Abstract group 960.10151: $S_3\times Q_8\times C_{20}$

• Label: 960.10151
• Order: \( 2^{6} \cdot 3 \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \cdot 5 \)
• $\card{Z(G)}$: \( 2^{3} \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{9} \cdot 3 \)
• Perm deg.: $20$
• Trans deg.: $480$
• Rank: $4$

Group information

Description:	$S_3\times Q_8\times C_{20}$
Order:	\(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$C_2^2\times C_4\times C_2^2:S_4.C_2^2\times S_3$, of order \(36864\)\(\medspace = 2^{12} \cdot 3^{2} \)
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	15	2	112	4	6	60	56	8	448	24	224	960
Conjugacy classes	1	7	1	32	4	3	28	16	4	128	12	64	300
Divisions	1	7	1	22	1	3	7	11	1	22	3	11	90
Autjugacy classes	1	4	1	6	1	3	4	3	1	6	3	3	36

Dimension	1	2	4	8	16	32
Irr. complex chars.	160	120	20	0	0	0	300
Irr. rational chars.	16	20	24	21	8	1	90

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$480$
Rank:	$4$
Inequivalent generating quadruples:	$6814080$

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:	$\langle a, b, c, d \mid b^{2}=c^{4}=d^{60}=[a,b]=[a,c]=[a,d]=[b,c]=1, a^{2}=c^{2}d^{30}, d^{b}=d^{41}, d^{c}=d^{31} \rangle$
Permutation group:	Degree $20$ $\langle(1,2)(3,5)(4,6)(7,8)(14,15), (1,3,2,5)(4,8,6,7)(9,10)(11,12), (1,4,2,6) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right), \left(\begin{array}{rr} 16 & 15 \\ 15 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 20 \\ 20 & 23 \end{array}\right), \left(\begin{array}{rr} 11 & 0 \\ 0 & 11 \end{array}\right), \left(\begin{array}{rr} 1 & 15 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 10 \\ 10 & 23 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 19 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/30\Z)$
Direct product:	$C_4$ $\, \times\, $ $C_5$ $\, \times\, $ $S_3$ $\, \times\, $ $Q_8$
Semidirect product:	$(C_{60}:Q_8)$ $\,\rtimes\,$ $C_2$	$(Q_8\times C_{60})$ $\,\rtimes\,$ $C_2$	$C_{60}$ $\,\rtimes\,$ $(C_2\times Q_8)$	$(C_{12}:Q_8)$ $\,\rtimes\,$ $C_{10}$	all 18
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(Q_8\times D_6)$ . $C_{10}$	$(Q_8\times C_{10})$ . $D_6$	$D_6$ . $(Q_8\times C_{10})$	$C_{10}$ . $(Q_8\times D_6)$	all 66

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

Homology

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

Subgroups

There are 1136 subgroups in 596 conjugacy classes, 338 normal (44 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_{20}$	$G/Z \simeq$ $C_2\times D_6$
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:	$\operatorname{Fit} \simeq$ $Q_8\times C_{60}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $S_3\times Q_8\times C_{20}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{30}$	$G/\operatorname{soc} \simeq$ $C_2^4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2.C_2^2$
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: $S_3\times Q_8\times C_{20}$

Order 480: $C_4:C_{20}\times S_3$ x 3, $C_3:(Q_8\times C_{20})$ x 3, $C_4\times S_3\times C_{20}$ x 3, $C_{60}:Q_8$ x 3, $Q_8\times C_{60}$, $C_3:C_{20}\times Q_8$, $C_{60}.C_2^3$

Order 320: $C_2\times Q_8\times C_{20}$

Order 240: $C_{12}:C_{20}$ x 9, $C_{20}.D_6$ x 8, $D_6\times C_{20}$ x 7, $C_{30}.D_4$ x 6, $C_4:C_{60}$ x 3, $C_4\times C_{60}$ x 3, $C_{12}:C_{20}$ x 3, $C_{30}:Q_8$ x 3, $Q_8\times C_{30}$

