Abstract group 480.1199: $C_2^3:D_{30}$

• Label: 480.1199
• Order: \( 2^{5} \cdot 3 \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{2} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3 \)
• Perm deg.: $13$
• Trans deg.: $60$
• Rank: $3$

Group information

Description:	$C_2^3:D_{30}$
Order:	\(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$F_5\times S_4^2$, of order \(11520\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5 \)
Composition factors:	$C_2$ x 5, $C_3$, $C_5$
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	5	6	10	15	30
Elements	1	135	8	120	4	24	60	32	96	480
Conjugacy classes	1	11	1	4	2	3	14	4	12	52
Divisions	1	11	1	4	1	3	7	1	3	32
Autjugacy classes	1	4	1	1	1	1	3	1	1	14

Dimension	1	2	3	4	6	8	12
Irr. complex chars.	8	28	8	0	8	0	0	52
Irr. rational chars.	8	4	8	4	0	4	4	32

Minimal presentations

Permutation degree:	$13$
Transitive degree:	$60$
Rank:	$3$
Inequivalent generating triples:	$2520$

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:	${\langle a, b, c, d, e \mid a^{2}=b^{6}=c^{10}=d^{2}=e^{2}=[a,d]=[a,e]=[b,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $13$ $\langle(2,3)(4,5)(6,7)(8,9)(11,12), (6,7)(8,9), (6,8)(7,9), (11,12,13), (1,2,4,5,3), (10,11)(12,13), (10,12)(11,13)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 11 & 0 \\ 10 & 11 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 11 & 15 \\ 5 & 16 \end{array}\right), \left(\begin{array}{rr} 14 & 13 \\ 5 & 6 \end{array}\right), \left(\begin{array}{rr} 1 & 10 \\ 10 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/20\Z)$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $(C_5:S_4)$
Semidirect product:	$C_2^4$ $\,\rtimes\,$ $D_{15}$	$C_2^3$ $\,\rtimes\,$ $D_{30}$	$(C_2\times C_{10})$ $\,\rtimes\,$ $S_4$	$C_{10}$ $\,\rtimes\,$ $(C_2\times S_4)$	all 15
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{20}\Z)$.

Homology

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

Subgroups

There are 2152 subgroups in 262 conjugacy classes, 41 normal (13 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_5:S_4$
Commutator:	$G' \simeq$ $C_5\times A_4$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2^3:D_{30}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^3\times C_{10}$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $C_2^3:D_{30}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3\times C_{10}$	$G/\operatorname{soc} \simeq$ $S_3$
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

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 480: $C_2^3:D_{30}$

Order 240: $C_{10}:S_4$ x 6, $C_2^3:C_{30}$

Order 160: $C_2^3:D_{10}$

Order 120: $C_5:S_4$ x 4, $C_{10}\times A_4$ x 3, $C_2\times D_{30}$

Order 96: $C_2^2\times S_4$

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

Order 60: $D_{30}$ x 6, $C_5\times A_4$, $C_2\times C_{30}$

Order 48: $C_2\times S_4$ x 6, $C_2^2\times A_4$

Order 40: $C_5:D_4$ x 16, $C_2\times D_{10}$ x 10, $C_2^2\times C_{10}$ x 7, $C_{10}:C_4$ x 6

Order 32: $C_2^2\times D_4$

Order 30: $D_{15}$ x 4, $C_{30}$ x 3

Order 24: $S_4$ x 4, $C_2\times A_4$ x 3, $C_2\times D_6$

Order 20: $D_{10}$ x 16, $C_2\times C_{10}$ x 12, $C_5:C_4$ 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 6, $C_2\times C_6$, $A_4$

Order 10: $C_{10}$ x 7, $D_5$ x 4

Order 8: $C_2^3$ x 17, $D_4$ x 16, $C_2\times C_4$ x 6

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

Order 5: $C_5$

Order 4: $C_2^2$ x 28, $C_4$ x 4

Order 3: $C_3$

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 480: $C_2^3:D_{30}$

Order 240: $C_2^3:C_{30}$, $C_{10}:S_4$

Order 160: $C_2^3:D_{10}$

Order 120: $C_2\times D_{30}$, $C_{10}\times A_4$, $C_5:S_4$

Order 96: $C_2^2\times S_4$

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

Order 60: $C_5\times A_4$, $C_2\times C_{30}$, $D_{30}$

Order 48: $C_2^2\times A_4$, $C_2\times S_4$

Order 40: $C_2^2\times C_{10}$ x 3, $C_2\times D_{10}$ x 3, $C_5:D_4$ x 2, $C_{10}:C_4$

