Abstract group 720.740: $C_{60}:D_6$

• Label: 720.740
• Order: \( 2^{4} \cdot 3^{2} \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 5 \)
• $\card{Z(G)}$: \( 2^{3} \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{10} \cdot 3 \)
• Perm deg.: $17$
• Trans deg.: $360$
• Rank: $3$

Group information

Description:	$C_{60}:D_6$
Order:	\(720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$\ASL(2,3).C_2^4.C_2^4$, of order \(55296\)\(\medspace = 2^{11} \cdot 3^{3} \)
Composition factors:	$C_2$ x 4, $C_3$ x 2, $C_5$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.

Group statistics

Order	1	2	3	4	5	6	10	12	15	20	30	60
Elements	1	39	8	40	4	24	156	32	32	160	96	128	720
Conjugacy classes	1	7	4	8	4	12	28	16	16	32	48	64	240
Divisions	1	7	4	4	1	12	7	8	4	4	12	8	72
Autjugacy classes	1	3	1	2	1	2	3	1	1	2	2	1	20

Dimension	1	2	4	8	16
Irr. complex chars.	80	160	0	0	0	240
Irr. rational chars.	8	20	16	20	8	72

Minimal presentations

Permutation degree:	$17$
Transitive degree:	$360$
Rank:	$3$
Inequivalent generating triples:	$1302$

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:	$\langle a, b, c \mid a^{2}=b^{6}=c^{60}=[b,c]=1, b^{a}=b^{5}, c^{a}=c^{41} \rangle$
Permutation group:	Degree $17$ $\langle(2,3)(7,8), (4,5), (14,15,16,17), (9,10,11,12,13), (14,16)(15,17), (1,2,3), (1,3,2)(6,7,8)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right), \left(\begin{array}{rr} 22 & 15 \\ 15 & 2 \end{array}\right), \left(\begin{array}{rr} 16 & 15 \\ 15 & 1 \end{array}\right), \left(\begin{array}{rr} 11 & 0 \\ 0 & 11 \end{array}\right), \left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 19 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/30\Z)$
Direct product:	$C_2$ $\, \times\, $ $C_4$ $\, \times\, $ $C_5$ $\, \times\, $ $(C_3:S_3)$
Semidirect product:	$C_{60}$ $\,\rtimes\,$ $D_6$	$(C_2\times C_{60})$ $\,\rtimes\,$ $S_3$	$C_{30}$ $\,\rtimes\,$ $(C_4\times S_3)$	$C_6$ $\,\rtimes\,$ $(S_3\times C_{20})$	all 20
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_{30}:D_6)$ . $C_2$	$(C_6:D_6)$ . $C_{10}$	$(C_2\times C_{30})$ . $D_6$	$C_{30}$ . $(C_2\times D_6)$	all 16

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

Homology

Abelianization:	$C_{2}^{2} \times C_{20} \simeq C_{2}^{2} \times C_{4} \times C_{5}$
Schur multiplier:	$C_{2}^{2} \times C_{6}$
Commutator length:	$1$

Subgroups

There are 932 subgroups in 324 conjugacy classes, 134 normal (22 characteristic).

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

Special subgroups

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

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

Order 720: $C_{60}:D_6$

Order 360: $C_{60}:S_3$ x 4, $C_{30}:D_6$, $C_6\times C_{60}$, $C_{30}.D_6$

Order 240: $D_6\times C_{20}$ x 4

Order 180: $C_{30}:S_3$ x 6, $C_3\times C_{60}$ x 2, $C_3^2:C_{20}$ x 2, $C_6\times C_{30}$

Order 144: $C_{12}:D_6$

Order 120: $S_3\times C_{20}$ x 16, $C_{10}\times D_6$ x 4, $C_2\times C_{60}$ x 4, $C_6:C_{20}$ x 4

Order 90: $C_{15}:S_3$ x 4, $C_3\times C_{30}$ x 3

Order 80: $C_2^2\times C_{20}$

Order 72: $C_{12}:S_3$ x 4, $C_6:D_6$, $C_6\times C_{12}$, $C_6.D_6$

Order 60: $S_3\times C_{10}$ x 24, $C_{60}$ x 8, $C_3:C_{20}$ x 8, $C_2\times C_{30}$ x 4

Order 48: $C_4\times D_6$ x 4

Order 45: $C_3\times C_{15}$

Order 40: $C_2\times C_{20}$ x 6, $C_2^2\times C_{10}$

Order 36: $C_6:S_3$ x 6, $C_3\times C_{12}$ x 2, $C_3^2:C_4$ x 2, $C_6^2$

Order 30: $C_5\times S_3$ x 16, $C_{30}$ x 12

Order 24: $C_4\times S_3$ x 16, $C_2\times C_{12}$ x 4, $C_6:C_4$ x 4, $C_2\times D_6$ x 4

