Abstract group 960.10064: $S_3\times C_2^3:C_{20}$

• Label: 960.10064
• Order: \( 2^{6} \cdot 3 \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \cdot 5 \)
• $\card{Z(G)}$: \( 2^{3} \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{15} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{12} \)
• Perm deg.: $18$
• Trans deg.: $240$
• Rank: $4$

Group information

Description:	$S_3\times C_2^3:C_{20}$
Order:	\(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$C_3:(C_2^7.C_2^4.C_2^4)$, of order \(98304\)\(\medspace = 2^{15} \cdot 3 \)
Composition factors:	$C_2$ x 6, $C_3$, $C_5$
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	10	12	15	20	30	60
Elements	1	63	2	64	4	30	252	32	8	256	120	128	960
Conjugacy classes	1	23	1	16	4	11	92	8	4	64	44	32	300
Divisions	1	23	1	8	1	11	23	4	1	8	11	4	96
Autjugacy classes	1	6	1	2	1	4	6	1	1	2	4	1	30

Dimension	1	2	4	8	16
Irr. complex chars.	160	120	20	0	0	300
Irr. rational chars.	16	24	24	24	8	96

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$240$
Rank:	$4$
Inequivalent generating quadruples:	$2555280$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d, e \mid a^{2}=b^{2}=c^{2}=d^{4}=e^{30}=[a,b]=[a,c]=[a,d]=[a,e]=[b,c]=[b,d]=[c,e]=[d,e]=1, e^{b}=e^{11}, d^{c}=de^{15} \rangle$
Permutation group:	Degree $18$ $\langle(1,2)(3,4)(7,8)(9,10)(12,13), (5,6), (1,2)(3,4)(9,10), (1,3,2,4)(5,6)(7,9)(8,10), (14,15,16,17,18), (1,2)(3,4), (7,8)(9,10), (11,12,13)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 25 & 4 \\ 16 & 29 \end{array}\right), \left(\begin{array}{rr} 23 & 0 \\ 0 & 23 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 1 & 11 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 21 & 0 \\ 0 & 21 \end{array}\right), \left(\begin{array}{rr} 1 & 22 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 3 & 16 \\ 16 & 19 \end{array}\right), \left(\begin{array}{rr} 23 & 22 \\ 22 & 23 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/44\Z)$
Direct product:	$C_2$ $\, \times\, $ $C_5$ $\, \times\, $ $S_3$ $\, \times\, $ $(C_2^2:C_4)$
Semidirect product:	$C_2^3$ $\,\rtimes\,$ $(S_3\times C_{20})$	$(C_{30}.D_4)$ $\,\rtimes\,$ $C_2^2$	$(D_6\times C_{20})$ $\,\rtimes\,$ $C_2^2$	$(D_6:C_{20})$ $\,\rtimes\,$ $C_2^2$	all 46
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_{10}\times D_6)$ . $D_4$	$D_6$ . $(D_4\times C_{10})$	$C_{10}$ . $(D_4\times D_6)$	$C_2^4$ . $(S_3\times C_{10})$	all 42

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{44}\Z)$.

Homology

Abelianization:	$C_{2}^{3} \times C_{20} \simeq C_{2}^{3} \times C_{4} \times C_{5}$
Schur multiplier:	$C_{2}^{7}$
Commutator length:	$1$

Subgroups

There are 3664 subgroups in 1348 conjugacy classes, 414 normal (34 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2\times C_{10}$	$G/Z \simeq$ $C_2\times D_6$
Commutator:	$G' \simeq$ $C_6$	$G/G' \simeq$ $C_2^3\times C_{20}$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_{15}:C_2^4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^3:C_{60}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $S_3\times C_2^3:C_{20}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_{30}$	$G/\operatorname{soc} \simeq$ $C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4:C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$

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

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 960: $S_3\times C_2^3:C_{20}$

Order 480: $C_2^2:C_{20}\times S_3$ x 8, $C_2\times D_6:C_{20}$ x 2, $C_{60}:C_2^3$ x 2, $C_2^3:C_{60}$, $C_6.D_4\times C_{10}$, $C_{15}:C_2^5$

