Abstract group $C_{20}.S_4$

• Label: 480.1024
• Order: \( 2^{5} \cdot 3 \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{8} \cdot 3 \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $17$
• Trans deg.: $120$
• Rank: $2$

Group information

Description:	$C_{20}.S_4$
Order:	\(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$D_4\times F_5\times S_4$, of order \(3840\)\(\medspace = 2^{8} \cdot 3 \cdot 5 \) (generators)
Outer automorphisms:	$C_2^2\times C_4$, of order \(16\)\(\medspace = 2^{4} \)
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	12	15	20	30	60
Elements	1	7	8	248	4	8	28	16	32	32	32	64	480
Conjugacy classes	1	3	1	6	2	1	6	2	4	8	4	8	46
Divisions	1	3	1	6	1	1	3	1	1	2	1	1	22
Autjugacy classes	1	3	1	4	1	1	3	1	1	2	1	1	20

Dimension	1	2	3	4	6	8	12	16	24
Irr. complex chars.	4	29	4	0	9	0	0	0	0	46
Irr. rational chars.	4	3	4	3	1	3	2	1	1	22

Minimal Presentations

Permutation degree:	$17$
Transitive degree:	$120$
Rank:	$2$
Inequivalent generating pairs:	$9$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	12	48
Arbitrary	5	7	11

Constructions

Presentation:	$\langle a, b, c, d \mid b^{12}=c^{2}=d^{10}=[a,c]=[c,d]=1, a^{2}=b^{6}, b^{a}=b^{11}, d^{a}=cd^{9}, c^{b}=d^{5}, d^{b}=cd \rangle$
Permutation group:	Degree $17$ $\langle(2,3)(4,5)(6,7,9,11)(8,12,13,10)(15,16), (6,8,9,13)(7,10,11,12), (6,9)(7,11)(8,13)(10,12), (15,16,17), (1,2,4,5,3), (14,15)(16,17), (14,16)(15,17)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 41 & 49 \\ 10 & 49 \end{array}\right), \left(\begin{array}{rr} 1 & 30 \\ 30 & 1 \end{array}\right), \left(\begin{array}{rr} 31 & 0 \\ 30 & 31 \end{array}\right), \left(\begin{array}{rr} 49 & 40 \\ 40 & 29 \end{array}\right), \left(\begin{array}{rr} 41 & 0 \\ 0 & 41 \end{array}\right), \left(\begin{array}{rr} 16 & 15 \\ 45 & 31 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/60\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_5\times A_4)$ $\,\rtimes\,$ $Q_8$	$A_4$ $\,\rtimes\,$ $(C_5:Q_8)$	$C_5$ $\,\rtimes\,$ $(A_4:Q_8)$	$C_2^2$ $\,\rtimes\,$ $(C_{15}:Q_8)$	all 5
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{20}$ . $S_4$	$C_2^3$ . $D_{30}$	$C_4$ . $(C_5:S_4)$	$(C_4\times A_4)$ . $D_5$	all 13

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

Homology

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

Subgroups

There are 588 subgroups in 84 conjugacy classes, 21 normal (19 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_{10}:S_4$
Commutator:	$G' \simeq$ $C_{10}\times A_4$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_{10}:S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times C_{20}$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $C_{20}.S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_{10}$	$G/\operatorname{soc} \simeq$ $D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2:Q_8$
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

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Subgroup information

Click on a subgroup in the diagram to see information about it.

Classes of subgroups up to conjugation

Order 480: $C_{20}.S_4$

Order 240: $C_{10}.S_4$ x 2, $A_4\times C_{20}$

Order 160: $C_{20}.D_4$

Order 120: $C_{10}\times A_4$, $C_{15}:Q_8$

Order 96: $A_4:Q_8$

Order 80: $C_{10}.D_4$ x 2, $C_{10}.D_4$ x 2, $C_2^2\times C_{20}$, $C_{10}:Q_8$, $C_{20}:C_4$

Order 60: $C_{15}:C_4$ x 2, $C_5\times A_4$, $C_{60}$

Order 48: $A_4:C_4$ x 2, $C_4\times A_4$

Order 40: $C_{10}:C_4$ x 4, $C_2\times C_{20}$ x 2, $C_5:Q_8$ x 2, $C_2^2\times C_{10}$

