Abstract group 960.11026: $C_{20}:C_4\times D_6$

• Label: 960.11026
• Order: \( 2^{6} \cdot 3 \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3 \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $14$
• Trans deg.: $120$
• Rank: $4$

Group information

Description:	$C_{20}:C_4\times D_6$
Order:	\(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$C_{15}:(C_2\times C_4\times C_2^5.C_2^3)$, of order \(30720\)\(\medspace = 2^{11} \cdot 3 \cdot 5 \)
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	95	2	416	4	46	60	208	8	64	24	32	960
Conjugacy classes	1	15	1	24	1	7	7	12	1	8	3	4	84
Divisions	1	15	1	16	1	7	7	8	1	4	3	2	66
Autjugacy classes	1	7	1	8	1	5	3	4	1	2	2	1	36

Dimension	1	2	4	8	16
Irr. complex chars.	32	24	20	8	0	84
Irr. rational chars.	16	24	16	8	2	66

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$120$
Rank:	$4$
Inequivalent generating quadruples:	$8124480$

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:	$\langle a, b, c, d \mid a^{2}=b^{2}=c^{4}=d^{60}=[a,b]=[a,c]=[b,c]=[b,d]=1, d^{a}=d^{41}, d^{c}=d^{23} \rangle$
Permutation group:	Degree $14$ $\langle(1,2)(3,4)(9,10,11,12)(13,14), (6,7), (13,14), (1,3,4,2), (9,11)(10,12), (1,4)(2,3), (5,6,7), (8,9,10,12,11)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 11 & 3 \\ 10 & 7 \end{array}\right), \left(\begin{array}{rr} 1 & 20 \\ 20 & 11 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 16 & 15 \\ 15 & 1 \end{array}\right), \left(\begin{array}{rr} 11 & 0 \\ 0 & 11 \end{array}\right), \left(\begin{array}{rr} 1 & 15 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 19 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/30\Z)$
Direct product:	$C_2$ $\, \times\, $ $S_3$ $\, \times\, $ $(C_{20}:C_4)$
Semidirect product:	$D_5$ $\,\rtimes\,$ $(D_6.D_4)$	$(C_4\times D_6)$ $\,\rtimes\,$ $F_5$	$C_4$ $\,\rtimes\,$ $(D_6\times F_5)$	$(D_6.D_{10})$ $\,\rtimes\,$ $C_4$	all 51
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_5$ . $(Q_8\times D_6)$	$D_5$ . $(D_4\times D_6)$	$(S_3\times D_{10})$ . $Q_8$	$D_{10}$ . $(S_3\times Q_8)$	all 51

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

Homology

Abelianization:	$C_{2}^{3} \times C_{4} $
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$1$

Subgroups

There are 4960 subgroups in 836 conjugacy classes, 242 normal (48 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $D_6\times F_5$
Commutator:	$G' \simeq$ $C_{30}$	$G/G' \simeq$ $C_2^3\times C_4$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2\times D_6\times F_5$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{60}$	$G/\operatorname{Fit} \simeq$ $C_2\times C_4$
Radical:	$R \simeq$ $C_{20}:C_4\times D_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{30}$	$G/\operatorname{soc} \simeq$ $C_2^2\times C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3.D_4$
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: $C_{20}:C_4\times D_6$

Order 480: $C_{20}:C_4\times S_3$ x 8, $C_2\times D_6\times F_5$ x 2, $C_2\times D_{10}.D_6$ x 2, $C_{60}:C_2^3$, $D_{10}.D_{12}$, $C_2\times C_{20}:C_{12}$

Order 320: $D_{10}.C_2^4$

Order 240: $D_6\times F_5$ x 16, $D_{10}.D_6$ x 8, $C_{12}:D_{10}$ x 8, $C_{60}:C_4$ x 4, $C_{20}:C_{12}$ x 4, $D_{10}.D_6$ x 2, $D_{10}:C_{12}$ x 2, $D_6\times D_{10}$, $C_4\times D_{30}$, $D_6\times C_{20}$, $C_{12}\times D_{10}$, $D_{30}:C_4$, $D_6.D_{10}$, $D_{10}.D_6$