Order 320: $C_2^4:C_{20}$

Order 240: $C_{15}:C_2^4$ x 22, $D_6\times C_{20}$ x 16, $D_6:C_{20}$ x 8, $C_2^2:C_{60}$ x 4, $C_{30}.D_4$ x 4, $C_2^2\times C_{60}$ x 2, $C_{30}.C_2^3$ x 2, $C_2^3\times C_{30}$

Order 192: $C_2^4.D_6$

Order 160: $C_2^3:C_{20}$ x 12, $C_2^3\times C_{20}$ x 2, $C_2^4\times C_{10}$

Order 120: $C_{10}\times D_6$ x 88, $S_3\times C_{20}$ x 16, $C_2^2\times C_{30}$ x 11, $C_2\times C_{60}$ x 8, $C_6:C_{20}$ x 8

Order 96: $D_6.D_4$ x 8, $C_{12}:C_2^3$ x 2, $C_2^2.D_{12}$ x 2, $C_2^3\times D_6$, $C_2^3:C_{12}$, $C_2^3.D_6$

Order 80: $C_2^3\times C_{10}$ x 23, $C_2^2\times C_{20}$ x 20, $C_2^2:C_{20}$ x 16

Order 64: $C_2^4:C_4$

Order 60: $S_3\times C_{10}$ x 76, $C_2\times C_{30}$ x 23, $C_{60}$ x 4, $C_3:C_{20}$ x 4

Order 48: $C_2^2\times D_6$ x 22, $C_4\times D_6$ x 16, $D_6:C_4$ x 8, $C_2^2:C_{12}$ x 4, $C_6.D_4$ x 4, $C_2^2\times C_{12}$ x 2, $C_6.C_2^3$ x 2, $C_2^3\times C_6$

Order 40: $C_2^2\times C_{10}$ x 99, $C_2\times C_{20}$ x 32

Order 32: $C_2^3:C_4$ x 12, $C_2^3\times C_4$ x 2, $C_2^5$

Order 30: $C_5\times S_3$ x 12, $C_{30}$ x 11

Order 24: $C_2\times D_6$ x 88, $C_4\times S_3$ x 16, $C_2^2\times C_6$ x 11, $C_2\times C_{12}$ x 8, $C_6:C_4$ x 8

Order 20: $C_2\times C_{10}$ x 99, $C_{20}$ x 8

Order 16: $C_2^4$ x 23, $C_2^2\times C_4$ x 20, $C_2^2:C_4$ x 16

Order 15: $C_{15}$

Order 12: $D_6$ x 76, $C_2\times C_6$ x 23, $C_{12}$ x 4, $C_3:C_4$ x 4

Order 10: $C_{10}$ x 23

Order 8: $C_2^3$ x 99, $C_2\times C_4$ x 32

Order 6: $S_3$ x 12, $C_6$ x 11

Order 5: $C_5$

Order 4: $C_2^2$ x 99, $C_4$ x 8

Order 3: $C_3$

Order 2: $C_2$ x 23

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 960: $S_3\times C_2^3:C_{20}$

Order 480: $C_2^3:C_{60}$, $C_6.D_4\times C_{10}$, $C_2\times D_6:C_{20}$, $C_2^2:C_{20}\times S_3$, $C_{15}:C_2^5$, $C_{60}:C_2^3$

Order 320: $C_2^4:C_{20}$

Order 240: $C_{15}:C_2^4$ x 5, $D_6\times C_{20}$ x 2, $C_2^2:C_{60}$, $C_{30}.D_4$, $D_6:C_{20}$, $C_2^3\times C_{30}$, $C_2^2\times C_{60}$, $C_{30}.C_2^3$

