Abstract group 320.1253: $C_4^2.D_{10}$

• Label: 320.1253
• Order: \( 2^{6} \cdot 5 \)
• Exponent: \( 2^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $21$
• Trans deg.: $160$
• Rank: $4$

Group information

Description:	$C_4^2.D_{10}$
Order:	\(320\)\(\medspace = 2^{6} \cdot 5 \)
Exponent:	\(20\)\(\medspace = 2^{2} \cdot 5 \)
Automorphism group:	Group of order \(10240\)\(\medspace = 2^{11} \cdot 5 \)
Composition factors:	$C_2$ x 6, $C_5$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.

Group statistics

Order	1	2	4	5	10	20
Elements	1	43	148	4	12	112	320
Conjugacy classes	1	6	18	2	6	32	65
Divisions	1	6	14	1	3	8	33
Autjugacy classes	1	5	9	1	3	5	24

Dimension	1	2	4	16
Irr. complex chars.	16	40	9	0	65
Irr. rational chars.	16	0	13	4	33

Minimal presentations

Permutation degree:	$21$
Transitive degree:	$160$
Rank:	$4$
Inequivalent generating quadruples:	$312480$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	4	8	12

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=c^{4}=d^{20}=[a,b]=[a,c]=[b,d]=[c,d]=1, b^{2}=c^{2}d^{10}, d^{a}=c^{2}d^{19}, c^{b}=c^{3} \rangle$
Permutation group:	Degree $21$ $\langle(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(18,19)(20,21), (1,3,2,5) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_4\times Q_8)$ $\,\rtimes\,$ $D_5$	$(D_{10}:Q_8)$ $\,\rtimes\,$ $C_2$ (2)	$(Q_8\times C_{20})$ $\,\rtimes\,$ $C_2$	$(C_{20}.Q_8)$ $\,\rtimes\,$ $C_2$	all 10
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_4^2$ . $D_{10}$ (3)	$(C_2\times Q_8)$ . $D_{10}$	$C_2$ . $(Q_8.D_{10})$	$(D_{10}.D_4)$ . $C_2$ (2)	all 28

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

Homology

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

Subgroups

There are 694 subgroups in 214 conjugacy classes, 97 normal (43 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_2^2\times D_{10}$
Commutator:	$G' \simeq$ $C_2\times C_{10}$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_2^2\times D_{10}$
Fitting:	$\operatorname{Fit} \simeq$ $Q_8\times C_{20}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_4^2.D_{10}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{10}$	$G/\operatorname{soc} \simeq$ $C_2^4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2.C_2^2$
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

Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

Order 320: $C_4^2.D_{10}$

Order 160: $C_4^2:D_5$ x 2, $C_4^2:D_5$ x 2, $D_{10}:Q_8$ x 2, $D_{10}.D_4$ x 2, $(C_2\times C_4).D_{10}$ x 2, $C_4\times D_{20}$, $C_{20}.Q_8$, $Q_8\times C_{20}$, $(C_4\times D_5):C_4$, $D_{10}.D_4$

Order 80: $C_{10}.D_4$ x 10, $D_{10}:C_4$ x 8, $C_4\times D_{10}$ x 4, $C_4:C_{20}$ x 3, $C_4\times C_{20}$ x 3, $C_{20}:C_4$ x 3, $C_{20}:C_4$ x 2, $Q_8\times C_{10}$, $C_2\times D_{20}$

Order 64: $C_4^2.C_2^2$

Order 40: $C_4\times D_5$ x 8, $C_2\times C_{20}$ x 7, $C_{10}:C_4$ x 6, $D_{20}$ x 2, $C_2\times D_{10}$ x 2, $C_5\times Q_8$ x 2

Order 32: $C_4.Q_8$ x 3, $C_4^2:C_2$ x 3, $C_4^2:C_2$ x 2, $C_2^2.D_4$ x 2, $C_2^2:Q_8$ x 2, $C_4\times Q_8$, $C_4\times D_4$, $C_2^2.D_4$