Order 192: $C_4:C_4\times D_6$

Order 160: $D_{10}.D_4$ x 12, $C_2^3\times F_5$ x 2, $C_{20}:C_2^3$

Order 120: $S_3\times F_5$ x 16, $S_3\times D_{10}$ x 12, $C_{30}:C_4$ x 8, $C_6\times F_5$ x 8, $C_6.D_{10}$ x 4, $C_6.D_{10}$ x 4, $C_4\times D_{15}$ x 4, $S_3\times C_{20}$ x 4, $D_5\times C_{12}$ x 4, $D_{15}:C_4$ x 4, $C_2\times D_{30}$, $C_{10}\times D_6$, $C_6\times D_{10}$, $C_2\times C_{60}$, $C_{30}:C_4$, $C_6:C_{20}$, $C_{10}:C_{12}$

Order 96: $D_6.D_4$ x 8, $C_{12}:C_2^3$ x 3, $C_2^3.D_6$ x 2, $C_2\times C_4:C_{12}$, $C_2^2.D_{12}$

Order 80: $C_2^2\times F_5$ x 20, $C_{20}:C_4$ x 16, $C_4\times D_{10}$ x 12, $C_2^2\times D_{10}$, $C_2^2\times C_{20}$, $C_2^2.D_{10}$

Order 64: $C_2^3.D_4$

Order 60: $S_3\times D_5$ x 16, $D_{30}$ x 6, $S_3\times C_{10}$ x 6, $C_3\times D_{10}$ x 6, $C_{15}:C_4$ x 4, $C_3\times F_5$ x 4, $C_{60}$ x 2, $C_{15}:C_4$ x 2, $C_5:C_{12}$ x 2, $C_3:C_{20}$ x 2, $C_2\times C_{30}$

Order 48: $C_4\times D_6$ x 28, $C_6.D_4$ x 8, $C_4:C_{12}$ x 4, $C_{12}:C_4$ x 4, $C_2^2\times C_{12}$ x 3, $C_6.C_2^3$ x 3, $C_2^2\times D_6$

Order 40: $C_2\times F_5$ x 32, $C_4\times D_5$ x 16, $C_2\times D_{10}$ x 14, $C_2\times C_{20}$ x 6, $C_{10}:C_4$ x 6, $C_2^2\times C_{10}$

Order 32: $C_2^2.D_4$ x 12, $C_2^3\times C_4$ x 3

Order 30: $D_{15}$ x 4, $C_3\times D_5$ x 4, $C_5\times S_3$ x 4, $C_{30}$ x 3

Order 24: $C_4\times S_3$ x 32, $C_2\times C_{12}$ x 14, $C_6:C_4$ x 14, $C_2\times D_6$ x 14, $C_2^2\times C_6$

Order 20: $D_{10}$ x 28, $F_5$ x 8, $C_2\times C_{10}$ x 7, $C_{20}$ x 4, $C_5:C_4$ x 4

Order 16: $C_2^2\times C_4$ x 34, $C_4:C_4$ x 16, $C_2^4$

Order 15: $C_{15}$

Order 12: $D_6$ x 28, $C_{12}$ x 8, $C_3:C_4$ x 8, $C_2\times C_6$ x 7

Order 10: $D_5$ x 8, $C_{10}$ x 7

Order 8: $C_2\times C_4$ x 60, $C_2^3$ x 15

Order 6: $S_3$ x 8, $C_6$ x 7

Order 5: $C_5$

Order 4: $C_2^2$ x 35, $C_4$ x 16

Order 3: $C_3$

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 960: $C_{20}:C_4\times D_6$

Order 480: $C_{20}:C_4\times S_3$, $C_2\times D_6\times F_5$, $C_{60}:C_2^3$, $D_{10}.D_{12}$, $C_2\times C_{20}:C_{12}$, $C_2\times D_{10}.D_6$

Order 320: $D_{10}.C_2^4$

Order 240: $D_6\times F_5$ x 2, $C_{12}:D_{10}$ x 2, $D_{10}.D_6$, $D_6\times D_{10}$, $D_{10}.D_6$, $D_{10}:C_{12}$, $C_4\times D_{30}$, $D_6\times C_{20}$, $C_{12}\times D_{10}$, $D_{30}:C_4$, $D_6.D_{10}$, $D_{10}.D_6$, $C_{60}:C_4$, $C_{20}:C_{12}$