Order 192: $C_2^4.D_6$

Order 160: $C_2^3:C_{20}$ x 4, $C_2^4\times C_{10}$, $C_2^3\times C_{20}$

Order 120: $C_{10}\times D_6$ x 9, $C_2^2\times C_{30}$ x 4, $C_2\times C_{60}$ x 2, $C_6:C_{20}$ x 2, $S_3\times C_{20}$

Order 96: $D_6.D_4$, $C_2^3\times D_6$, $C_{12}:C_2^3$, $C_2^3:C_{12}$, $C_2^3.D_6$, $C_2^2.D_{12}$

Order 80: $C_2^3\times C_{10}$ x 6, $C_2^2\times C_{20}$ x 4, $C_2^2:C_{20}$ x 3

Order 64: $C_2^4:C_4$

Order 60: $S_3\times C_{10}$ x 7, $C_2\times C_{30}$ x 6, $C_{60}$, $C_3:C_{20}$

Order 48: $C_2^2\times D_6$ x 5, $C_4\times D_6$ x 2, $C_2^3\times C_6$, $C_2^2\times C_{12}$, $C_6.C_2^3$, $C_2^2:C_{12}$, $C_6.D_4$, $D_6:C_4$

Order 40: $C_2^2\times C_{10}$ x 13, $C_2\times C_{20}$ x 5

Order 32: $C_2^3:C_4$ x 4, $C_2^5$, $C_2^3\times C_4$

Order 30: $C_{30}$ x 4, $C_5\times S_3$ x 2

Order 24: $C_2\times D_6$ x 9, $C_2^2\times C_6$ x 4, $C_2\times C_{12}$ x 2, $C_6:C_4$ x 2, $C_4\times S_3$

Order 20: $C_2\times C_{10}$ x 13, $C_{20}$ x 2

Order 16: $C_2^4$ x 6, $C_2^2\times C_4$ x 4, $C_2^2:C_4$ x 3

Order 15: $C_{15}$

Order 12: $D_6$ x 7, $C_2\times C_6$ x 6, $C_{12}$, $C_3:C_4$

Order 10: $C_{10}$ x 6

Order 8: $C_2^3$ x 13, $C_2\times C_4$ x 5

Order 6: $C_6$ x 4, $S_3$ x 2

Order 5: $C_5$

Order 4: $C_2^2$ x 13, $C_4$ x 2

Order 3: $C_3$

Order 2: $C_2$ x 6

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 960: $S_3\times C_2^3:C_{20}$ ($C_1$)

Order 480: $C_2^2:C_{20}\times S_3$ ($C_2$) x 8, $C_2\times D_6:C_{20}$ ($C_2$) x 2, $C_{60}:C_2^3$ ($C_2$) x 2, $C_2^3:C_{60}$ ($C_2$), $C_6.D_4\times C_{10}$ ($C_2$), $C_{15}:C_2^5$ ($C_2$)

Order 240: $D_6:C_{20}$ ($C_2^2$) x 8, $C_{15}:C_2^4$ ($C_4$) x 8, $D_6\times C_{20}$ ($C_2^2$) x 8, $C_{15}:C_2^4$ ($C_2^2$) x 6, $C_2^2:C_{60}$ ($C_2^2$) x 4, $C_{30}.D_4$ ($C_2^2$) x 4, $C_2^2\times C_{60}$ ($C_2^2$) x 2, $C_{30}.C_2^3$ ($C_2^2$) x 2, $C_2^3\times C_{30}$ ($C_2^2$)

Order 192: $C_2^4.D_6$ ($C_5$)

Order 160: $C_2^3:C_{20}$ ($S_3$)

Order 120: $C_{10}\times D_6$ ($C_2\times C_4$) x 24, $C_{10}\times D_6$ ($D_4$) x 8, $C_2^2\times C_{30}$ ($C_2\times C_4$) x 4, $C_{10}\times D_6$ ($C_2^3$) x 4, $C_2\times C_{60}$ ($C_2^3$) x 4, $C_6:C_{20}$ ($C_2^3$) x 4, $C_2^2\times C_{30}$ ($C_2^3$) x 3

