Abstract group 480.1154: $C_{60}:C_2^3$

• Label: 480.1154
• Order: \( 2^{5} \cdot 3 \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 5 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \)
• Perm deg.: $14$
• Trans deg.: $120$
• Rank: $4$

Group information

Description:	$C_{60}:C_2^3$
Order:	\(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$C_3:(C_2^6.C_2^5)$, of order \(6144\)\(\medspace = 2^{11} \cdot 3 \)
Composition factors:	$C_2$ x 5, $C_3$, $C_5$
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	47	2	16	4	22	188	8	8	64	88	32	480
Conjugacy classes	1	15	1	4	4	7	60	2	4	16	28	8	150
Divisions	1	15	1	4	1	7	15	2	1	4	7	2	60
Autjugacy classes	1	5	1	2	1	3	5	1	1	2	3	1	26

Dimension	1	2	4	8	16
Irr. complex chars.	80	60	10	0	0	150
Irr. rational chars.	16	12	18	12	2	60

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$120$
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:	$\langle a, b, c, d \mid a^{2}=b^{2}=c^{2}=d^{60}=[a,b]=[a,c]=[b,c]=[c,d]=1, d^{a}=d^{41}, d^{b}=d^{31} \rangle$
Permutation group:	Degree $14$ $\langle(2,3)(9,10)(11,12)(13,14), (13,14), (11,12), (9,11)(10,12), (4,5,6,7,8), (9,10)(11,12), (1,2,3)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 20 \\ 20 & 11 \end{array}\right), \left(\begin{array}{rr} 16 & 15 \\ 15 & 1 \end{array}\right), \left(\begin{array}{rr} 11 & 0 \\ 0 & 11 \end{array}\right), \left(\begin{array}{rr} 1 & 15 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 29 & 0 \\ 10 & 19 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 19 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/30\Z)$
Direct product:	$C_2$ $\, \times\, $ $C_5$ $\, \times\, $ $S_3$ $\, \times\, $ $D_4$
Semidirect product:	$C_{60}$ $\,\rtimes\,$ $C_2^3$	$(C_{30}:D_4)$ $\,\rtimes\,$ $C_2$	$(D_4\times C_{30})$ $\,\rtimes\,$ $C_2$	$(D_6\times C_{20})$ $\,\rtimes\,$ $C_2$	all 46
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{30}$ . $C_2^4$	$(S_3\times C_{10})$ . $C_2^3$	$(C_{10}\times D_6)$ . $C_2^2$	$D_6$ . $(C_2^2\times C_{10})$	all 12
Aut. group:	$\Aut(C_{33}:Q_8)$	$\Aut(C_{11}\times D_{12})$	$\Aut(C_6:C_{44})$

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

Homology

Abelianization:	$C_{2}^{3} \times C_{10} \simeq C_{2}^{4} \times C_{5}$
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$1$

Subgroups

There are 1124 subgroups in 472 conjugacy classes, 194 normal (30 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_{10}$	$G/Z \simeq$ $C_2\times D_6$
Commutator:	$G' \simeq$ $C_6$	$G/G' \simeq$ $C_2^3\times C_{10}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_{15}:C_2^4$
Fitting:	$\operatorname{Fit} \simeq$ $D_4\times C_{30}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{60}:C_2^3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{30}$	$G/\operatorname{soc} \simeq$ $C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times D_4$
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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Subgroup information

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Classes of subgroups up to conjugation

Order 480: $C_{60}:C_2^3$

Order 240: $C_{20}:D_6$ x 8, $C_{15}:C_2^4$ x 2, $C_{30}:D_4$ x 2, $D_4\times C_{30}$, $C_{10}\times D_{12}$, $D_6\times C_{20}$

Order 160: $C_{20}:C_2^3$

Order 120: $C_{10}\times D_6$ x 19, $C_{15}:D_4$ x 8, $D_4\times C_{15}$ x 4, $C_5\times D_{12}$ x 4, $S_3\times C_{20}$ x 4, $C_2^2\times C_{30}$ x 2, $C_2\times C_{60}$, $C_6:C_{20}$

Order 96: $D_4\times D_6$

Order 80: $D_4\times C_{10}$ x 12, $C_2^3\times C_{10}$ x 2, $C_2^2\times C_{20}$

Order 60: $S_3\times C_{10}$ x 30, $C_2\times C_{30}$ x 9, $C_{60}$ x 2, $C_3:C_{20}$ x 2

Order 48: $S_3\times D_4$ x 8, $C_2^2\times D_6$ x 2, $C_6:D_4$ x 2, $C_6\times D_4$, $C_2\times D_{12}$, $C_4\times D_6$

Order 40: $C_2^2\times C_{10}$ x 21, $C_5\times D_4$ x 16, $C_2\times C_{20}$ x 6

Order 32: $C_2^2\times D_4$

Order 30: $C_5\times S_3$ x 8, $C_{30}$ x 7

Order 24: $C_2\times D_6$ x 19, $C_3:D_4$ x 8, $D_{12}$ x 4, $C_4\times S_3$ x 4, $C_3\times D_4$ x 4, $C_2^2\times C_6$ x 2, $C_2\times C_{12}$, $C_6:C_4$

