Abstract group 320.1570: $C_{40}.C_2^3$

• Label: 320.1570
• Order: \( 2^{6} \cdot 5 \)
• Exponent: \( 2^{3} \cdot 5 \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \cdot 5 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \cdot 3 \)
• Perm deg.: $21$
• Trans deg.: $80$
• Rank: $4$

Group information

Description:	$C_{40}.C_2^3$
Order:	\(320\)\(\medspace = 2^{6} \cdot 5 \)
Exponent:	\(40\)\(\medspace = 2^{3} \cdot 5 \)
Automorphism group:	$C_2^3:A_4.C_4.C_2^4$, of order \(6144\)\(\medspace = 2^{11} \cdot 3 \)
Composition factors:	$C_2$ x 6, $C_5$
Nilpotency class:	$2$
Derived length:	$2$

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

Group statistics

Order	1	2	4	5	8	10	20	40
Elements	1	15	16	4	32	60	64	128	320
Conjugacy classes	1	8	9	4	16	32	36	64	170
Divisions	1	8	8	1	8	8	8	8	50
Autjugacy classes	1	3	3	1	1	3	3	1	16

Dimension	1	2	4	8	32
Irr. complex chars.	160	0	10	0	0	170
Irr. rational chars.	16	8	16	9	1	50

Minimal presentations

Permutation degree:	$21$
Transitive degree:	$80$
Rank:	$4$
Inequivalent generating quadruples:	$524160$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	32
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{2}=c^{2}=d^{40}=[a,b]=[a,c]=[c,d]=1, d^{a}=d^{21}, c^{b}=cd^{20}, d^{b}=d^{21} \rangle$
Permutation group:	Degree $21$ $\langle(1,12,4,10,2,11,3,9)(5,16,8,14,6,15,7,13), (1,7)(2,8)(3,6)(4,5)(9,15)(10,16) \!\cdots\! \rangle$
Direct product:	$C_5$ $\, \times\, $ $(C_8.C_2^3)$
Semidirect product:	$(C_2\times C_{40})$ $\,\rtimes\,$ $C_2^2$	$(C_{10}\times \OD_{16})$ $\,\rtimes\,$ $C_2$	$(\OD_{16}:C_{10})$ $\,\rtimes\,$ $C_2$	$(C_2\times \OD_{16})$ $\,\rtimes\,$ $C_{10}$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{40}$ . $C_2^3$	$C_{20}$ . $C_2^4$	$(Q_8\times C_{10})$ . $C_4$	$(C_2\times Q_8)$ . $C_{20}$	all 34

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

Homology

Abelianization:	$C_{2}^{3} \times C_{20} \simeq C_{2}^{3} \times C_{4} \times C_{5}$
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 290 subgroups in 258 conjugacy classes, 238 normal (18 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{20}$	$G/Z \simeq$ $C_2^4$
Commutator:	$G' \simeq$ $C_2$	$G/G' \simeq$ $C_2^3\times C_{20}$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_2^3\times C_{10}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{40}.C_2^3$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_{40}.C_2^3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{10}$	$G/\operatorname{soc} \simeq$ $C_2^3\times C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8.C_2^3$
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

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 320: $C_{40}.C_2^3$

Order 160: $\OD_{16}:C_{10}$ x 8, $C_{10}\times \OD_{16}$ x 6, $C_{10}.C_2^4$

Order 80: $C_5\times \OD_{16}$ x 16, $C_2\times C_{40}$ x 12, $D_4:C_{10}$ x 8, $D_4\times C_{10}$ x 3, $C_2^2\times C_{20}$ x 3, $Q_8\times C_{10}$

Order 64: $C_8.C_2^3$

Order 40: $C_2\times C_{20}$ x 16, $C_5\times D_4$ x 12, $C_{40}$ x 8, $C_5\times Q_8$ x 4, $C_2^2\times C_{10}$ x 3

