Abstract group 320.850: $C_2^4.D_{10}$

• Label: 320.850
• Order: \( 2^{6} \cdot 5 \)
• Exponent: \( 2^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{9} \)
• Perm deg.: $17$
• Trans deg.: $160$
• Rank: $3$

Group information

Description:	$C_2^4.D_{10}$
Order:	\(320\)\(\medspace = 2^{6} \cdot 5 \)
Exponent:	\(20\)\(\medspace = 2^{2} \cdot 5 \)
Automorphism group:	$C_5:(C_2\times C_4\times C_2^7.C_2^2)$, of order \(20480\)\(\medspace = 2^{12} \cdot 5 \)
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	63	128	4	92	32	320
Conjugacy classes	1	13	8	2	30	8	62
Divisions	1	13	7	1	11	2	35
Autjugacy classes	1	7	3	1	6	1	19

Dimension	1	2	4	8
Irr. complex chars.	8	46	8	0	62
Irr. rational chars.	8	12	5	10	35

Minimal presentations

Permutation degree:	$17$
Transitive degree:	$160$
Rank:	$3$
Inequivalent generating triples:	$504$

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:	$\langle a, b, c, d \mid a^{2}=b^{4}=c^{2}=d^{20}=[a,c]=[b,c]=[c,d]=1, b^{a}=b^{3}, d^{a}=d^{11}, d^{b}=cd^{9} \rangle$
Permutation group:	Degree $17$ $\langle(2,3)(4,5)(6,7)(8,9)(11,14)(15,17), (6,8)(10,11)(12,14)(13,15)(16,17), (7,9) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 30 \\ 30 & 1 \end{array}\right), \left(\begin{array}{rr} 31 & 12 \\ 30 & 39 \end{array}\right), \left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 21 & 10 \\ 20 & 11 \end{array}\right), \left(\begin{array}{rr} 21 & 0 \\ 0 & 21 \end{array}\right), \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 21 & 20 \\ 20 & 21 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/40\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_2\times D_{10})$ $\,\rtimes\,$ $D_4$	$(C_2\times C_{20})$ $\,\rtimes\,$ $D_4$	$(C_{10}:C_4)$ $\,\rtimes\,$ $D_4$	$(C_2^2\times D_4)$ $\,\rtimes\,$ $D_5$	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^4$ . $D_{10}$	$C_2$ . $(D_{10}:D_4)$	$C_{10}$ . $(C_4:D_4)$	$C_{10}$ . $(C_4:D_4)$ (2)	all 22

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$2$

Subgroups

There are 1270 subgroups in 322 conjugacy classes, 69 normal (25 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^3$	$G/Z \simeq$ $C_2\times D_{10}$
Commutator:	$G' \simeq$ $C_2^2\times C_{10}$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2^3$	$G/\Phi \simeq$ $C_2\times D_{10}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{20}:C_2^3$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_2^4.D_{10}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_{10}$	$G/\operatorname{soc} \simeq$ $C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3:D_4$
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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Classes of subgroups up to conjugation

Order 320: $C_2^4.D_{10}$

Order 160: $C_2^3:D_{10}$ x 2, $C_2^3.D_{10}$ x 2, $C_{10}.C_4^2$, $C_{20}:C_2^3$, $C_2^2.D_{20}$

Order 80: $C_{10}:D_4$ x 12, $D_4\times C_{10}$ x 6, $C_{10}.D_4$ x 4, $C_2^2.D_{10}$ x 3, $C_2^3\times C_{10}$ x 2, $D_{10}:C_4$ x 2, $C_2^2\times D_{10}$, $C_2^2\times C_{20}$

Order 64: $C_2^3:D_4$

Order 40: $C_2^2\times C_{10}$ x 17, $C_5:D_4$ x 16, $C_{10}:C_4$ x 11, $C_2\times D_{10}$ x 8, $C_5\times D_4$ x 8, $C_2\times C_{20}$ x 4

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

Order 20: $C_2\times C_{10}$ x 27, $D_{10}$ x 10, $C_5:C_4$ x 5, $C_{20}$ x 2

Order 16: $C_2\times D_4$ x 18, $C_2^2:C_4$ x 6, $C_2^2\times C_4$ x 4, $C_2^4$ x 3

Order 10: $C_{10}$ x 11, $D_5$ x 2

Order 8: $C_2^3$ x 25, $D_4$ x 24, $C_2\times C_4$ x 15

Order 5: $C_5$

Order 4: $C_2^2$ x 37, $C_4$ x 7

Order 2: $C_2$ x 13

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 320: $C_2^4.D_{10}$

Order 160: $C_{10}.C_4^2$, $C_{20}:C_2^3$, $C_2^3:D_{10}$, $C_2^3.D_{10}$, $C_2^2.D_{20}$

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

Order 64: $C_2^3:D_4$

Order 40: $C_2^2\times C_{10}$ x 8, $C_{10}:C_4$ x 5, $C_2\times D_{10}$ x 5, $C_2\times C_{20}$ x 2, $C_5:D_4$, $C_5\times D_4$

