Abstract group 480.919: $C_2\times C_4\times C_{60}$

• Label: 480.919
• Order: \( 2^{5} \cdot 3 \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3 \)
• Perm deg.: $18$
• Trans deg.: $480$
• Rank: $3$

Group information

Description:	$C_2\times C_4\times C_{60}$
Order:	\(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$C_2\times C_4\times C_2^6.S_4$, of order \(12288\)\(\medspace = 2^{12} \cdot 3 \)
Composition factors:	$C_2$ x 5, $C_3$, $C_5$
Nilpotency class:	$1$
Derived length:	$1$

This group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and elementary for $p = 2$ (hence hyperelementary).

Group statistics

Order	1	2	3	4	5	6	10	12	15	20	30	60
Elements	1	7	2	24	4	14	28	48	8	96	56	192	480
Conjugacy classes	1	7	2	24	4	14	28	48	8	96	56	192	480
Divisions	1	7	1	12	1	7	7	12	1	12	7	12	80
Autjugacy classes	1	2	1	1	1	2	2	1	1	1	2	1	16

Dimension	1	2	4	8	16
Irr. complex chars.	480	0	0	0	0	480
Irr. rational chars.	8	20	20	20	12	80

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$480$
Rank:	$3$
Inequivalent generating triples:	$2821$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	3	not computed	not computed

Constructions

Presentation:	Abelian group $\langle a, b, c \mid a^{2}=b^{4}=c^{60}=1 \rangle$
Permutation group:	Degree $18$ $\langle(3,6,4,5), (7,10,8,9), (1,2), (11,13,12), (14,18,17,16,15), (3,4)(5,6), (7,8)(9,10)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 17 & 0 \\ 0 & 17 \end{array}\right), \left(\begin{array}{rr} 29 & 30 \\ 10 & 39 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 11 & 15 \\ 5 & 36 \end{array}\right), \left(\begin{array}{rr} 21 & 0 \\ 0 & 21 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right), \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/40\Z)$
Direct product:	$C_2$ $\, \times\, $ $C_4$ ${}^2$ $\, \times\, $ $C_3$ $\, \times\, $ $C_5$
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2^2\times C_{60})$ . $C_2$	$(C_2\times C_{60})$ . $C_2^2$	$(C_2^2\times C_{20})$ . $C_6$	$(C_2\times C_{30})$ . $C_2^3$	all 24
Aut. group:	$\Aut(C_{715})$	$\Aut(C_{915})$	$\Aut(C_{975})$	$\Aut(C_{976})$	all 9

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

Homology

Primary decomposition:	$C_{2} \times C_{4}^{2} \times C_{3} \times C_{5}$
Schur multiplier:	$C_{2}^{2} \times C_{4}$
Commutator length:	$0$

Subgroups

There are 216 subgroups, all normal (16 characteristic).

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

Special subgroups

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

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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Classes of subgroups up to conjugation

Order 480: $C_2\times C_4\times C_{60}$

Order 240: $C_4\times C_{60}$ x 4, $C_2^2\times C_{60}$ x 3

Order 160: $C_2\times C_4\times C_{20}$

Order 120: $C_2\times C_{60}$ x 18, $C_2^2\times C_{30}$

Order 96: $C_2\times C_4\times C_{12}$

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

Order 60: $C_{60}$ x 12, $C_2\times C_{30}$ x 7

Order 48: $C_4\times C_{12}$ x 4, $C_2^2\times C_{12}$ x 3

Order 40: $C_2\times C_{20}$ x 18, $C_2^2\times C_{10}$

Order 32: $C_2\times C_4^2$

Order 30: $C_{30}$ x 7

Order 24: $C_2\times C_{12}$ x 18, $C_2^2\times C_6$

Order 20: $C_{20}$ x 12, $C_2\times C_{10}$ x 7

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

Order 15: $C_{15}$

Order 12: $C_{12}$ x 12, $C_2\times C_6$ x 7

Order 10: $C_{10}$ x 7

Order 8: $C_2\times C_4$ x 18, $C_2^3$

Order 6: $C_6$ x 7

Order 5: $C_5$

Order 4: $C_4$ x 12, $C_2^2$ x 7

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 480: $C_2\times C_4\times C_{60}$

Order 240: $C_4\times C_{60}$, $C_2^2\times C_{60}$

Order 160: $C_2\times C_4\times C_{20}$

Order 120: $C_2\times C_{60}$ x 2, $C_2^2\times C_{30}$

Order 96: $C_2\times C_4\times C_{12}$

Order 80: $C_2^2\times C_{20}$, $C_4\times C_{20}$

Order 60: $C_2\times C_{30}$ x 2, $C_{60}$

Order 48: $C_2^2\times C_{12}$, $C_4\times C_{12}$

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

Order 32: $C_2\times C_4^2$

Order 30: $C_{30}$ x 2

Order 24: $C_2\times C_{12}$ x 2, $C_2^2\times C_6$

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

Order 16: $C_4^2$, $C_2^2\times C_4$

Order 15: $C_{15}$

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

Order 10: $C_{10}$ x 2

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

Order 6: $C_6$ x 2

Order 5: $C_5$

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

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 480: $C_2\times C_4\times C_{60}$ ($C_1$)

Order 240: $C_4\times C_{60}$ ($C_2$) x 4, $C_2^2\times C_{60}$ ($C_2$) x 3

Order 160: $C_2\times C_4\times C_{20}$ ($C_3$)

Order 120: $C_2\times C_{60}$ ($C_4$) x 12, $C_2\times C_{60}$ ($C_2^2$) x 6, $C_2^2\times C_{30}$ ($C_2^2$)

Order 96: $C_2\times C_4\times C_{12}$ ($C_5$)