Order 96: $D_6.D_4$ ($C_{10}$) x 8, $C_{12}:C_2^3$ ($C_{10}$) x 2, $C_2^2.D_{12}$ ($C_{10}$) x 2, $C_2^3\times D_6$ ($C_{10}$), $C_2^3:C_{12}$ ($C_{10}$), $C_2^3.D_6$ ($C_{10}$)

Order 80: $C_2^2:C_{20}$ ($D_6$) x 4, $C_2^2\times C_{20}$ ($D_6$) x 2, $C_2^3\times C_{10}$ ($D_6$)

Order 60: $S_3\times C_{10}$ ($C_2^2:C_4$) x 16, $S_3\times C_{10}$ ($C_2\times D_4$) x 8, $S_3\times C_{10}$ ($C_2^2\times C_4$) x 8, $C_2\times C_{30}$ ($C_2^2\times C_4$) x 6, $C_2\times C_{30}$ ($C_2\times D_4$) x 4, $C_2\times C_{30}$ ($C_2^4$)

Order 48: $C_2^2\times D_6$ ($C_{20}$) x 8, $C_4\times D_6$ ($C_2\times C_{10}$) x 8, $D_6:C_4$ ($C_2\times C_{10}$) x 8, $C_2^2\times D_6$ ($C_2\times C_{10}$) x 6, $C_2^2:C_{12}$ ($C_2\times C_{10}$) x 4, $C_6.D_4$ ($C_2\times C_{10}$) x 4, $C_2^2\times C_{12}$ ($C_2\times C_{10}$) x 2, $C_6.C_2^3$ ($C_2\times C_{10}$) x 2, $C_2^3\times C_6$ ($C_2\times C_{10}$)

Order 40: $C_2\times C_{20}$ ($C_2\times D_6$) x 4, $C_2^2\times C_{10}$ ($C_4\times S_3$) x 4, $C_2^2\times C_{10}$ ($C_2\times D_6$) x 3

Order 32: $C_2^3:C_4$ ($C_5\times S_3$)

Order 30: $C_5\times S_3$ ($C_2^3:C_4$) x 8, $C_{30}$ ($C_2^3:C_4$) x 4, $C_{30}$ ($C_2^2\times D_4$) x 2, $C_{30}$ ($C_2^3\times C_4$)

Order 24: $C_2\times D_6$ ($C_2\times C_{20}$) x 24, $C_2\times D_6$ ($C_5\times D_4$) x 8, $C_2\times C_{12}$ ($C_2^2\times C_{10}$) x 4, $C_6:C_4$ ($C_2^2\times C_{10}$) x 4, $C_2^2\times C_6$ ($C_2\times C_{20}$) x 4, $C_2\times D_6$ ($C_2^2\times C_{10}$) x 4, $C_2^2\times C_6$ ($C_2^2\times C_{10}$) x 3

Order 20: $C_2\times C_{10}$ ($C_4\times D_6$) x 6, $C_2\times C_{10}$ ($S_3\times D_4$) x 4, $C_2\times C_{10}$ ($C_2^2\times D_6$)

Order 16: $C_2^2:C_4$ ($S_3\times C_{10}$) x 4, $C_2^2\times C_4$ ($S_3\times C_{10}$) x 2, $C_2^4$ ($S_3\times C_{10}$)

Order 15: $C_{15}$ ($C_2^4:C_4$)

Order 12: $D_6$ ($C_2^2:C_{20}$) x 16, $D_6$ ($D_4\times C_{10}$) x 8, $D_6$ ($C_2^2\times C_{20}$) x 8, $C_2\times C_6$ ($C_2^2\times C_{20}$) x 6, $C_2\times C_6$ ($D_4\times C_{10}$) x 4, $C_2\times C_6$ ($C_2^3\times C_{10}$)

Order 10: $C_{10}$ ($D_6.D_4$) x 4, $C_{10}$ ($D_4\times D_6$) x 2, $C_{10}$ ($C_{12}:C_2^3$)