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

Order 20: $C_2\times C_{10}$ x 12, $D_{10}$ x 5, $C_5:C_4$ x 2, $C_{20}$

Order 16: $C_2\times D_4$ x 5, $C_2^2\times C_4$ x 3, $C_2^2:C_4$ x 2, $C_2^4$ x 2

Order 10: $C_{10}$ x 6, $D_5$

Order 8: $C_2^3$ x 13, $C_2\times C_4$ x 7, $D_4$ x 2

Order 5: $C_5$

Order 4: $C_2^2$ x 17, $C_4$ x 3

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 320: $C_2^4.D_{10}$ ($C_1$)

Order 160: $C_2^3:D_{10}$ ($C_2$) x 2, $C_2^3.D_{10}$ ($C_2$) x 2, $C_{10}.C_4^2$ ($C_2$), $C_{20}:C_2^3$ ($C_2$), $C_2^2.D_{20}$ ($C_2$)

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

Order 40: $C_{10}:C_4$ ($D_4$) x 4, $C_2^2\times C_{10}$ ($D_4$) x 4, $C_2\times C_{20}$ ($D_4$) x 2, $C_2\times D_{10}$ ($D_4$) x 2, $C_2^2\times C_{10}$ ($C_2^3$)

Order 32: $C_2^2\times D_4$ ($D_5$)

Order 20: $C_2\times C_{10}$ ($C_2\times D_4$) x 6, $C_2\times C_{10}$ ($D_4:C_2$)

Order 16: $C_2^4$ ($D_{10}$) x 2, $C_2^2\times C_4$ ($D_{10}$)

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

Order 8: $C_2^3$ ($C_5:D_4$) x 4, $C_2\times C_4$ ($C_5:D_4$) x 2, $C_2^3$ ($C_2\times D_{10}$)

Order 5: $C_5$ ($C_2^3:D_4$)

Order 4: $C_2^2$ ($C_{10}:D_4$) x 3, $C_2^2$ ($D_4\times D_5$) x 3, $C_2^2$ ($D_4:D_5$)

Order 2: $C_2$ ($C_2^3.D_{10}$) x 2, $C_2$ ($D_{10}:D_4$) x 2, $C_2$ ($C_2^4:D_5$), $C_2$ ($C_{20}:D_4$), $C_2$ ($C_{20}:D_4$)

Order 1: $C_1$ ($C_2^4.D_{10}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 320: $C_2^4.D_{10}$ ($C_1$)

Order 160: $C_{10}.C_4^2$ ($C_2$), $C_{20}:C_2^3$ ($C_2$), $C_2^3:D_{10}$ ($C_2$), $C_2^3.D_{10}$ ($C_2$), $C_2^2.D_{20}$ ($C_2$)

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

Order 40: $C_2\times C_{20}$ ($D_4$), $C_{10}:C_4$ ($D_4$), $C_2^2\times C_{10}$ ($C_2^3$), $C_2^2\times C_{10}$ ($D_4$), $C_2\times D_{10}$ ($D_4$)

Order 32: $C_2^2\times D_4$ ($D_5$)

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

Order 16: $C_2^4$ ($D_{10}$), $C_2^2\times C_4$ ($D_{10}$)

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

Order 8: $C_2^3$ ($C_5:D_4$), $C_2^3$ ($C_2\times D_{10}$), $C_2\times C_4$ ($C_5:D_4$)

Order 5: $C_5$ ($C_2^3:D_4$)

Order 4: $C_2^2$ ($C_{10}:D_4$) x 2, $C_2^2$ ($D_4\times D_5$) x 2, $C_2^2$ ($D_4:D_5$)

Order 2: $C_2$ ($C_2^4:D_5$), $C_2$ ($C_{20}:D_4$), $C_2$ ($C_2^3.D_{10}$), $C_2$ ($C_{20}:D_4$), $C_2$ ($D_{10}:D_4$)

Order 1: $C_1$ ($C_2^4.D_{10}$)

Series

Derived series

$C_2^4.D_{10}$

$\rhd$

$C_2^2\times C_{10}$

$\rhd$

$C_1$

Chief series

$C_2^4.D_{10}$

$\rhd$

$C_2^3:D_{10}$

$\rhd$

$C_2^3\times C_{10}$

$\rhd$

$C_2^2\times C_{10}$

$\rhd$

$C_2\times C_{10}$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_2^4.D_{10}$

$\rhd$

$C_2^2\times C_{10}$

$\rhd$

$C_5$

Upper central series

$C_1$

$\lhd$

$C_2^3$

$\lhd$

$C_2^2\times D_4$

Supergroups

This group is a maximal subgroup of 13 larger groups in the database.

This group is a maximal quotient of 13 larger groups in the database.

Character theory

Complex character table

See the $62 \times 62$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $35 \times 35$ rational character table.