Order 80: $C_4\times C_{20}$ ($C_6$) x 4, $C_2^2\times C_{20}$ ($C_6$) x 3

Order 60: $C_{60}$ ($C_2\times C_4$) x 12, $C_2\times C_{30}$ ($C_2\times C_4$) x 6, $C_2\times C_{30}$ ($C_2^3$)

Order 48: $C_4\times C_{12}$ ($C_{10}$) x 4, $C_2^2\times C_{12}$ ($C_{10}$) x 3

Order 40: $C_2\times C_{20}$ ($C_{12}$) x 12, $C_2\times C_{20}$ ($C_2\times C_6$) x 6, $C_2^2\times C_{10}$ ($C_2\times C_6$)

Order 32: $C_2\times C_4^2$ ($C_{15}$)

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

Order 24: $C_2\times C_{12}$ ($C_{20}$) x 12, $C_2\times C_{12}$ ($C_2\times C_{10}$) x 6, $C_2^2\times C_6$ ($C_2\times C_{10}$)

Order 20: $C_{20}$ ($C_2\times C_{12}$) x 12, $C_2\times C_{10}$ ($C_2\times C_{12}$) x 6, $C_2\times C_{10}$ ($C_2^2\times C_6$)

Order 16: $C_4^2$ ($C_{30}$) x 4, $C_2^2\times C_4$ ($C_{30}$) x 3

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

Order 12: $C_{12}$ ($C_2\times C_{20}$) x 12, $C_2\times C_6$ ($C_2\times C_{20}$) x 6, $C_2\times C_6$ ($C_2^2\times C_{10}$)

Order 10: $C_{10}$ ($C_4\times C_{12}$) x 4, $C_{10}$ ($C_2^2\times C_{12}$) x 3

Order 8: $C_2\times C_4$ ($C_{60}$) x 12, $C_2\times C_4$ ($C_2\times C_{30}$) x 6, $C_2^3$ ($C_2\times C_{30}$)

Order 6: $C_6$ ($C_4\times C_{20}$) x 4, $C_6$ ($C_2^2\times C_{20}$) x 3

Order 5: $C_5$ ($C_2\times C_4\times C_{12}$)

Order 4: $C_4$ ($C_2\times C_{60}$) x 12, $C_2^2$ ($C_2\times C_{60}$) x 6, $C_2^2$ ($C_2^2\times C_{30}$)

Order 3: $C_3$ ($C_2\times C_4\times C_{20}$)

Order 2: $C_2$ ($C_4\times C_{60}$) x 4, $C_2$ ($C_2^2\times C_{60}$) x 3

Order 1: $C_1$ ($C_2\times C_4\times C_{60}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 480: $C_2\times C_4\times C_{60}$ ($C_1$)

Order 240: $C_4\times C_{60}$ ($C_2$), $C_2^2\times C_{60}$ ($C_2$)

Order 160: $C_2\times C_4\times C_{20}$ ($C_3$)

Order 120: $C_2^2\times C_{30}$ ($C_2^2$), $C_2\times C_{60}$ ($C_2^2$), $C_2\times C_{60}$ ($C_4$)

Order 96: $C_2\times C_4\times C_{12}$ ($C_5$)

Order 80: $C_2^2\times C_{20}$ ($C_6$), $C_4\times C_{20}$ ($C_6$)

Order 60: $C_{60}$ ($C_2\times C_4$), $C_2\times C_{30}$ ($C_2^3$), $C_2\times C_{30}$ ($C_2\times C_4$)

Order 48: $C_2^2\times C_{12}$ ($C_{10}$), $C_4\times C_{12}$ ($C_{10}$)

Order 40: $C_2\times C_{20}$ ($C_2\times C_6$), $C_2\times C_{20}$ ($C_{12}$), $C_2^2\times C_{10}$ ($C_2\times C_6$)

Order 32: $C_2\times C_4^2$ ($C_{15}$)

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

Order 24: $C_2\times C_{12}$ ($C_2\times C_{10}$), $C_2\times C_{12}$ ($C_{20}$), $C_2^2\times C_6$ ($C_2\times C_{10}$)

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

Order 16: $C_4^2$ ($C_{30}$), $C_2^2\times C_4$ ($C_{30}$)

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

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

Order 10: $C_{10}$ ($C_2^2\times C_{12}$), $C_{10}$ ($C_4\times C_{12}$)

Order 8: $C_2^3$ ($C_2\times C_{30}$), $C_2\times C_4$ ($C_{60}$), $C_2\times C_4$ ($C_2\times C_{30}$)

Order 6: $C_6$ ($C_2^2\times C_{20}$), $C_6$ ($C_4\times C_{20}$)

Order 5: $C_5$ ($C_2\times C_4\times C_{12}$)

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

Order 3: $C_3$ ($C_2\times C_4\times C_{20}$)

Order 2: $C_2$ ($C_4\times C_{60}$), $C_2$ ($C_2^2\times C_{60}$)

Order 1: $C_1$ ($C_2\times C_4\times C_{60}$)

Series

Derived series

$C_2\times C_4\times C_{60}$

$\rhd$

$C_1$

Chief series

$C_2\times C_4\times C_{60}$

$\rhd$

$C_2^2\times C_{60}$

$\rhd$

$C_2^2\times C_{30}$

$\rhd$

$C_2\times C_{30}$

$\rhd$

$C_{30}$

$\rhd$

$C_{15}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_2\times C_4\times C_{60}$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2\times C_4\times C_{60}$

Supergroups

This group is a maximal subgroup of 149 larger groups in the database.

This group is a maximal quotient of 14 larger groups in the database.

Character theory

Complex character table

See the $480 \times 480$ 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 $80 \times 80$ rational character table.