Order 20: $C_2\times C_{10}$ x 39, $C_{20}$ x 4

Order 16: $C_2\times D_4$ x 12, $C_2^4$ x 2, $C_2^2\times C_4$

Order 15: $C_{15}$

Order 12: $D_6$ x 30, $C_2\times C_6$ x 9, $C_{12}$ x 2, $C_3:C_4$ x 2

Order 10: $C_{10}$ x 15

Order 8: $C_2^3$ x 21, $D_4$ x 16, $C_2\times C_4$ x 6

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

Order 5: $C_5$

Order 4: $C_2^2$ x 39, $C_4$ x 4

Order 3: $C_3$

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 480: $C_{60}:C_2^3$

Order 240: $C_{15}:C_2^4$, $D_4\times C_{30}$, $C_{30}:D_4$, $C_{20}:D_6$, $C_{10}\times D_{12}$, $D_6\times C_{20}$

Order 160: $C_{20}:C_2^3$

Order 120: $C_{10}\times D_6$ x 4, $C_2^2\times C_{30}$, $D_4\times C_{15}$, $C_2\times C_{60}$, $C_{15}:D_4$, $C_6:C_{20}$, $C_5\times D_{12}$, $S_3\times C_{20}$

Order 96: $D_4\times D_6$

Order 80: $D_4\times C_{10}$ x 4, $C_2^3\times C_{10}$, $C_2^2\times C_{20}$

Order 60: $S_3\times C_{10}$ x 5, $C_2\times C_{30}$ x 3, $C_{60}$, $C_3:C_{20}$

Order 48: $C_2^2\times D_6$, $C_6\times D_4$, $C_6:D_4$, $S_3\times D_4$, $C_2\times D_{12}$, $C_4\times D_6$

Order 40: $C_2^2\times C_{10}$ x 5, $C_2\times C_{20}$ x 3, $C_5\times D_4$ x 3

Order 32: $C_2^2\times D_4$

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

Order 24: $C_2\times D_6$ x 4, $C_2\times C_{12}$, $C_3:D_4$, $C_6:C_4$, $D_{12}$, $C_4\times S_3$, $C_2^2\times C_6$, $C_3\times D_4$

Order 20: $C_2\times C_{10}$ x 8, $C_{20}$ x 2

Order 16: $C_2\times D_4$ x 4, $C_2^4$, $C_2^2\times C_4$

Order 15: $C_{15}$

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

Order 10: $C_{10}$ x 5

Order 8: $C_2^3$ x 5, $D_4$ x 3, $C_2\times C_4$ x 3

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

Order 5: $C_5$

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

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 480: $C_{60}:C_2^3$ ($C_1$)

Order 240: $C_{20}:D_6$ ($C_2$) x 8, $C_{15}:C_2^4$ ($C_2$) x 2, $C_{30}:D_4$ ($C_2$) x 2, $D_4\times C_{30}$ ($C_2$), $C_{10}\times D_{12}$ ($C_2$), $D_6\times C_{20}$ ($C_2$)

Order 120: $C_{10}\times D_6$ ($C_2^2$) x 11, $C_{15}:D_4$ ($C_2^2$) x 8, $D_4\times C_{15}$ ($C_2^2$) x 4, $C_5\times D_{12}$ ($C_2^2$) x 4, $S_3\times C_{20}$ ($C_2^2$) x 4, $C_2^2\times C_{30}$ ($C_2^2$) x 2, $C_2\times C_{60}$ ($C_2^2$), $C_6:C_{20}$ ($C_2^2$)

Order 96: $D_4\times D_6$ ($C_5$)

Order 80: $D_4\times C_{10}$ ($S_3$)

Order 60: $S_3\times C_{10}$ ($C_2^3$) x 6, $C_2\times C_{30}$ ($C_2^3$) x 5, $S_3\times C_{10}$ ($D_4$) x 4, $C_{60}$ ($C_2^3$) x 2, $C_3:C_{20}$ ($C_2^3$) x 2

Order 48: $S_3\times D_4$ ($C_{10}$) x 8, $C_2^2\times D_6$ ($C_{10}$) x 2, $C_6:D_4$ ($C_{10}$) x 2, $C_6\times D_4$ ($C_{10}$), $C_2\times D_{12}$ ($C_{10}$), $C_4\times D_6$ ($C_{10}$)

Order 40: $C_5\times D_4$ ($D_6$) x 4, $C_2^2\times C_{10}$ ($D_6$) x 2, $C_2\times C_{20}$ ($D_6$)

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

Order 24: $C_2\times D_6$ ($C_2\times C_{10}$) x 11, $C_3:D_4$ ($C_2\times C_{10}$) x 8, $D_{12}$ ($C_2\times C_{10}$) x 4, $C_4\times S_3$ ($C_2\times C_{10}$) x 4, $C_3\times D_4$ ($C_2\times C_{10}$) x 4, $C_2^2\times C_6$ ($C_2\times C_{10}$) x 2, $C_2\times C_{12}$ ($C_2\times C_{10}$), $C_6:C_4$ ($C_2\times C_{10}$)