Order 192: $C_4:C_4\times D_6$

Order 160: $D_{10}.D_4$ x 4, $C_2^3\times F_5$, $C_{20}:C_2^3$

Order 120: $S_3\times D_{10}$ x 4, $C_6.D_{10}$ x 2, $C_{30}:C_4$ x 2, $C_6\times F_5$ x 2, $D_5\times C_{12}$ x 2, $C_6.D_{10}$, $C_2\times D_{30}$, $C_{10}\times D_6$, $C_6\times D_{10}$, $S_3\times F_5$, $C_2\times C_{60}$, $C_{30}:C_4$, $C_4\times D_{15}$, $C_6:C_{20}$, $S_3\times C_{20}$, $C_{10}:C_{12}$, $D_{15}:C_4$

Order 96: $C_{12}:C_2^3$ x 2, $D_6.D_4$, $C_2\times C_4:C_{12}$, $C_2^2.D_{12}$, $C_2^3.D_6$

Order 80: $C_4\times D_{10}$ x 6, $C_2^2\times F_5$ x 4, $C_{20}:C_4$ x 3, $C_2^2\times D_{10}$, $C_2^2\times C_{20}$, $C_2^2.D_{10}$

Order 64: $C_2^3.D_4$

Order 60: $C_3\times D_{10}$ x 4, $S_3\times D_5$ x 3, $D_{30}$ x 2, $S_3\times C_{10}$ x 2, $C_{15}:C_4$, $C_3\times F_5$, $C_{60}$, $C_{15}:C_4$, $C_5:C_{12}$, $C_2\times C_{30}$, $C_3:C_{20}$

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

Order 40: $C_4\times D_5$ x 6, $C_2\times D_{10}$ x 6, $C_2\times F_5$ x 5, $C_2\times C_{20}$ x 3, $C_{10}:C_4$ x 3, $C_2^2\times C_{10}$

Order 32: $C_2^2.D_4$ x 4, $C_2^3\times C_4$ x 2

Order 30: $C_3\times D_5$ x 3, $C_{30}$ x 2, $D_{15}$, $C_5\times S_3$

Order 24: $C_2\times C_{12}$ x 6, $C_6:C_4$ x 6, $C_2\times D_6$ x 6, $C_4\times S_3$ x 5, $C_2^2\times C_6$

Order 20: $D_{10}$ x 9, $C_2\times C_{10}$ x 3, $F_5$ x 2, $C_{20}$ x 2, $C_5:C_4$ x 2

Order 16: $C_2^2\times C_4$ x 12, $C_4:C_4$ x 3, $C_2^4$

Order 15: $C_{15}$

Order 12: $D_6$ x 7, $C_2\times C_6$ x 5, $C_{12}$ x 3, $C_3:C_4$ x 3

Order 10: $D_5$ x 4, $C_{10}$ x 3

Order 8: $C_2\times C_4$ x 17, $C_2^3$ x 7

Order 6: $C_6$ x 5, $S_3$ x 2

Order 5: $C_5$

Order 4: $C_2^2$ x 12, $C_4$ x 6

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 960: $C_{20}:C_4\times D_6$ ($C_1$)

Order 480: $C_{20}:C_4\times S_3$ ($C_2$) x 8, $C_2\times D_6\times F_5$ ($C_2$) x 2, $C_2\times D_{10}.D_6$ ($C_2$) x 2, $C_{60}:C_2^3$ ($C_2$), $D_{10}.D_{12}$ ($C_2$), $C_2\times C_{20}:C_{12}$ ($C_2$)

Order 240: $D_{10}.D_6$ ($C_2^2$) x 8, $D_6\times F_5$ ($C_2^2$) x 8, $C_{12}:D_{10}$ ($C_2^2$) x 4, $C_{12}:D_{10}$ ($C_4$) x 4, $C_{60}:C_4$ ($C_2^2$) x 4, $C_{20}:C_{12}$ ($C_2^2$) x 4, $D_{10}.D_6$ ($C_2^2$) x 2, $D_{10}:C_{12}$ ($C_2^2$) x 2, $D_6\times D_{10}$ ($C_2^2$), $C_4\times D_{30}$ ($C_4$), $D_6\times C_{20}$ ($C_4$), $C_{12}\times D_{10}$ ($C_2^2$), $D_{30}:C_4$ ($C_4$), $D_6.D_{10}$ ($C_4$), $D_{10}.D_6$ ($C_2^2$)