Order 8: $C_2^3$ ($S_3\times C_{20}$) x 4, $C_2\times C_4$ ($C_{10}\times D_6$) x 4, $C_2^3$ ($C_{10}\times D_6$) x 3

Order 6: $S_3$ ($C_2^3:C_{20}$) x 8, $C_6$ ($C_2^3:C_{20}$) x 4, $C_6$ ($C_{20}:C_2^3$) x 2, $C_6$ ($C_2^3\times C_{20}$)

Order 5: $C_5$ ($C_2^4.D_6$)

Order 4: $C_2^2$ ($D_6\times C_{20}$) x 6, $C_2^2$ ($C_{20}:D_6$) x 4, $C_2^2$ ($C_{15}:C_2^4$)

Order 3: $C_3$ ($C_2^4:C_{20}$)

Order 2: $C_2$ ($C_2^2:C_{20}\times S_3$) x 4, $C_2$ ($C_{60}:C_2^3$) x 2, $C_2$ ($C_{60}:C_2^3$)

Order 1: $C_1$ ($S_3\times C_2^3:C_{20}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 960: $S_3\times C_2^3:C_{20}$ ($C_1$)

Order 480: $C_2^3:C_{60}$ ($C_2$), $C_6.D_4\times C_{10}$ ($C_2$), $C_2\times D_6:C_{20}$ ($C_2$), $C_2^2:C_{20}\times S_3$ ($C_2$), $C_{15}:C_2^5$ ($C_2$), $C_{60}:C_2^3$ ($C_2$)

Order 240: $C_{15}:C_2^4$ ($C_2^2$) x 3, $C_2^2:C_{60}$ ($C_2^2$), $C_{30}.D_4$ ($C_2^2$), $D_6:C_{20}$ ($C_2^2$), $C_2^3\times C_{30}$ ($C_2^2$), $C_{15}:C_2^4$ ($C_4$), $C_2^2\times C_{60}$ ($C_2^2$), $C_{30}.C_2^3$ ($C_2^2$), $D_6\times C_{20}$ ($C_2^2$)

Order 192: $C_2^4.D_6$ ($C_5$)

Order 160: $C_2^3:C_{20}$ ($S_3$)

Order 120: $C_{10}\times D_6$ ($C_2\times C_4$) x 3, $C_2^2\times C_{30}$ ($C_2^3$) x 2, $C_{10}\times D_6$ ($C_2^3$) x 2, $C_2^2\times C_{30}$ ($C_2\times C_4$), $C_{10}\times D_6$ ($D_4$), $C_2\times C_{60}$ ($C_2^3$), $C_6:C_{20}$ ($C_2^3$)

Order 96: $D_6.D_4$ ($C_{10}$), $C_2^3\times D_6$ ($C_{10}$), $C_{12}:C_2^3$ ($C_{10}$), $C_2^3:C_{12}$ ($C_{10}$), $C_2^3.D_6$ ($C_{10}$), $C_2^2.D_{12}$ ($C_{10}$)

Order 80: $C_2^3\times C_{10}$ ($D_6$), $C_2^2\times C_{20}$ ($D_6$), $C_2^2:C_{20}$ ($D_6$)

Order 60: $C_2\times C_{30}$ ($C_2^2\times C_4$) x 2, $S_3\times C_{10}$ ($C_2^2\times C_4$) x 2, $C_2\times C_{30}$ ($C_2^4$), $C_2\times C_{30}$ ($C_2\times D_4$), $S_3\times C_{10}$ ($C_2^2:C_4$), $S_3\times C_{10}$ ($C_2\times D_4$)

Order 48: $C_2^2\times D_6$ ($C_2\times C_{10}$) x 3, $C_2^3\times C_6$ ($C_2\times C_{10}$), $C_2^2\times D_6$ ($C_{20}$), $C_2^2\times C_{12}$ ($C_2\times C_{10}$), $C_6.C_2^3$ ($C_2\times C_{10}$), $C_4\times D_6$ ($C_2\times C_{10}$), $C_2^2:C_{12}$ ($C_2\times C_{10}$), $C_6.D_4$ ($C_2\times C_{10}$), $D_6:C_4$ ($C_2\times C_{10}$)