Order 20: $C_2\times C_{10}$ ($C_2\times D_6$) x 5, $C_{20}$ ($C_2\times D_6$) x 2

Order 16: $C_2\times D_4$ ($C_5\times S_3$)

Order 15: $C_{15}$ ($C_2^2\times D_4$)

Order 12: $D_6$ ($C_2^2\times C_{10}$) x 6, $C_2\times C_6$ ($C_2^2\times C_{10}$) x 5, $D_6$ ($C_5\times D_4$) x 4, $C_{12}$ ($C_2^2\times C_{10}$) x 2, $C_3:C_4$ ($C_2^2\times C_{10}$) x 2

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

Order 8: $D_4$ ($S_3\times C_{10}$) x 4, $C_2^3$ ($S_3\times C_{10}$) x 2, $C_2\times C_4$ ($S_3\times C_{10}$)

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

Order 5: $C_5$ ($D_4\times D_6$)

Order 4: $C_2^2$ ($C_{10}\times D_6$) x 5, $C_4$ ($C_{10}\times D_6$) x 2

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

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

Order 1: $C_1$ ($C_{60}:C_2^3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 480: $C_{60}:C_2^3$ ($C_1$)

Order 240: $C_{15}:C_2^4$ ($C_2$), $D_4\times C_{30}$ ($C_2$), $C_{30}:D_4$ ($C_2$), $C_{20}:D_6$ ($C_2$), $C_{10}\times D_{12}$ ($C_2$), $D_6\times C_{20}$ ($C_2$)

Order 120: $C_{10}\times D_6$ ($C_2^2$) x 3, $C_2^2\times C_{30}$ ($C_2^2$), $D_4\times C_{15}$ ($C_2^2$), $C_2\times C_{60}$ ($C_2^2$), $C_{15}:D_4$ ($C_2^2$), $C_6:C_{20}$ ($C_2^2$), $C_5\times D_{12}$ ($C_2^2$), $S_3\times C_{20}$ ($C_2^2$)

Order 96: $D_4\times D_6$ ($C_5$)

Order 80: $D_4\times C_{10}$ ($S_3$)

Order 60: $C_2\times C_{30}$ ($C_2^3$) x 2, $S_3\times C_{10}$ ($C_2^3$) x 2, $C_{60}$ ($C_2^3$), $S_3\times C_{10}$ ($D_4$), $C_3:C_{20}$ ($C_2^3$)

Order 48: $C_2^2\times D_6$ ($C_{10}$), $C_6\times D_4$ ($C_{10}$), $C_6:D_4$ ($C_{10}$), $S_3\times D_4$ ($C_{10}$), $C_2\times D_{12}$ ($C_{10}$), $C_4\times D_6$ ($C_{10}$)

Order 40: $C_2\times C_{20}$ ($D_6$), $C_2^2\times C_{10}$ ($D_6$), $C_5\times D_4$ ($D_6$)

Order 30: $C_{30}$ ($C_2^4$), $C_{30}$ ($C_2\times D_4$), $C_5\times S_3$ ($C_2\times D_4$)

Order 24: $C_2\times D_6$ ($C_2\times C_{10}$) x 3, $C_2\times C_{12}$ ($C_2\times C_{10}$), $C_3:D_4$ ($C_2\times C_{10}$), $C_6:C_4$ ($C_2\times C_{10}$), $D_{12}$ ($C_2\times C_{10}$), $C_4\times S_3$ ($C_2\times C_{10}$), $C_2^2\times C_6$ ($C_2\times C_{10}$), $C_3\times D_4$ ($C_2\times C_{10}$)

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

Order 16: $C_2\times D_4$ ($C_5\times S_3$)

Order 15: $C_{15}$ ($C_2^2\times D_4$)

Order 12: $C_2\times C_6$ ($C_2^2\times C_{10}$) x 2, $D_6$ ($C_2^2\times C_{10}$) x 2, $D_6$ ($C_5\times D_4$), $C_{12}$ ($C_2^2\times C_{10}$), $C_3:C_4$ ($C_2^2\times C_{10}$)

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

Order 8: $C_2^3$ ($S_3\times C_{10}$), $D_4$ ($S_3\times C_{10}$), $C_2\times C_4$ ($S_3\times C_{10}$)

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

Order 5: $C_5$ ($D_4\times D_6$)

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

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

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

Order 1: $C_1$ ($C_{60}:C_2^3$)

Series

Derived series

$C_{60}:C_2^3$

$\rhd$

$C_6$

$\rhd$

$C_1$

Chief series

$C_{60}:C_2^3$

$\rhd$

$C_{15}:C_2^4$

$\rhd$

$C_{10}\times D_6$

$\rhd$

$S_3\times C_{10}$

$\rhd$

$C_{30}$

$\rhd$

$C_{15}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_{60}:C_2^3$

$\rhd$

$C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2\times C_{10}$

$\lhd$

$D_4\times C_{10}$

Supergroups

This group is a maximal subgroup of 97 larger groups in the database.

This group is a maximal quotient of 115 larger groups in the database.

Character theory

Complex character table

See the $150 \times 150$ 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 $60 \times 60$ rational character table.