Order 32: $\OD_{16}:C_2$ x 8, $C_2\times \OD_{16}$ x 6, $D_4:C_2^2$

Order 20: $C_2\times C_{10}$ x 10, $C_{20}$ x 8

Order 16: $\OD_{16}$ x 16, $C_2\times C_8$ x 12, $D_4:C_2$ x 8, $C_2\times D_4$ x 3, $C_2^2\times C_4$ x 3, $C_2\times Q_8$

Order 10: $C_{10}$ x 8

Order 8: $C_2\times C_4$ x 16, $D_4$ x 12, $C_8$ x 8, $Q_8$ x 4, $C_2^3$ x 3

Order 5: $C_5$

Order 4: $C_2^2$ x 10, $C_4$ x 8

Order 2: $C_2$ x 8

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 320: $C_{40}.C_2^3$

Order 160: $C_{10}.C_2^4$, $\OD_{16}:C_{10}$, $C_{10}\times \OD_{16}$

Order 80: $D_4:C_{10}$ x 2, $C_5\times \OD_{16}$ x 2, $Q_8\times C_{10}$, $D_4\times C_{10}$, $C_2^2\times C_{20}$, $C_2\times C_{40}$

Order 64: $C_8.C_2^3$

Order 40: $C_2\times C_{20}$ x 4, $C_{40}$, $C_2^2\times C_{10}$, $C_5\times Q_8$, $C_5\times D_4$

Order 32: $D_4:C_2^2$, $\OD_{16}:C_2$, $C_2\times \OD_{16}$

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

Order 16: $\OD_{16}$ x 2, $D_4:C_2$ x 2, $C_2\times C_8$, $C_2\times Q_8$, $C_2\times D_4$, $C_2^2\times C_4$

Order 10: $C_{10}$ x 3

Order 8: $C_2\times C_4$ x 4, $C_2^3$, $Q_8$, $D_4$, $C_8$

Order 5: $C_5$

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 320: $C_{40}.C_2^3$ ($C_1$)

Order 160: $\OD_{16}:C_{10}$ ($C_2$) x 8, $C_{10}\times \OD_{16}$ ($C_2$) x 6, $C_{10}.C_2^4$ ($C_2$)

Order 80: $C_5\times \OD_{16}$ ($C_2^2$) x 16, $C_2\times C_{40}$ ($C_2^2$) x 12, $D_4:C_{10}$ ($C_2^2$) x 4, $D_4:C_{10}$ ($C_4$) x 4, $D_4\times C_{10}$ ($C_4$) x 3, $C_2^2\times C_{20}$ ($C_2^2$) x 3, $Q_8\times C_{10}$ ($C_4$)

Order 64: $C_8.C_2^3$ ($C_5$)

Order 40: $C_5\times D_4$ ($C_2\times C_4$) x 12, $C_2\times C_{20}$ ($C_2\times C_4$) x 9, $C_{40}$ ($C_2^3$) x 8, $C_2\times C_{20}$ ($C_2^3$) x 7, $C_5\times Q_8$ ($C_2\times C_4$) x 4, $C_2^2\times C_{10}$ ($C_2\times C_4$) x 3

Order 32: $\OD_{16}:C_2$ ($C_{10}$) x 8, $C_2\times \OD_{16}$ ($C_{10}$) x 6, $D_4:C_2^2$ ($C_{10}$)

Order 20: $C_2\times C_{10}$ ($C_2^2\times C_4$) x 7, $C_{20}$ ($C_2^2\times C_4$) x 7, $C_{20}$ ($C_2^4$)

Order 16: $\OD_{16}$ ($C_2\times C_{10}$) x 16, $C_2\times C_8$ ($C_2\times C_{10}$) x 12, $D_4:C_2$ ($C_2\times C_{10}$) x 4, $D_4:C_2$ ($C_{20}$) x 4, $C_2\times D_4$ ($C_{20}$) x 3, $C_2^2\times C_4$ ($C_2\times C_{10}$) x 3, $C_2\times Q_8$ ($C_{20}$)