Order 32: $C_2^2:Q_8$

Order 30: $C_{30}$

Order 24: $C_3:Q_8$, $C_2\times A_4$

Order 20: $C_5:C_4$ x 4, $C_2\times C_{10}$ x 3, $C_{20}$ x 2

Order 16: $C_4:C_4$ x 3, $C_2^2:C_4$ x 2, $C_2\times Q_8$, $C_2^2\times C_4$

Order 15: $C_{15}$

Order 12: $C_3:C_4$ x 2, $A_4$, $C_{12}$

Order 10: $C_{10}$ x 3

Order 8: $C_2\times C_4$ x 6, $Q_8$ x 2, $C_2^3$

Order 6: $C_6$

Order 5: $C_5$

Order 4: $C_4$ x 6, $C_2^2$ x 3

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 480: $C_{20}.S_4$

Order 240: $A_4\times C_{20}$, $C_{10}.S_4$

Order 160: $C_{20}.D_4$

Order 120: $C_{10}\times A_4$, $C_{15}:Q_8$

Order 96: $A_4:Q_8$

Order 80: $C_2^2\times C_{20}$, $C_{10}:Q_8$, $C_{10}.D_4$, $C_{20}:C_4$, $C_{10}.D_4$

Order 60: $C_5\times A_4$, $C_{60}$, $C_{15}:C_4$

Order 48: $C_4\times A_4$, $A_4:C_4$

Order 40: $C_2\times C_{20}$ x 2, $C_{10}:C_4$ x 2, $C_5:Q_8$ x 2, $C_2^2\times C_{10}$

Order 32: $C_2^2:Q_8$

Order 30: $C_{30}$

Order 24: $C_3:Q_8$, $C_2\times A_4$

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

Order 16: $C_4:C_4$ x 2, $C_2^2:C_4$, $C_2\times Q_8$, $C_2^2\times C_4$

Order 15: $C_{15}$

Order 12: $A_4$, $C_{12}$, $C_3:C_4$

Order 10: $C_{10}$ x 3

Order 8: $C_2\times C_4$ x 4, $Q_8$ x 2, $C_2^3$

Order 6: $C_6$

Order 5: $C_5$

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 480: $C_{20}.S_4$ ($C_1$)

Order 240: $C_{10}.S_4$ ($C_2$) x 2, $A_4\times C_{20}$ ($C_2$)

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

Order 80: $C_2^2\times C_{20}$ ($S_3$)

Order 60: $C_5\times A_4$ ($Q_8$)

Order 48: $C_4\times A_4$ ($D_5$)

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

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

Order 20: $C_2\times C_{10}$ ($C_3:Q_8$), $C_{20}$ ($S_4$)

Order 16: $C_2^2\times C_4$ ($D_{15}$)

Order 12: $A_4$ ($C_5:Q_8$)

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

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

Order 5: $C_5$ ($A_4:Q_8$)

Order 4: $C_2^2$ ($C_{15}:Q_8$), $C_4$ ($C_5:S_4$)

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

Order 1: $C_1$ ($C_{20}.S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 480: $C_{20}.S_4$ ($C_1$)

Order 240: $A_4\times C_{20}$ ($C_2$), $C_{10}.S_4$ ($C_2$)

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

Order 80: $C_2^2\times C_{20}$ ($S_3$)

Order 60: $C_5\times A_4$ ($Q_8$)

Order 48: $C_4\times A_4$ ($D_5$)

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

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

Order 20: $C_2\times C_{10}$ ($C_3:Q_8$), $C_{20}$ ($S_4$)

Order 16: $C_2^2\times C_4$ ($D_{15}$)

Order 12: $A_4$ ($C_5:Q_8$)

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

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

Order 5: $C_5$ ($A_4:Q_8$)

Order 4: $C_2^2$ ($C_{15}:Q_8$), $C_4$ ($C_5:S_4$)

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

Order 1: $C_1$ ($C_{20}.S_4$)

Series

Derived series

$C_{20}.S_4$

$\rhd$

$C_{10}\times A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_{20}.S_4$

$\rhd$

$C_{10}.S_4$

$\rhd$

$C_{10}\times A_4$

$\rhd$

$C_5\times A_4$

$\rhd$

$C_2\times C_{10}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_{20}.S_4$

$\rhd$

$C_{10}\times A_4$

$\rhd$

$C_5\times A_4$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_4$

Supergroups

This group is a maximal subgroup of 19 larger groups in the database.

This group is a maximal quotient of 10 larger groups in the database.

Character theory

Complex character table

See the $46 \times 46$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $22 \times 22$ rational character table.