Order 20: $C_{20}$ x 8, $D_{10}$ x 7, $C_5:C_4$ x 6, $C_2\times C_{10}$

Order 16: $C_4:C_4$ x 16, $C_2^2:C_4$ x 8, $C_4^2$ x 5, $C_2^2\times C_4$ x 4, $C_2\times Q_8$, $C_2\times D_4$

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

Order 8: $C_2\times C_4$ x 21, $C_2^3$ x 2, $Q_8$ x 2, $D_4$ x 2

Order 5: $C_5$

Order 4: $C_4$ x 14, $C_2^2$ x 8

Order 2: $C_2$ x 6

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 320: $C_4^2.D_{10}$

Order 160: $D_{10}:Q_8$ x 2, $C_4^2:D_5$, $C_4\times D_{20}$, $C_4^2:D_5$, $C_{20}.Q_8$, $Q_8\times C_{20}$, $D_{10}.D_4$, $(C_4\times D_5):C_4$, $D_{10}.D_4$, $(C_2\times C_4).D_{10}$

Order 80: $D_{10}:C_4$ x 5, $C_{10}.D_4$ x 5, $C_4\times D_{10}$ x 3, $C_{20}:C_4$ x 3, $C_4:C_{20}$ x 2, $C_4\times C_{20}$ x 2, $Q_8\times C_{10}$, $C_2\times D_{20}$, $C_{20}:C_4$

Order 64: $C_4^2.C_2^2$

Order 40: $C_2\times C_{20}$ x 5, $C_{10}:C_4$ x 4, $C_4\times D_5$ x 3, $C_2\times D_{10}$ x 2, $D_{20}$, $C_5\times Q_8$

Order 32: $C_4.Q_8$ x 2, $C_2^2:Q_8$ x 2, $C_4^2:C_2$ x 2, $C_4^2:C_2$, $C_2^2.D_4$, $C_4\times Q_8$, $C_4\times D_4$, $C_2^2.D_4$

Order 20: $D_{10}$ x 6, $C_{20}$ x 5, $C_5:C_4$ x 4, $C_2\times C_{10}$

Order 16: $C_4:C_4$ x 10, $C_2^2:C_4$ x 5, $C_4^2$ x 3, $C_2^2\times C_4$ x 3, $C_2\times Q_8$, $C_2\times D_4$

Order 10: $C_{10}$ x 3, $D_5$ x 2

Order 8: $C_2\times C_4$ x 12, $C_2^3$ x 2, $Q_8$, $D_4$

Order 5: $C_5$

Order 4: $C_4$ x 9, $C_2^2$ x 7

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 320: $C_4^2.D_{10}$ ($C_1$)

Order 160: $C_4^2:D_5$ ($C_2$) x 2, $C_4^2:D_5$ ($C_2$) x 2, $D_{10}:Q_8$ ($C_2$) x 2, $D_{10}.D_4$ ($C_2$) x 2, $(C_2\times C_4).D_{10}$ ($C_2$) x 2, $C_4\times D_{20}$ ($C_2$), $C_{20}.Q_8$ ($C_2$), $Q_8\times C_{20}$ ($C_2$), $(C_4\times D_5):C_4$ ($C_2$), $D_{10}.D_4$ ($C_2$)

Order 80: $C_{10}.D_4$ ($C_2^2$) x 10, $D_{10}:C_4$ ($C_2^2$) x 8, $C_4\times D_{10}$ ($C_2^2$) x 4, $C_4:C_{20}$ ($C_2^2$) x 3, $C_4\times C_{20}$ ($C_2^2$) x 3, $C_{20}:C_4$ ($C_2^2$) x 3, $C_{20}:C_4$ ($C_2^2$) x 2, $Q_8\times C_{10}$ ($C_2^2$), $C_2\times D_{20}$ ($C_2^2$)

Order 40: $C_2\times C_{20}$ ($C_2^3$) x 7, $C_{10}:C_4$ ($C_2^3$) x 6, $C_2\times D_{10}$ ($C_2^3$) x 2

Order 32: $C_4\times Q_8$ ($D_5$)