Order 20: $C_2\times C_{10}$ x 7, $C_{20}$ x 4

Order 18: $C_3:S_3$ x 4, $C_3\times C_6$ x 3

Order 16: $C_2^2\times C_4$

Order 15: $C_{15}$ x 4

Order 12: $D_6$ x 24, $C_{12}$ x 8, $C_3:C_4$ x 8, $C_2\times C_6$ x 4

Order 10: $C_{10}$ x 7

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 6, $C_2^3$

Order 6: $S_3$ x 16, $C_6$ x 12

Order 5: $C_5$

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

Order 3: $C_3$ x 4

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 720: $C_{60}:D_6$

Order 360: $C_{30}:D_6$, $C_6\times C_{60}$, $C_{30}.D_6$, $C_{60}:S_3$

Order 240: $D_6\times C_{20}$

Order 180: $C_{30}:S_3$ x 2, $C_6\times C_{30}$, $C_3\times C_{60}$, $C_3^2:C_{20}$

Order 144: $C_{12}:D_6$

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

Order 90: $C_3\times C_{30}$ x 2, $C_{15}:S_3$

Order 80: $C_2^2\times C_{20}$

Order 72: $C_6:D_6$, $C_6\times C_{12}$, $C_6.D_6$, $C_{12}:S_3$

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

Order 48: $C_4\times D_6$

Order 45: $C_3\times C_{15}$

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

Order 36: $C_6:S_3$ x 2, $C_3\times C_{12}$, $C_3^2:C_4$, $C_6^2$

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

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

Order 20: $C_2\times C_{10}$ x 3, $C_{20}$ x 2

Order 18: $C_3\times C_6$ x 2, $C_3:S_3$

Order 16: $C_2^2\times C_4$

Order 15: $C_{15}$

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

Order 10: $C_{10}$ x 3

Order 9: $C_3^2$

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

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

Order 5: $C_5$

Order 4: $C_2^2$ x 3, $C_4$ x 2

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 720: $C_{60}:D_6$ ($C_1$)

Order 360: $C_{60}:S_3$ ($C_2$) x 4, $C_{30}:D_6$ ($C_2$), $C_6\times C_{60}$ ($C_2$), $C_{30}.D_6$ ($C_2$)

Order 180: $C_{30}:S_3$ ($C_4$) x 4, $C_{30}:S_3$ ($C_2^2$) x 2, $C_3\times C_{60}$ ($C_2^2$) x 2, $C_3^2:C_{20}$ ($C_2^2$) x 2, $C_6\times C_{30}$ ($C_2^2$)

Order 144: $C_{12}:D_6$ ($C_5$)

Order 120: $C_2\times C_{60}$ ($S_3$) x 4

Order 90: $C_{15}:S_3$ ($C_2\times C_4$) x 4, $C_3\times C_{30}$ ($C_2\times C_4$) x 2, $C_3\times C_{30}$ ($C_2^3$)

Order 72: $C_{12}:S_3$ ($C_{10}$) x 4, $C_6:D_6$ ($C_{10}$), $C_6\times C_{12}$ ($C_{10}$), $C_6.D_6$ ($C_{10}$)

Order 60: $C_{60}$ ($D_6$) x 8, $C_2\times C_{30}$ ($D_6$) x 4

Order 45: $C_3\times C_{15}$ ($C_2^2\times C_4$)

Order 40: $C_2\times C_{20}$ ($C_3:S_3$)

Order 36: $C_6:S_3$ ($C_{20}$) x 4, $C_3\times C_{12}$ ($C_2\times C_{10}$) x 2, $C_3^2:C_4$ ($C_2\times C_{10}$) x 2, $C_6:S_3$ ($C_2\times C_{10}$) x 2, $C_6^2$ ($C_2\times C_{10}$)

Order 30: $C_{30}$ ($C_4\times S_3$) x 8, $C_{30}$ ($C_2\times D_6$) x 4

Order 24: $C_2\times C_{12}$ ($C_5\times S_3$) x 4

Order 20: $C_{20}$ ($C_6:S_3$) x 2, $C_2\times C_{10}$ ($C_6:S_3$)

Order 18: $C_3:S_3$ ($C_2\times C_{20}$) x 4, $C_3\times C_6$ ($C_2\times C_{20}$) x 2, $C_3\times C_6$ ($C_2^2\times C_{10}$)

Order 15: $C_{15}$ ($C_4\times D_6$) x 4

Order 12: $C_{12}$ ($S_3\times C_{10}$) x 8, $C_2\times C_6$ ($S_3\times C_{10}$) x 4