Order 160: $D_{10}.D_4$ ($S_3$)

Order 120: $C_6.D_{10}$ ($C_2\times C_4$) x 4, $S_3\times D_{10}$ ($Q_8$) x 4, $S_3\times D_{10}$ ($D_4$) x 4, $C_{30}:C_4$ ($C_2^3$) x 4, $C_6\times F_5$ ($C_2^3$) x 4, $C_4\times D_{15}$ ($C_2\times C_4$) x 4, $S_3\times C_{20}$ ($C_2\times C_4$) x 4, $D_{15}:C_4$ ($C_2\times C_4$) x 4, $C_6.D_{10}$ ($C_2^3$) x 2, $C_6.D_{10}$ ($C_2\times C_4$) x 2, $S_3\times D_{10}$ ($C_2^3$) x 2, $S_3\times D_{10}$ ($C_2\times C_4$) x 2, $D_5\times C_{12}$ ($C_2^3$) x 2, $D_5\times C_{12}$ ($C_2\times C_4$) x 2, $C_2\times D_{30}$ ($C_2\times C_4$), $C_{10}\times D_6$ ($C_2\times C_4$), $C_6\times D_{10}$ ($C_2^3$), $C_2\times C_{60}$ ($C_2\times C_4$), $C_{30}:C_4$ ($C_2\times C_4$), $C_6:C_{20}$ ($C_2\times C_4$), $C_{10}:C_{12}$ ($C_2\times C_4$)

Order 80: $C_{20}:C_4$ ($D_6$) x 4, $C_2^2\times F_5$ ($D_6$) x 2, $C_4\times D_{10}$ ($D_6$)

Order 60: $S_3\times D_5$ ($C_4:C_4$) x 8, $S_3\times D_5$ ($C_2\times Q_8$) x 4, $S_3\times D_5$ ($C_2\times D_4$) x 4, $D_{30}$ ($C_4:C_4$) x 4, $S_3\times C_{10}$ ($C_4:C_4$) x 4, $C_{60}$ ($C_2^2\times C_4$) x 2, $C_{15}:C_4$ ($C_2^2\times C_4$) x 2, $C_5:C_{12}$ ($C_2^2\times C_4$) x 2, $D_{30}$ ($C_2^2\times C_4$) x 2, $S_3\times C_{10}$ ($C_2^2\times C_4$) x 2, $C_3\times D_{10}$ ($C_2\times Q_8$) x 2, $C_3\times D_{10}$ ($C_2\times D_4$) x 2, $C_3:C_{20}$ ($C_2^2\times C_4$) x 2, $C_2\times C_{30}$ ($C_2^2\times C_4$), $C_3\times D_{10}$ ($C_2^4$), $C_3\times D_{10}$ ($C_2^2\times C_4$)

Order 48: $C_4\times D_6$ ($F_5$)

Order 40: $C_2\times F_5$ ($C_2\times D_6$) x 4, $C_4\times D_5$ ($C_4\times S_3$) x 2, $C_4\times D_5$ ($C_2\times D_6$) x 2, $C_2\times C_{20}$ ($C_4\times S_3$), $C_{10}:C_4$ ($C_4\times S_3$), $C_2\times D_{10}$ ($C_2\times D_6$)

Order 30: $D_{15}$ ($C_2^2.D_4$) x 4, $C_5\times S_3$ ($C_2^2.D_4$) x 4, $C_{30}$ ($C_2^2.D_4$) x 2, $C_3\times D_5$ ($C_2^2.D_4$) x 2, $C_{30}$ ($C_2^3\times C_4$), $C_3\times D_5$ ($C_2^2\times Q_8$), $C_3\times D_5$ ($C_2^2\times D_4$)

Order 24: $C_4\times S_3$ ($C_2\times F_5$) x 4, $C_2\times C_{12}$ ($C_2\times F_5$), $C_6:C_4$ ($C_2\times F_5$), $C_2\times D_6$ ($C_2\times F_5$)

Order 20: $D_{10}$ ($S_3\times Q_8$) x 2, $D_{10}$ ($S_3\times D_4$) x 2, $C_{20}$ ($C_4\times D_6$) x 2, $C_5:C_4$ ($C_4\times D_6$) x 2, $C_2\times C_{10}$ ($C_4\times D_6$), $D_{10}$ ($C_2^2\times D_6$), $D_{10}$ ($C_4\times D_6$)