Order 192: $C_4^2.D_6$

Order 160: $Q_8\times C_{20}$ x 8, $C_2\times C_4:C_{20}$ x 3, $C_2\times C_4\times C_{20}$ x 3, $C_{10}.C_2^4$

Order 120: $S_3\times C_{20}$ x 22, $C_{15}:Q_8$ x 12, $C_2\times C_{60}$ x 7, $C_6:C_{20}$ x 7, $Q_8\times C_{15}$ x 4, $C_{10}\times D_6$

Order 96: $D_6.D_4$ x 3, $C_3:(C_4\times Q_8)$ x 3, $S_3\times C_4^2$ x 3, $C_{12}:Q_8$ x 3, $Q_8\times D_6$, $Q_8\times C_{12}$, $C_3:C_4\times Q_8$

Order 80: $Q_8\times C_{10}$ x 12, $C_4:C_{20}$ x 12, $C_4\times C_{20}$ x 12, $C_2^2\times C_{20}$ x 7

Order 64: $C_4^2.C_2^2$

Order 60: $C_{60}$ x 11, $C_3:C_{20}$ x 11, $S_3\times C_{10}$ x 6, $C_2\times C_{30}$

Order 48: $C_{12}:C_4$ x 9, $S_3\times Q_8$ x 8, $C_4\times D_6$ x 7, $C_6.D_4$ x 6, $C_6:Q_8$ x 3, $C_4:C_{12}$ x 3, $C_4\times C_{12}$ x 3, $C_{12}:C_4$ x 3, $C_6\times Q_8$

Order 40: $C_2\times C_{20}$ x 36, $C_5\times Q_8$ x 16, $C_2^2\times C_{10}$

Order 32: $C_4\times Q_8$ x 8, $C_2^2.D_4$ x 3, $C_2\times C_4^2$ x 3, $C_2^2\times Q_8$

Order 30: $C_5\times S_3$ x 4, $C_{30}$ x 3

Order 24: $C_4\times S_3$ x 22, $C_3:Q_8$ x 12, $C_2\times C_{12}$ x 7, $C_6:C_4$ x 7, $C_3\times Q_8$ x 4, $C_2\times D_6$

Order 20: $C_{20}$ x 22, $C_2\times C_{10}$ x 7

Order 16: $C_4:C_4$ x 12, $C_4^2$ x 12, $C_2\times Q_8$ x 12, $C_2^2\times C_4$ x 7

Order 15: $C_{15}$

Order 12: $C_{12}$ x 11, $C_3:C_4$ x 11, $D_6$ x 6, $C_2\times C_6$

Order 10: $C_{10}$ x 7

Order 8: $C_2\times C_4$ x 36, $Q_8$ x 16, $C_2^3$

Order 6: $S_3$ x 4, $C_6$ x 3

Order 5: $C_5$

Order 4: $C_4$ x 22, $C_2^2$ x 7

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 960: $S_3\times Q_8\times C_{20}$

Order 480: $Q_8\times C_{60}$, $C_3:C_{20}\times Q_8$, $C_4:C_{20}\times S_3$, $C_3:(Q_8\times C_{20})$, $C_4\times S_3\times C_{20}$, $C_{60}:Q_8$, $C_{60}.C_2^3$

Order 320: $C_2\times Q_8\times C_{20}$

Order 240: $C_{12}:C_{20}$ x 3, $D_6\times C_{20}$ x 3, $C_4:C_{60}$, $C_4\times C_{60}$, $C_{12}:C_{20}$, $C_{30}.D_4$, $Q_8\times C_{30}$, $C_{20}.D_6$, $C_{30}:Q_8$

Order 192: $C_4^2.D_6$

Order 160: $Q_8\times C_{20}$ x 4, $C_{10}.C_2^4$, $C_2\times C_4:C_{20}$, $C_2\times C_4\times C_{20}$