Order 10: $C_{10}$ ($C_{12}:S_3$) x 2, $C_{10}$ ($C_6:D_6$)

Order 9: $C_3^2$ ($C_2^2\times C_{20}$)

Order 8: $C_2\times C_4$ ($C_{15}:S_3$)

Order 6: $C_6$ ($S_3\times C_{20}$) x 8, $C_6$ ($C_{10}\times D_6$) x 4

Order 5: $C_5$ ($C_{12}:D_6$)

Order 4: $C_4$ ($C_{30}:S_3$) x 2, $C_2^2$ ($C_{30}:S_3$)

Order 3: $C_3$ ($D_6\times C_{20}$) x 4

Order 2: $C_2$ ($C_{60}:S_3$) x 2, $C_2$ ($C_{30}:D_6$)

Order 1: $C_1$ ($C_{60}:D_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 720: $C_{60}:D_6$ ($C_1$)

Order 360: $C_{30}:D_6$ ($C_2$), $C_6\times C_{60}$ ($C_2$), $C_{30}.D_6$ ($C_2$), $C_{60}:S_3$ ($C_2$)

Order 180: $C_6\times C_{30}$ ($C_2^2$), $C_{30}:S_3$ ($C_2^2$), $C_{30}:S_3$ ($C_4$), $C_3\times C_{60}$ ($C_2^2$), $C_3^2:C_{20}$ ($C_2^2$)

Order 144: $C_{12}:D_6$ ($C_5$)

Order 120: $C_2\times C_{60}$ ($S_3$)

Order 90: $C_{15}:S_3$ ($C_2\times C_4$), $C_3\times C_{30}$ ($C_2^3$), $C_3\times C_{30}$ ($C_2\times C_4$)

Order 72: $C_6:D_6$ ($C_{10}$), $C_6\times C_{12}$ ($C_{10}$), $C_6.D_6$ ($C_{10}$), $C_{12}:S_3$ ($C_{10}$)

Order 60: $C_{60}$ ($D_6$), $C_2\times C_{30}$ ($D_6$)

Order 45: $C_3\times C_{15}$ ($C_2^2\times C_4$)

Order 40: $C_2\times C_{20}$ ($C_3:S_3$)

Order 36: $C_3\times C_{12}$ ($C_2\times C_{10}$), $C_3^2:C_4$ ($C_2\times C_{10}$), $C_6^2$ ($C_2\times C_{10}$), $C_6:S_3$ ($C_2\times C_{10}$), $C_6:S_3$ ($C_{20}$)

Order 30: $C_{30}$ ($C_4\times S_3$), $C_{30}$ ($C_2\times D_6$)

Order 24: $C_2\times C_{12}$ ($C_5\times S_3$)

Order 20: $C_2\times C_{10}$ ($C_6:S_3$), $C_{20}$ ($C_6:S_3$)

Order 18: $C_3\times C_6$ ($C_2\times C_{20}$), $C_3\times C_6$ ($C_2^2\times C_{10}$), $C_3:S_3$ ($C_2\times C_{20}$)

Order 15: $C_{15}$ ($C_4\times D_6$)

Order 12: $C_2\times C_6$ ($S_3\times C_{10}$), $C_{12}$ ($S_3\times C_{10}$)

Order 10: $C_{10}$ ($C_6:D_6$), $C_{10}$ ($C_{12}:S_3$)

Order 9: $C_3^2$ ($C_2^2\times C_{20}$)

Order 8: $C_2\times C_4$ ($C_{15}:S_3$)

Order 6: $C_6$ ($C_{10}\times D_6$), $C_6$ ($S_3\times C_{20}$)

Order 5: $C_5$ ($C_{12}:D_6$)

Order 4: $C_2^2$ ($C_{30}:S_3$), $C_4$ ($C_{30}:S_3$)

Order 3: $C_3$ ($D_6\times C_{20}$)

Order 2: $C_2$ ($C_{30}:D_6$), $C_2$ ($C_{60}:S_3$)

Order 1: $C_1$ ($C_{60}:D_6$)

Series

Derived series

$C_{60}:D_6$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Chief series

$C_{60}:D_6$

$\rhd$

$C_{60}:S_3$

$\rhd$

$C_3\times C_{60}$

$\rhd$

$C_{60}$

$\rhd$

$C_{30}$

$\rhd$

$C_{15}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_{60}:D_6$

$\rhd$

$C_3^2$

Upper central series

$C_1$

$\lhd$

$C_2\times C_{20}$

Supergroups

This group is a maximal subgroup of 119 larger groups in the database.

This group is a maximal quotient of 21 larger groups in the database.

Character theory

Complex character table

See the $240 \times 240$ 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 $72 \times 72$ rational character table.