Order 40: $C_2^2\times C_{10}$ ($C_2\times D_6$) x 2, $C_2\times C_{20}$ ($C_2\times D_6$), $C_2^2\times C_{10}$ ($C_4\times S_3$)

Order 32: $C_2^3:C_4$ ($C_5\times S_3$)

Order 30: $C_{30}$ ($C_2^2\times D_4$), $C_{30}$ ($C_2^3\times C_4$), $C_{30}$ ($C_2^3:C_4$), $C_5\times S_3$ ($C_2^3:C_4$)

Order 24: $C_2\times D_6$ ($C_2\times C_{20}$) x 3, $C_2^2\times C_6$ ($C_2^2\times C_{10}$) x 2, $C_2\times D_6$ ($C_2^2\times C_{10}$) x 2, $C_2\times C_{12}$ ($C_2^2\times C_{10}$), $C_6:C_4$ ($C_2^2\times C_{10}$), $C_2^2\times C_6$ ($C_2\times C_{20}$), $C_2\times D_6$ ($C_5\times D_4$)

Order 20: $C_2\times C_{10}$ ($C_4\times D_6$) x 2, $C_2\times C_{10}$ ($C_2^2\times D_6$), $C_2\times C_{10}$ ($S_3\times D_4$)

Order 16: $C_2^2:C_4$ ($S_3\times C_{10}$), $C_2^4$ ($S_3\times C_{10}$), $C_2^2\times C_4$ ($S_3\times C_{10}$)

Order 15: $C_{15}$ ($C_2^4:C_4$)

Order 12: $C_2\times C_6$ ($C_2^2\times C_{20}$) x 2, $D_6$ ($C_2^2\times C_{20}$) x 2, $C_2\times C_6$ ($C_2^3\times C_{10}$), $C_2\times C_6$ ($D_4\times C_{10}$), $D_6$ ($D_4\times C_{10}$), $D_6$ ($C_2^2:C_{20}$)

Order 10: $C_{10}$ ($D_6.D_4$), $C_{10}$ ($D_4\times D_6$), $C_{10}$ ($C_{12}:C_2^3$)

Order 8: $C_2^3$ ($C_{10}\times D_6$) x 2, $C_2^3$ ($S_3\times C_{20}$), $C_2\times C_4$ ($C_{10}\times D_6$)

Order 6: $C_6$ ($C_{20}:C_2^3$), $C_6$ ($C_2^3\times C_{20}$), $C_6$ ($C_2^3:C_{20}$), $S_3$ ($C_2^3:C_{20}$)

Order 5: $C_5$ ($C_2^4.D_6$)

Order 4: $C_2^2$ ($D_6\times C_{20}$) x 2, $C_2^2$ ($C_{15}:C_2^4$), $C_2^2$ ($C_{20}:D_6$)

Order 3: $C_3$ ($C_2^4:C_{20}$)

Order 2: $C_2$ ($C_2^2:C_{20}\times S_3$), $C_2$ ($C_{60}:C_2^3$), $C_2$ ($C_{60}:C_2^3$)

Order 1: $C_1$ ($S_3\times C_2^3:C_{20}$)

Series

Derived series

$S_3\times C_2^3:C_{20}$

$\rhd$

$C_6$

$\rhd$

$C_1$

Chief series

$S_3\times C_2^3:C_{20}$

$\rhd$

$C_{15}:C_2^5$

$\rhd$

$C_{15}:C_2^4$

$\rhd$

$C_2^2\times C_{30}$

$\rhd$

$C_2\times C_{30}$

$\rhd$

$C_{30}$

$\rhd$

$C_{15}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$S_3\times C_2^3:C_{20}$

$\rhd$

$C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2^2\times C_{10}$

$\lhd$

$C_2^3:C_{20}$

Character theory

Complex character table

See the $300 \times 300$ 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 $96 \times 96$ rational character table.