Order 32: $C_2^2\times D_4$

Order 30: $C_{30}$, $D_{15}$

Order 24: $C_2\times D_6$, $C_2\times A_4$, $S_4$

Order 20: $C_2\times C_{10}$ x 6, $D_{10}$ x 3, $C_5:C_4$

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

Order 15: $C_{15}$

Order 12: $C_2\times C_6$, $D_6$, $A_4$

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

Order 8: $C_2^3$ x 6, $D_4$ x 2, $C_2\times C_4$

Order 6: $C_6$, $S_3$

Order 5: $C_5$

Order 4: $C_2^2$ x 9, $C_4$

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 480: $C_2^3:D_{30}$ ($C_1$)

Order 240: $C_{10}:S_4$ ($C_2$) x 6, $C_2^3:C_{30}$ ($C_2$)

Order 120: $C_5:S_4$ ($C_2^2$) x 4, $C_{10}\times A_4$ ($C_2^2$) x 3

Order 80: $C_2^3\times C_{10}$ ($S_3$)

Order 60: $C_5\times A_4$ ($C_2^3$)

Order 48: $C_2^2\times A_4$ ($D_5$)

Order 40: $C_2^2\times C_{10}$ ($D_6$) x 3

Order 24: $C_2\times A_4$ ($D_{10}$) x 3

Order 20: $C_2\times C_{10}$ ($C_2\times D_6$), $C_2\times C_{10}$ ($S_4$)

Order 16: $C_2^4$ ($D_{15}$)

Order 12: $A_4$ ($C_2\times D_{10}$)

Order 10: $C_{10}$ ($C_2\times S_4$) x 3

Order 8: $C_2^3$ ($D_{30}$) x 3

Order 5: $C_5$ ($C_2^2\times S_4$)

Order 4: $C_2^2$ ($C_2\times D_{30}$), $C_2^2$ ($C_5:S_4$)

Order 2: $C_2$ ($C_{10}:S_4$) x 3

Order 1: $C_1$ ($C_2^3:D_{30}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 480: $C_2^3:D_{30}$ ($C_1$)

Order 240: $C_2^3:C_{30}$ ($C_2$), $C_{10}:S_4$ ($C_2$)

Order 120: $C_{10}\times A_4$ ($C_2^2$), $C_5:S_4$ ($C_2^2$)

Order 80: $C_2^3\times C_{10}$ ($S_3$)

Order 60: $C_5\times A_4$ ($C_2^3$)

Order 48: $C_2^2\times A_4$ ($D_5$)

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

Order 24: $C_2\times A_4$ ($D_{10}$)

Order 20: $C_2\times C_{10}$ ($C_2\times D_6$), $C_2\times C_{10}$ ($S_4$)

Order 16: $C_2^4$ ($D_{15}$)

Order 12: $A_4$ ($C_2\times D_{10}$)

Order 10: $C_{10}$ ($C_2\times S_4$)

Order 8: $C_2^3$ ($D_{30}$)

Order 5: $C_5$ ($C_2^2\times S_4$)

Order 4: $C_2^2$ ($C_2\times D_{30}$), $C_2^2$ ($C_5:S_4$)

Order 2: $C_2$ ($C_{10}:S_4$)

Order 1: $C_1$ ($C_2^3:D_{30}$)

Series

Derived series

$C_2^3:D_{30}$

$\rhd$

$C_5\times A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_2^3:D_{30}$

$\rhd$

$C_{10}:S_4$

$\rhd$

$C_5:S_4$

$\rhd$

$C_5\times A_4$

$\rhd$

$C_2\times C_{10}$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_2^3:D_{30}$

$\rhd$

$C_5\times A_4$

Upper central series

$C_1$

$\lhd$

$C_2^2$

Supergroups

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

This group is a maximal quotient of 27 larger groups in the database.

Character theory

Complex character table

See the $52 \times 52$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $32 \times 32$ rational character table.