Order 15: $C_{15}$ ($C_2^3.D_4$)

Order 12: $D_6$ ($C_{20}:C_4$) x 4, $D_6$ ($C_2^2\times F_5$) x 2, $C_{12}$ ($C_2^2\times F_5$) x 2, $C_3:C_4$ ($C_2^2\times F_5$) x 2, $C_2\times C_6$ ($C_2^2\times F_5$)

Order 10: $C_{10}$ ($D_6.D_4$) x 2, $D_5$ ($D_6.D_4$) x 2, $C_{10}$ ($C_{12}:C_2^3$), $D_5$ ($Q_8\times D_6$), $D_5$ ($D_4\times D_6$)

Order 8: $C_2\times C_4$ ($S_3\times F_5$)

Order 6: $S_3$ ($D_{10}.D_4$) x 4, $C_6$ ($D_{10}.D_4$) x 2, $C_6$ ($C_2^3\times F_5$)

Order 5: $C_5$ ($C_4:C_4\times D_6$)

Order 4: $C_4$ ($D_6\times F_5$) x 2, $C_2^2$ ($D_6\times F_5$)

Order 3: $C_3$ ($D_{10}.C_2^4$)

Order 2: $C_2$ ($C_{20}:C_4\times S_3$) x 2, $C_2$ ($C_2\times D_6\times F_5$)

Order 1: $C_1$ ($C_{20}:C_4\times D_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 960: $C_{20}:C_4\times D_6$ ($C_1$)

Order 480: $C_{20}:C_4\times S_3$ ($C_2$), $C_2\times D_6\times F_5$ ($C_2$), $C_{60}:C_2^3$ ($C_2$), $D_{10}.D_{12}$ ($C_2$), $C_2\times C_{20}:C_{12}$ ($C_2$), $C_2\times D_{10}.D_6$ ($C_2$)

Order 240: $D_{10}.D_6$ ($C_2^2$), $D_6\times D_{10}$ ($C_2^2$), $D_{10}.D_6$ ($C_2^2$), $D_{10}:C_{12}$ ($C_2^2$), $D_6\times F_5$ ($C_2^2$), $C_4\times D_{30}$ ($C_4$), $D_6\times C_{20}$ ($C_4$), $C_{12}\times D_{10}$ ($C_2^2$), $D_{30}:C_4$ ($C_4$), $D_6.D_{10}$ ($C_4$), $D_{10}.D_6$ ($C_2^2$), $C_{12}:D_{10}$ ($C_2^2$), $C_{12}:D_{10}$ ($C_4$), $C_{60}:C_4$ ($C_2^2$), $C_{20}:C_{12}$ ($C_2^2$)

Order 160: $D_{10}.D_4$ ($S_3$)

Order 120: $C_6.D_{10}$ ($C_2\times C_4$), $C_6.D_{10}$ ($C_2^3$), $C_6.D_{10}$ ($C_2\times C_4$), $C_2\times D_{30}$ ($C_2\times C_4$), $C_{10}\times D_6$ ($C_2\times C_4$), $C_6\times D_{10}$ ($C_2^3$), $S_3\times D_{10}$ ($C_2^3$), $S_3\times D_{10}$ ($Q_8$), $S_3\times D_{10}$ ($D_4$), $S_3\times D_{10}$ ($C_2\times C_4$), $C_{30}:C_4$ ($C_2^3$), $C_6\times F_5$ ($C_2^3$), $C_2\times C_{60}$ ($C_2\times C_4$), $C_{30}:C_4$ ($C_2\times C_4$), $C_4\times D_{15}$ ($C_2\times C_4$), $C_6:C_{20}$ ($C_2\times C_4$), $S_3\times C_{20}$ ($C_2\times C_4$), $C_{10}:C_{12}$ ($C_2\times C_4$), $D_5\times C_{12}$ ($C_2^3$), $D_5\times C_{12}$ ($C_2\times C_4$), $D_{15}:C_4$ ($C_2\times C_4$)

Order 80: $C_2^2\times F_5$ ($D_6$), $C_4\times D_{10}$ ($D_6$), $C_{20}:C_4$ ($D_6$)

