}\times C_{60}$ ($C_2^2$), $C_{11}:C_{120}$ ($C_2^2$), $C_{1320}$ ($C_2^2$), $C_5\times D_{132}$ ($C_2^2$), $C_{165}:Q_8$ ($C_2^2$), $D_{12}\times C_{55}$ ($C_2^2$), $C_{165}:Q_8$ ($C_2^2$)

Order 1056: $C_{24}:D_{22}$ ($C_5$)

Order 880: $D_{11}\times C_{40}$ ($S_3$)

Order 660: $C_{11}:C_{60}$ ($D_4$), $C_{15}\times D_{22}$ ($D_4$), $C_{660}$ ($C_2^3$)

Order 528: $C_{12}.D_{22}$ ($C_{10}$), $C_{264}:C_2$ ($C_{10}$), $C_{24}:C_{22}$ ($C_{10}$), $D_{11}\times C_{24}$ ($C_{10}$), $C_{33}:\SD_{16}$ ($C_{10}$), $C_{33}:\SD_{16}$ ($C_{10}$), $D_{11}\times D_{12}$ ($C_{10}$)

Order 440: $C_{440}$ ($D_6$), $C_{11}:C_{40}$ ($D_6$), $C_{20}\times D_{11}$ ($D_6$)

Order 330: $C_{15}\times D_{11}$ ($\SD_{16}$) x 2, $C_{330}$ ($C_2\times D_4$)

Order 264: $C_{264}$ ($C_2\times C_{10}$), $D_{132}$ ($C_2\times C_{10}$), $C_{33}:Q_8$ ($C_2\times C_{10}$), $C_{11}\times D_{12}$ ($C_2\times C_{10}$), $C_{11}:C_{24}$ ($C_2\times C_{10}$), $C_{33}:Q_8$ ($C_2\times C_{10}$), $C_{12}\times D_{11}$ ($C_2\times C_{10}$)

Order 240: $C_{120}:C_2$ ($D_{11}$)

Order 220: $C_{220}$ ($C_2\times D_6$), $C_{11}:C_{20}$ ($D_{12}$), $C_5\times D_{22}$ ($D_{12}$)

Order 176: $C_8\times D_{11}$ ($C_5\times S_3$)

Order 165: $C_{165}$ ($C_2\times \SD_{16}$)

Order 132: $C_3\times D_{22}$ ($C_5\times D_4$), $C_{132}$ ($C_2^2\times C_{10}$), $C_{11}:C_{12}$ ($C_5\times D_4$)

Order 120: $C_{120}$ ($D_{22}$), $C_5\times D_{12}$ ($D_{22}$), $C_{15}:Q_8$ ($D_{22}$)

Order 110: $C_5\times D_{11}$ ($C_{24}:C_2$) x 2, $C_{110}$ ($C_2\times D_{12}$)

Order 88: $C_4\times D_{11}$ ($S_3\times C_{10}$), $C_{88}$ ($S_3\times C_{10}$), $C_{11}:C_8$ ($S_3\times C_{10}$)

Order 66: $C_3\times D_{11}$ ($C_5\times \SD_{16}$) x 2, $C_{66}$ ($D_4\times C_{10}$)

Order 60: $C_{60}$ ($C_2\times D_{22}$)

Order 55: $C_{55}$ ($C_8:D_6$)

Order 48: $C_{24}:C_2$ ($C_5\times D_{11}$)

Order 44: $D_{22}$ ($C_5\times D_{12}$), $C_{44}$ ($C_{10}\times D_6$), $C_{11}:C_4$ ($C_5\times D_{12}$)

Order 40: $C_{40}$ ($S_3\times D_{11}$)

Order 33: $C_{33}$ ($C_{10}\times \SD_{16}$)

Order 30: $C_{30}$ ($D_4\times D_{11}$)

Order 24: $D_{12}$ ($C_5\times D_{22}$), $C_3:Q_8$ ($C_5\times D_{22}$), $C_{24}$ ($C_5\times D_{22}$)

Order 22: $D_{11}$ ($C_{120}:C_2$) x 2, $C_{22}$ ($C_{10}\times D_{12}$)

Order 20: $C_{20}$ ($S_3\times D_{22}$)

Order 15: $C_{15}$ ($\SD_{16}\times D_{11}$)

Order 12: $C_{12}$ ($C_{10}\times D_{22}$)

Order 11: $C_{11}$ ($C_{40}:D_6$)

Order 10: $C_{10}$ ($D_{11}\times D_{12}$)

Order 8: $C_8$ ($C_{15}:D_{22}$)

Order 6: $C_6$ ($C_{20}:D_{22}$)

Order 5: $C_5$ ($C_{24}:D_{22}$)

Order 4: $C_4$ ($C_{30}:D_{22}$)

Order 3: $C_3$ ($C_{40}:D_{22}$)

Order 2: $C_2$ ($C_{220}:D_6$)

Order 1: $C_1$ ($C_{120}:D_{22}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 5280: $C_{120}:D_{22}$ ($C_1$)

Order 2640: $C_{24}:C_{110}$ ($C_2$), $C_{1320}:C_2$ ($C_2$), $D_{11}\times C_{120}$ ($C_2$), $C_{220}:D_6$ ($C_2$), $C_{165}:\SD_{16}$ ($C_2$), $C_{220}.D_6$ ($C_2$), $C_{165}:\SD_{16}$ ($C_2$)

Order 1320: $D_{11}\times C_{60}$ ($C_2^2$), $C_{11}:C_{120}$ ($C_2^2$), $C_{1320}$ ($C_2^2$), $C_5\times D_{132}$ ($C_2^2$), $C_{165}:Q_8$ ($C_2^2$), $D_{12}\times C_{55}$ ($C_2^2$), $C_{165}:Q_8$ ($C_2^2$)