Order 120: $C_2\times C_{60}$ x 3, $C_6:C_{20}$ x 3, $S_3\times C_{20}$ x 3, $C_{10}\times D_6$, $Q_8\times C_{15}$, $C_{15}:Q_8$

Order 96: $D_6.D_4$, $C_3:(C_4\times Q_8)$, $S_3\times C_4^2$, $C_{12}:Q_8$, $Q_8\times D_6$, $Q_8\times C_{12}$, $C_3:C_4\times Q_8$

Order 80: $C_4\times C_{20}$ x 4, $Q_8\times C_{10}$ x 3, $C_2^2\times C_{20}$ x 3, $C_4:C_{20}$ x 3

Order 64: $C_4^2.C_2^2$

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

Order 48: $C_4\times D_6$ x 3, $C_{12}:C_4$ x 3, $C_6\times Q_8$, $S_3\times Q_8$, $C_6:Q_8$, $C_4:C_{12}$, $C_4\times C_{12}$, $C_{12}:C_4$, $C_6.D_4$

Order 40: $C_2\times C_{20}$ x 9, $C_5\times Q_8$ x 2, $C_2^2\times C_{10}$

Order 32: $C_4\times Q_8$ x 4, $C_2^2\times Q_8$, $C_2^2.D_4$, $C_2\times C_4^2$

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

Order 24: $C_2\times C_{12}$ x 3, $C_6:C_4$ x 3, $C_4\times S_3$ x 3, $C_3:Q_8$, $C_2\times D_6$, $C_3\times Q_8$

Order 20: $C_{20}$ x 6, $C_2\times C_{10}$ x 4

Order 16: $C_4^2$ x 4, $C_4:C_4$ x 3, $C_2\times Q_8$ x 3, $C_2^2\times C_4$ x 3

Order 15: $C_{15}$

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

Order 10: $C_{10}$ x 4

Order 8: $C_2\times C_4$ x 9, $Q_8$ x 2, $C_2^3$

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

Order 5: $C_5$

Order 4: $C_4$ x 6, $C_2^2$ x 4

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 960: $S_3\times Q_8\times C_{20}$ ($C_1$)

Order 480: $C_4:C_{20}\times S_3$ ($C_2$) x 3, $C_3:(Q_8\times C_{20})$ ($C_2$) x 3, $C_4\times S_3\times C_{20}$ ($C_2$) x 3, $C_{60}:Q_8$ ($C_2$) x 3, $Q_8\times C_{60}$ ($C_2$), $C_3:C_{20}\times Q_8$ ($C_2$), $C_{60}.C_2^3$ ($C_2$)

Order 240: $C_{12}:C_{20}$ ($C_2^2$) x 9, $C_{20}.D_6$ ($C_4$) x 8, $D_6\times C_{20}$ ($C_2^2$) x 7, $C_{30}.D_4$ ($C_2^2$) x 6, $C_4:C_{60}$ ($C_2^2$) x 3, $C_4\times C_{60}$ ($C_2^2$) x 3, $C_{12}:C_{20}$ ($C_2^2$) x 3, $C_{30}:Q_8$ ($C_2^2$) x 3, $Q_8\times C_{30}$ ($C_2^2$)

Order 192: $C_4^2.D_6$ ($C_5$)

Order 160: $Q_8\times C_{20}$ ($S_3$)

Order 120: $S_3\times C_{20}$ ($C_2\times C_4$) x 12, $C_{15}:Q_8$ ($C_2\times C_4$) x 12, $C_2\times C_{60}$ ($C_2^3$) x 7, $C_6:C_{20}$ ($C_2^3$) x 7, $Q_8\times C_{15}$ ($C_2\times C_4$) x 4, $S_3\times C_{20}$ ($Q_8$) x 4, $C_{10}\times D_6$ ($C_2^3$)