Order 60: $S_3\times D_5$ ($C_4:C_4$), $S_3\times D_5$ ($C_2\times Q_8$), $S_3\times D_5$ ($C_2\times D_4$), $C_{60}$ ($C_2^2\times C_4$), $C_{15}:C_4$ ($C_2^2\times C_4$), $C_5:C_{12}$ ($C_2^2\times C_4$), $C_2\times C_{30}$ ($C_2^2\times C_4$), $D_{30}$ ($C_4:C_4$), $D_{30}$ ($C_2^2\times C_4$), $S_3\times C_{10}$ ($C_4:C_4$), $S_3\times C_{10}$ ($C_2^2\times C_4$), $C_3\times D_{10}$ ($C_2^4$), $C_3\times D_{10}$ ($C_2\times Q_8$), $C_3\times D_{10}$ ($C_2\times D_4$), $C_3\times D_{10}$ ($C_2^2\times C_4$), $C_3:C_{20}$ ($C_2^2\times C_4$)

Order 48: $C_4\times D_6$ ($F_5$)

Order 40: $C_2\times C_{20}$ ($C_4\times S_3$), $C_{10}:C_4$ ($C_4\times S_3$), $C_4\times D_5$ ($C_4\times S_3$), $C_4\times D_5$ ($C_2\times D_6$), $C_2\times D_{10}$ ($C_2\times D_6$), $C_2\times F_5$ ($C_2\times D_6$)

Order 30: $C_{30}$ ($C_2^3\times C_4$), $C_{30}$ ($C_2^2.D_4$), $D_{15}$ ($C_2^2.D_4$), $C_3\times D_5$ ($C_2^2\times Q_8$), $C_3\times D_5$ ($C_2^2\times D_4$), $C_3\times D_5$ ($C_2^2.D_4$), $C_5\times S_3$ ($C_2^2.D_4$)

Order 24: $C_2\times C_{12}$ ($C_2\times F_5$), $C_6:C_4$ ($C_2\times F_5$), $C_4\times S_3$ ($C_2\times F_5$), $C_2\times D_6$ ($C_2\times F_5$)

Order 20: $C_2\times C_{10}$ ($C_4\times D_6$), $D_{10}$ ($C_2^2\times D_6$), $D_{10}$ ($S_3\times Q_8$), $D_{10}$ ($S_3\times D_4$), $D_{10}$ ($C_4\times D_6$), $C_{20}$ ($C_4\times D_6$), $C_5:C_4$ ($C_4\times D_6$)

Order 15: $C_{15}$ ($C_2^3.D_4$)

Order 12: $C_2\times C_6$ ($C_2^2\times F_5$), $D_6$ ($C_2^2\times F_5$), $D_6$ ($C_{20}:C_4$), $C_{12}$ ($C_2^2\times F_5$), $C_3:C_4$ ($C_2^2\times F_5$)

Order 10: $C_{10}$ ($D_6.D_4$), $C_{10}$ ($C_{12}:C_2^3$), $D_5$ ($D_6.D_4$), $D_5$ ($Q_8\times D_6$), $D_5$ ($D_4\times D_6$)

Order 8: $C_2\times C_4$ ($S_3\times F_5$)

Order 6: $C_6$ ($C_2^3\times F_5$), $C_6$ ($D_{10}.D_4$), $S_3$ ($D_{10}.D_4$)

Order 5: $C_5$ ($C_4:C_4\times D_6$)

Order 4: $C_2^2$ ($D_6\times F_5$), $C_4$ ($D_6\times F_5$)

Order 3: $C_3$ ($D_{10}.C_2^4$)

Order 2: $C_2$ ($C_{20}:C_4\times S_3$), $C_2$ ($C_2\times D_6\times F_5$)

Order 1: $C_1$ ($C_{20}:C_4\times D_6$)

Series

Derived series

$C_{20}:C_4\times D_6$

$\rhd$

$C_{30}$

$\rhd$

$C_1$

Chief series

$C_{20}:C_4\times D_6$

$\rhd$

$C_2\times D_6\times F_5$

$\rhd$

$D_6\times D_{10}$

$\rhd$

$S_3\times D_{10}$

$\rhd$

$C_3\times D_{10}$

$\rhd$

$C_3\times D_5$

$\rhd$

$C_{15}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_{20}:C_4\times D_6$

$\rhd$

$C_{30}$

$\rhd$

$C_{15}$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2\times C_4$

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 $84 \times 84$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $66 \times 66$ rational character table.