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

Order 16: $C_4:C_4$ ($D_{10}$) x 3, $C_4^2$ ($D_{10}$) x 3, $C_2\times Q_8$ ($D_{10}$)

Order 10: $C_{10}$ ($D_4:C_2^2$) x 2, $C_{10}$ ($C_4.C_2^3$)

Order 8: $C_2\times C_4$ ($C_2\times D_{10}$) x 7

Order 5: $C_5$ ($C_4^2.C_2^2$)

Order 4: $C_4$ ($D_{20}:C_2$) x 2, $C_2^2$ ($C_2^2\times D_{10}$)

Order 2: $C_2$ ($D_4:D_{10}$), $C_2$ ($Q_8.D_{10}$), $C_2$ ($D_{20}:C_2^2$)

Order 1: $C_1$ ($C_4^2.D_{10}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 320: $C_4^2.D_{10}$ ($C_1$)

Order 160: $D_{10}:Q_8$ ($C_2$) x 2, $C_4^2:D_5$ ($C_2$), $C_4\times D_{20}$ ($C_2$), $C_4^2:D_5$ ($C_2$), $C_{20}.Q_8$ ($C_2$), $Q_8\times C_{20}$ ($C_2$), $D_{10}.D_4$ ($C_2$), $(C_4\times D_5):C_4$ ($C_2$), $D_{10}.D_4$ ($C_2$), $(C_2\times C_4).D_{10}$ ($C_2$)

Order 80: $D_{10}:C_4$ ($C_2^2$) x 5, $C_{10}.D_4$ ($C_2^2$) x 5, $C_4\times D_{10}$ ($C_2^2$) x 3, $C_{20}:C_4$ ($C_2^2$) x 3, $C_4:C_{20}$ ($C_2^2$) x 2, $C_4\times C_{20}$ ($C_2^2$) x 2, $Q_8\times C_{10}$ ($C_2^2$), $C_2\times D_{20}$ ($C_2^2$), $C_{20}:C_4$ ($C_2^2$)

Order 40: $C_2\times C_{20}$ ($C_2^3$) x 5, $C_{10}:C_4$ ($C_2^3$) x 4, $C_2\times D_{10}$ ($C_2^3$) x 2

Order 32: $C_4\times Q_8$ ($D_5$)

Order 20: $C_2\times C_{10}$ ($C_2^4$), $D_{10}$ ($D_4:C_2$), $C_{20}$ ($D_4:C_2$)

Order 16: $C_4:C_4$ ($D_{10}$) x 2, $C_4^2$ ($D_{10}$) x 2, $C_2\times Q_8$ ($D_{10}$)

Order 10: $C_{10}$ ($D_4:C_2^2$) x 2, $C_{10}$ ($C_4.C_2^3$)

Order 8: $C_2\times C_4$ ($C_2\times D_{10}$) x 5

Order 5: $C_5$ ($C_4^2.C_2^2$)

Order 4: $C_2^2$ ($C_2^2\times D_{10}$), $C_4$ ($D_{20}:C_2$)

Order 2: $C_2$ ($D_4:D_{10}$), $C_2$ ($Q_8.D_{10}$), $C_2$ ($D_{20}:C_2^2$)

Order 1: $C_1$ ($C_4^2.D_{10}$)

Series

Derived series

$C_4^2.D_{10}$

$\rhd$

$C_2\times C_{10}$

$\rhd$

$C_1$

Chief series

$C_4^2.D_{10}$

$\rhd$

$D_{10}.D_4$

$\rhd$

$C_4\times D_{10}$

$\rhd$

$C_2\times D_{10}$

$\rhd$

$C_2\times C_{10}$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_4^2.D_{10}$

$\rhd$

$C_2\times C_{10}$

$\rhd$

$C_5$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_4\times Q_8$

Supergroups

This group is a maximal subgroup of 25 larger groups in the database.

This group is a maximal quotient of 25 larger groups in the database.

Character theory

Complex character table

See the $65 \times 65$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $33 \times 33$ rational character table.