Order 96: $D_6.D_4$ ($C_{10}$) x 3, $C_3:(C_4\times Q_8)$ ($C_{10}$) x 3, $S_3\times C_4^2$ ($C_{10}$) x 3, $C_{12}:Q_8$ ($C_{10}$) x 3, $Q_8\times D_6$ ($C_{10}$), $Q_8\times C_{12}$ ($C_{10}$), $C_3:C_4\times Q_8$ ($C_{10}$)

Order 80: $C_4:C_{20}$ ($D_6$) x 3, $C_4\times C_{20}$ ($D_6$) x 3, $Q_8\times C_{10}$ ($D_6$)

Order 60: $C_{60}$ ($C_2^2\times C_4$) x 6, $C_3:C_{20}$ ($C_2^2\times C_4$) x 6, $C_{60}$ ($C_2\times Q_8$) x 2, $S_3\times C_{10}$ ($D_4:C_2$) x 2, $S_3\times C_{10}$ ($C_2\times Q_8$) x 2, $S_3\times C_{10}$ ($C_2^2\times C_4$) x 2, $C_3:C_{20}$ ($C_2\times Q_8$) x 2, $C_2\times C_{30}$ ($C_2^4$)

Order 48: $C_{12}:C_4$ ($C_2\times C_{10}$) x 9, $S_3\times Q_8$ ($C_{20}$) x 8, $C_4\times D_6$ ($C_2\times C_{10}$) x 7, $C_6.D_4$ ($C_2\times C_{10}$) x 6, $C_6:Q_8$ ($C_2\times C_{10}$) x 3, $C_4:C_{12}$ ($C_2\times C_{10}$) x 3, $C_4\times C_{12}$ ($C_2\times C_{10}$) x 3, $C_{12}:C_4$ ($C_2\times C_{10}$) x 3, $C_6\times Q_8$ ($C_2\times C_{10}$)

Order 40: $C_2\times C_{20}$ ($C_2\times D_6$) x 7, $C_5\times Q_8$ ($C_4\times S_3$) x 4

Order 32: $C_4\times Q_8$ ($C_5\times S_3$)

Order 30: $C_5\times S_3$ ($C_4\times Q_8$) x 4, $C_{30}$ ($D_4:C_2^2$), $C_{30}$ ($C_2^2\times Q_8$), $C_{30}$ ($C_2^3\times C_4$)

Order 24: $C_4\times S_3$ ($C_2\times C_{20}$) x 12, $C_3:Q_8$ ($C_2\times C_{20}$) x 12, $C_2\times C_{12}$ ($C_2^2\times C_{10}$) x 7, $C_6:C_4$ ($C_2^2\times C_{10}$) x 7, $C_4\times S_3$ ($C_5\times Q_8$) x 4, $C_3\times Q_8$ ($C_2\times C_{20}$) x 4, $C_2\times D_6$ ($C_2^2\times C_{10}$)

Order 20: $C_{20}$ ($C_4\times D_6$) x 6, $C_{20}$ ($S_3\times Q_8$) x 2, $C_2\times C_{10}$ ($C_2^2\times D_6$)

Order 16: $C_4:C_4$ ($S_3\times C_{10}$) x 3, $C_4^2$ ($S_3\times C_{10}$) x 3, $C_2\times Q_8$ ($S_3\times C_{10}$)

Order 15: $C_{15}$ ($C_4^2.C_2^2$)

Order 12: $C_{12}$ ($C_2^2\times C_{20}$) x 6, $C_3:C_4$ ($C_2^2\times C_{20}$) x 6, $D_6$ ($D_4:C_{10}$) x 2, $D_6$ ($Q_8\times C_{10}$) x 2, $D_6$ ($C_2^2\times C_{20}$) x 2, $C_{12}$ ($Q_8\times C_{10}$) x 2, $C_3:C_4$ ($Q_8\times C_{10}$) x 2, $C_2\times C_6$ ($C_2^3\times C_{10}$)

Order 10: $C_{10}$ ($D_4:D_6$), $C_{10}$ ($Q_8\times D_6$), $C_{10}$ ($C_{12}:C_2^3$)