Order 1056: $C_{24}:D_{22}$ ($C_5$)

Order 880: $D_{11}\times C_{40}$ ($S_3$)

Order 660: $C_{11}:C_{60}$ ($D_4$), $C_{15}\times D_{22}$ ($D_4$), $C_{660}$ ($C_2^3$)

Order 528: $C_{12}.D_{22}$ ($C_{10}$), $C_{264}:C_2$ ($C_{10}$), $C_{24}:C_{22}$ ($C_{10}$), $D_{11}\times C_{24}$ ($C_{10}$), $C_{33}:\SD_{16}$ ($C_{10}$), $C_{33}:\SD_{16}$ ($C_{10}$), $D_{11}\times D_{12}$ ($C_{10}$)

Order 440: $C_{440}$ ($D_6$), $C_{11}:C_{40}$ ($D_6$), $C_{20}\times D_{11}$ ($D_6$)

Order 330: $C_{15}\times D_{11}$ ($\SD_{16}$), $C_{330}$ ($C_2\times D_4$)

Order 264: $C_{264}$ ($C_2\times C_{10}$), $D_{132}$ ($C_2\times C_{10}$), $C_{33}:Q_8$ ($C_2\times C_{10}$), $C_{11}\times D_{12}$ ($C_2\times C_{10}$), $C_{11}:C_{24}$ ($C_2\times C_{10}$), $C_{33}:Q_8$ ($C_2\times C_{10}$), $C_{12}\times D_{11}$ ($C_2\times C_{10}$)

Order 240: $C_{120}:C_2$ ($D_{11}$)

Order 220: $C_{220}$ ($C_2\times D_6$), $C_{11}:C_{20}$ ($D_{12}$), $C_5\times D_{22}$ ($D_{12}$)

Order 176: $C_8\times D_{11}$ ($C_5\times S_3$)

Order 165: $C_{165}$ ($C_2\times \SD_{16}$)

Order 132: $C_3\times D_{22}$ ($C_5\times D_4$), $C_{132}$ ($C_2^2\times C_{10}$), $C_{11}:C_{12}$ ($C_5\times D_4$)

Order 120: $C_{120}$ ($D_{22}$), $C_5\times D_{12}$ ($D_{22}$), $C_{15}:Q_8$ ($D_{22}$)

Order 110: $C_{110}$ ($C_2\times D_{12}$), $C_5\times D_{11}$ ($C_{24}:C_2$)

Order 88: $C_4\times D_{11}$ ($S_3\times C_{10}$), $C_{88}$ ($S_3\times C_{10}$), $C_{11}:C_8$ ($S_3\times C_{10}$)

Order 66: $C_{66}$ ($D_4\times C_{10}$), $C_3\times D_{11}$ ($C_5\times \SD_{16}$)

Order 60: $C_{60}$ ($C_2\times D_{22}$)

Order 55: $C_{55}$ ($C_8:D_6$)

Order 48: $C_{24}:C_2$ ($C_5\times D_{11}$)

Order 44: $D_{22}$ ($C_5\times D_{12}$), $C_{44}$ ($C_{10}\times D_6$), $C_{11}:C_4$ ($C_5\times D_{12}$)

Order 40: $C_{40}$ ($S_3\times D_{11}$)

Order 33: $C_{33}$ ($C_{10}\times \SD_{16}$)

Order 30: $C_{30}$ ($D_4\times D_{11}$)

Order 24: $D_{12}$ ($C_5\times D_{22}$), $C_3:Q_8$ ($C_5\times D_{22}$), $C_{24}$ ($C_5\times D_{22}$)

Order 22: $C_{22}$ ($C_{10}\times D_{12}$), $D_{11}$ ($C_{120}:C_2$)

Order 20: $C_{20}$ ($S_3\times D_{22}$)

Order 15: $C_{15}$ ($\SD_{16}\times D_{11}$)

Order 12: $C_{12}$ ($C_{10}\times D_{22}$)

Order 11: $C_{11}$ ($C_{40}:D_6$)

Order 10: $C_{10}$ ($D_{11}\times D_{12}$)

Order 8: $C_8$ ($C_{15}:D_{22}$)

Order 6: $C_6$ ($C_{20}:D_{22}$)

Order 5: $C_5$ ($C_{24}:D_{22}$)

Order 4: $C_4$ ($C_{30}:D_{22}$)

Order 3: $C_3$ ($C_{40}:D_{22}$)

Order 2: $C_2$ ($C_{220}:D_6$)

Order 1: $C_1$ ($C_{120}:D_{22}$)

Series

Derived series

$C_{120}:D_{22}$

$\rhd$

$C_{132}$

$\rhd$

$C_1$

Chief series

$C_{120}:D_{22}$

$\rhd$

$D_{11}\times C_{120}$

$\rhd$

$D_{11}\times C_{24}$

$\rhd$

$C_{264}$

$\rhd$

$C_{132}$

$\rhd$

$C_{66}$

$\rhd$

$C_{33}$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

Lower central series

$C_{120}:D_{22}$

$\rhd$

$C_{132}$

$\rhd$

$C_{66}$

$\rhd$

$C_{33}$

Upper central series

$C_1$

$\lhd$

$C_{10}$

$\lhd$

$C_{20}$

$\lhd$

$C_{40}$

Supergroups

This group is a maximal subgroup of 2 larger groups in the database.

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

Character theory

Complex character table

The $525 \times 525$ character table is not available for this group.

Rational character table

See the $60 \times 60$ rational character table.