Order 10: $C_{10}$ ($C_2^3\times C_4$)

Order 8: $D_4$ ($C_2\times C_{20}$) x 12, $C_2\times C_4$ ($C_2\times C_{20}$) x 9, $C_8$ ($C_2^2\times C_{10}$) x 8, $C_2\times C_4$ ($C_2^2\times C_{10}$) x 7, $Q_8$ ($C_2\times C_{20}$) x 4, $C_2^3$ ($C_2\times C_{20}$) x 3

Order 5: $C_5$ ($C_8.C_2^3$)

Order 4: $C_2^2$ ($C_2^2\times C_{20}$) x 7, $C_4$ ($C_2^2\times C_{20}$) x 7, $C_4$ ($C_2^3\times C_{10}$)

Order 2: $C_2$ ($C_2^3\times C_{20}$)

Order 1: $C_1$ ($C_{40}.C_2^3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 320: $C_{40}.C_2^3$ ($C_1$)

Order 160: $C_{10}.C_2^4$ ($C_2$), $\OD_{16}:C_{10}$ ($C_2$), $C_{10}\times \OD_{16}$ ($C_2$)

Order 80: $C_5\times \OD_{16}$ ($C_2^2$) x 2, $D_4:C_{10}$ ($C_2^2$), $D_4:C_{10}$ ($C_4$), $Q_8\times C_{10}$ ($C_4$), $D_4\times C_{10}$ ($C_4$), $C_2^2\times C_{20}$ ($C_2^2$), $C_2\times C_{40}$ ($C_2^2$)

Order 64: $C_8.C_2^3$ ($C_5$)

Order 40: $C_2\times C_{20}$ ($C_2^3$) x 2, $C_2\times C_{20}$ ($C_2\times C_4$) x 2, $C_{40}$ ($C_2^3$), $C_2^2\times C_{10}$ ($C_2\times C_4$), $C_5\times Q_8$ ($C_2\times C_4$), $C_5\times D_4$ ($C_2\times C_4$)

Order 32: $D_4:C_2^2$ ($C_{10}$), $\OD_{16}:C_2$ ($C_{10}$), $C_2\times \OD_{16}$ ($C_{10}$)

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

Order 16: $\OD_{16}$ ($C_2\times C_{10}$) x 2, $C_2\times C_8$ ($C_2\times C_{10}$), $D_4:C_2$ ($C_2\times C_{10}$), $D_4:C_2$ ($C_{20}$), $C_2\times Q_8$ ($C_{20}$), $C_2\times D_4$ ($C_{20}$), $C_2^2\times C_4$ ($C_2\times C_{10}$)

Order 10: $C_{10}$ ($C_2^3\times C_4$)

Order 8: $C_2\times C_4$ ($C_2\times C_{20}$) x 2, $C_2\times C_4$ ($C_2^2\times C_{10}$) x 2, $C_2^3$ ($C_2\times C_{20}$), $Q_8$ ($C_2\times C_{20}$), $D_4$ ($C_2\times C_{20}$), $C_8$ ($C_2^2\times C_{10}$)

Order 5: $C_5$ ($C_8.C_2^3$)

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

Order 2: $C_2$ ($C_2^3\times C_{20}$)

Order 1: $C_1$ ($C_{40}.C_2^3$)

Series

Derived series

$C_{40}.C_2^3$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$C_{40}.C_2^3$

$\rhd$

$C_{10}.C_2^4$

$\rhd$

$C_2^2\times C_{20}$

$\rhd$

$C_2\times C_{20}$

$\rhd$

$C_{20}$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_{40}.C_2^3$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_{20}$

$\lhd$

$C_{40}.C_2^3$

Supergroups

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

This group is a maximal quotient of 12 larger groups in the database.

Character theory

Complex character table

See the $170 \times 170$ 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 $50 \times 50$ rational character table.