Order 8: $C_2\times C_4$ ($C_{10}\times D_6$) x 7, $Q_8$ ($S_3\times C_{20}$) x 4

Order 6: $S_3$ ($Q_8\times C_{20}$) x 4, $C_6$ ($C_{10}.C_2^4$), $C_6$ ($C_{10}.C_2^4$), $C_6$ ($C_2^3\times C_{20}$)

Order 5: $C_5$ ($C_4^2.D_6$)

Order 4: $C_4$ ($D_6\times C_{20}$) x 6, $C_4$ ($C_{20}.D_6$) x 2, $C_2^2$ ($C_{15}:C_2^4$)

Order 3: $C_3$ ($C_2\times Q_8\times C_{20}$)

Order 2: $C_2$ ($C_{60}.C_2^3$), $C_2$ ($C_{60}.C_2^3$), $C_2$ ($C_{60}:C_2^3$)

Order 1: $C_1$ ($S_3\times Q_8\times C_{20}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 960: $S_3\times Q_8\times C_{20}$ ($C_1$)

Order 480: $Q_8\times C_{60}$ ($C_2$), $C_3:C_{20}\times Q_8$ ($C_2$), $C_4:C_{20}\times S_3$ ($C_2$), $C_3:(Q_8\times C_{20})$ ($C_2$), $C_4\times S_3\times C_{20}$ ($C_2$), $C_{60}:Q_8$ ($C_2$), $C_{60}.C_2^3$ ($C_2$)

Order 240: $C_{12}:C_{20}$ ($C_2^2$) x 3, $D_6\times C_{20}$ ($C_2^2$) x 3, $C_4:C_{60}$ ($C_2^2$), $C_4\times C_{60}$ ($C_2^2$), $C_{12}:C_{20}$ ($C_2^2$), $C_{30}.D_4$ ($C_2^2$), $Q_8\times C_{30}$ ($C_2^2$), $C_{20}.D_6$ ($C_4$), $C_{30}:Q_8$ ($C_2^2$)

Order 192: $C_4^2.D_6$ ($C_5$)

Order 160: $Q_8\times C_{20}$ ($S_3$)

Order 120: $C_2\times C_{60}$ ($C_2^3$) x 3, $C_6:C_{20}$ ($C_2^3$) x 3, $C_{10}\times D_6$ ($C_2^3$), $Q_8\times C_{15}$ ($C_2\times C_4$), $S_3\times C_{20}$ ($Q_8$), $S_3\times C_{20}$ ($C_2\times C_4$), $C_{15}:Q_8$ ($C_2\times C_4$)

Order 96: $D_6.D_4$ ($C_{10}$), $C_3:(C_4\times Q_8)$ ($C_{10}$), $S_3\times C_4^2$ ($C_{10}$), $C_{12}:Q_8$ ($C_{10}$), $Q_8\times D_6$ ($C_{10}$), $Q_8\times C_{12}$ ($C_{10}$), $C_3:C_4\times Q_8$ ($C_{10}$)

Order 80: $Q_8\times C_{10}$ ($D_6$), $C_4:C_{20}$ ($D_6$), $C_4\times C_{20}$ ($D_6$)

Order 60: $C_{60}$ ($C_2\times Q_8$), $C_{60}$ ($C_2^2\times C_4$), $C_2\times C_{30}$ ($C_2^4$), $S_3\times C_{10}$ ($D_4:C_2$), $S_3\times C_{10}$ ($C_2\times Q_8$), $S_3\times C_{10}$ ($C_2^2\times C_4$), $C_3:C_{20}$ ($C_2\times Q_8$), $C_3:C_{20}$ ($C_2^2\times C_4$)

Order 48: $C_4\times D_6$ ($C_2\times C_{10}$) x 3, $C_{12}:C_4$ ($C_2\times C_{10}$) x 3, $C_6\times Q_8$ ($C_2\times C_{10}$), $S_3\times Q_8$ ($C_{20}$), $C_6:Q_8$ ($C_2\times C_{10}$), $C_4:C_{12}$ ($C_2\times C_{10}$), $C_4\times C_{12}$ ($C_2\times C_{10}$), $C_{12}:C_4$ ($C_2\times C_{10}$), $C_6.D_4$ ($C_2\times C_{10}$)

Order 40: $C_2\times C_{20}$ ($C_2\times D_6$) x 3, $C_5\times Q_8$ ($C_4\times S_3$)

Order 32: $C_4\times Q_8$ ($C_5\times S_3$)

Order 30: $C_{30}$ ($D_4:C_2^2$), $C_{30}$ ($C_2^2\times Q_8$), $C_{30}$ ($C_2^3\times C_4$), $C_5\times S_3$ ($C_4\times Q_8$)

Order 24: $C_2\times C_{12}$ ($C_2^2\times C_{10}$) x 3, $C_6:C_4$ ($C_2^2\times C_{10}$) x 3, $C_4\times S_3$ ($C_2\times C_{20}$), $C_4\times S_3$ ($C_5\times Q_8$), $C_3:Q_8$ ($C_2\times C_{20}$), $C_2\times D_6$ ($C_2^2\times C_{10}$), $C_3\times Q_8$ ($C_2\times C_{20}$)

Order 20: $C_2\times C_{10}$ ($C_2^2\times D_6$), $C_{20}$ ($S_3\times Q_8$), $C_{20}$ ($C_4\times D_6$)

Order 16: $C_4:C_4$ ($S_3\times C_{10}$), $C_4^2$ ($S_3\times C_{10}$), $C_2\times Q_8$ ($S_3\times C_{10}$)

Order 15: $C_{15}$ ($C_4^2.C_2^2$)

Order 12: $C_2\times C_6$ ($C_2^3\times C_{10}$), $D_6$ ($D_4:C_{10}$), $D_6$ ($Q_8\times C_{10}$), $D_6$ ($C_2^2\times C_{20}$), $C_{12}$ ($Q_8\times C_{10}$), $C_{12}$ ($C_2^2\times C_{20}$), $C_3:C_4$ ($Q_8\times C_{10}$), $C_3:C_4$ ($C_2^2\times C_{20}$)

Order 10: $C_{10}$ ($D_4:D_6$), $C_{10}$ ($Q_8\times D_6$), $C_{10}$ ($C_{12}:C_2^3$)

Order 8: $C_2\times C_4$ ($C_{10}\times D_6$) x 3, $Q_8$ ($S_3\times C_{20}$)

Order 6: $C_6$ ($C_{10}.C_2^4$), $C_6$ ($C_{10}.C_2^4$), $C_6$ ($C_2^3\times C_{20}$), $S_3$ ($Q_8\times C_{20}$)

Order 5: $C_5$ ($C_4^2.D_6$)

Order 4: $C_2^2$ ($C_{15}:C_2^4$), $C_4$ ($C_{20}.D_6$), $C_4$ ($D_6\times C_{20}$)

Order 3: $C_3$ ($C_2\times Q_8\times C_{20}$)

Order 2: $C_2$ ($C_{60}.C_2^3$), $C_2$ ($C_{60}.C_2^3$), $C_2$ ($C_{60}:C_2^3$)

Order 1: $C_1$ ($S_3\times Q_8\times C_{20}$)

Series

Derived series

$S_3\times Q_8\times C_{20}$

$\rhd$

$C_6$

$\rhd$

$C_1$

Chief series

$S_3\times Q_8\times C_{20}$

$\rhd$

$C_4\times S_3\times C_{20}$

$\rhd$

$D_6\times C_{20}$

$\rhd$

$C_2\times C_{60}$

$\rhd$

$C_2\times C_{30}$

$\rhd$

$C_{30}$

$\rhd$

$C_{15}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$S_3\times Q_8\times C_{20}$

$\rhd$

$C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2\times C_{20}$

$\lhd$

$Q_8\times C_{20}$

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 $90 \times 90$ rational character table.