Abstract group 320.625: $C_4.(C_4\times D_{10})$

• Label: 320.625
• Order: \( 2^{6} \cdot 5 \)
• Exponent: \( 2^{3} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \)
• Perm deg.: $25$
• Trans deg.: $160$
• Rank: $3$

Group information

Description:	$C_4.(C_4\times D_{10})$
Order:	\(320\)\(\medspace = 2^{6} \cdot 5 \)
Exponent:	\(40\)\(\medspace = 2^{3} \cdot 5 \)
Automorphism group:	$C_5:(C_2^6.C_2^5)$, 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	8	10	20
Elements	1	7	104	4	80	28	96	320
Conjugacy classes	1	5	14	2	8	10	28	68
Divisions	1	5	8	1	4	5	6	30
Autjugacy classes	1	3	5	1	2	3	4	19

Dimension	1	2	4	8	16
Irr. complex chars.	16	44	8	0	0	68
Irr. rational chars.	8	8	4	8	2	30

Minimal presentations

Permutation degree:	$25$
Transitive degree:	$160$
Rank:	$3$
Inequivalent generating triples:	$1008$

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:	$\langle a, b, c \mid a^{4}=b^{8}=c^{10}=1, b^{a}=b^{7}, c^{a}=b^{4}c, c^{b}=c^{9} \rangle$
Permutation group:	Degree $25$ $\langle(2,3)(4,5)(10,11,13,16,14,17,22,25)(12,18,20,23,19,24,21,15), (12,19)(15,23) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 68 & 20 \\ 45 & 68 \end{array}\right), \left(\begin{array}{rr} 36 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 1 & 17 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 16 & 0 \\ 0 & 16 \end{array}\right), \left(\begin{array}{rr} 16 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 81 & 0 \\ 0 & 81 \end{array}\right), \left(\begin{array}{rr} 69 & 0 \\ 0 & 69 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/85\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_{20}.Q_8)$ $\,\rtimes\,$ $C_2$ (2)	$(C_{10}.D_8)$ $\,\rtimes\,$ $C_2$ (2)	$(C_{10}:C_8)$ $\,\rtimes\,$ $C_4$ (4)	$C_5$ $\,\rtimes\,$ $(C_2^3.D_4)$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2\times C_{20})$ . $Q_8$ (2)	$C_{20}$ . $(C_2\times Q_8)$	$C_4$ . $(C_{10}:Q_8)$	$(C_2\times C_{20})$ . $D_4$	all 35

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

Homology

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

Subgroups

There are 302 subgroups in 114 conjugacy classes, 63 normal (25 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_4$	$G/Z \simeq$ $C_5:D_4$
Commutator:	$G' \simeq$ $C_{20}$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $C_2\times D_{10}$
Fitting:	$\operatorname{Fit} \simeq$ $C_4^2:C_{10}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_4.(C_4\times D_{10})$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{10}$	$G/\operatorname{soc} \simeq$ $C_2\times D_4$
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

No diagram available

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 320: $C_4.(C_4\times D_{10})$

Order 160: $C_{20}.Q_8$ x 2, $C_{10}.D_8$ x 2, $C_4^2:C_{10}$, $C_2^3.D_{10}$, $C_{20}.C_2^3$

Order 80: $C_{10}:C_8$ x 6, $C_4:C_{20}$ x 2, $C_{20}:C_4$ x 2, $C_2^2\times C_{20}$, $C_2^2:C_{20}$, $C_4\times C_{20}$, $C_{10}.D_4$, $C_{20}:C_4$

Order 64: $C_2^3.D_4$

Order 40: $C_2\times C_{20}$ x 8, $C_5:C_8$ x 4, $C_{10}:C_4$ x 2, $C_2^2\times C_{10}$

Order 32: $C_4^2:C_2$ x 2, $C_8:C_4$ x 2, $C_8:C_4$ x 2, $C_2^2\times C_8$

Order 20: $C_{20}$ x 6, $C_2\times C_{10}$ x 5, $C_5:C_4$ x 2

Order 16: $C_2\times C_8$ x 6, $C_4:C_4$ x 4, $C_2^2:C_4$ x 2, $C_4^2$ x 2, $C_2^2\times C_4$

Order 10: $C_{10}$ x 5

Order 8: $C_2\times C_4$ x 10, $C_8$ x 4, $C_2^3$

Order 5: $C_5$

Order 4: $C_4$ x 8, $C_2^2$ x 5

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 320: $C_4.(C_4\times D_{10})$

Order 160: $C_4^2:C_{10}$, $C_{20}.Q_8$, $C_2^3.D_{10}$, $C_{20}.C_2^3$, $C_{10}.D_8$

Order 80: $C_{10}:C_8$ x 3, $C_2^2\times C_{20}$, $C_4:C_{20}$, $C_2^2:C_{20}$, $C_4\times C_{20}$, $C_{10}.D_4$, $C_{20}:C_4$, $C_{20}:C_4$

Order 64: $C_2^3.D_4$

Order 40: $C_2\times C_{20}$ x 5, $C_5:C_8$ x 2, $C_{10}:C_4$, $C_2^2\times C_{10}$

Order 32: $C_4^2:C_2$ x 2, $C_2^2\times C_8$, $C_8:C_4$, $C_8:C_4$

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

Order 16: $C_2\times C_8$ x 3, $C_4:C_4$ x 2, $C_2^2:C_4$ x 2, $C_4^2$ x 2, $C_2^2\times C_4$

Order 10: $C_{10}$ x 3

Order 8: $C_2\times C_4$ x 6, $C_8$ x 2, $C_2^3$

Order 5: $C_5$

Order 4: $C_4$ x 5, $C_2^2$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 320: $C_4.(C_4\times D_{10})$ ($C_1$)

Order 160: $C_{20}.Q_8$ ($C_2$) x 2, $C_{10}.D_8$ ($C_2$) x 2, $C_4^2:C_{10}$ ($C_2$), $C_2^3.D_{10}$ ($C_2$), $C_{20}.C_2^3$ ($C_2$)

Order 80: $C_{10}:C_8$ ($C_4$) x 4, $C_{10}:C_8$ ($C_2^2$) x 2, $C_4:C_{20}$ ($C_2^2$) x 2, $C_{20}:C_4$ ($C_2^2$) x 2, $C_2^2\times C_{20}$ ($C_2^2$)

Order 40: $C_5:C_8$ ($C_2\times C_4$) x 4, $C_2\times C_{20}$ ($Q_8$) x 2, $C_2\times C_{20}$ ($C_2\times C_4$) x 2, $C_2\times C_{20}$ ($C_2^3$), $C_2\times C_{20}$ ($D_4$), $C_2^2\times C_{10}$ ($D_4$)

Order 32: $C_4^2:C_2$ ($D_5$)

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

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

Order 10: $C_{10}$ ($D_8:C_2$) x 2, $C_{10}$ ($C_2^2.D_4$)

Order 8: $C_2\times C_4$ ($C_4\times D_5$) x 2, $C_2\times C_4$ ($C_5:Q_8$) x 2, $C_2^3$ ($C_5:D_4$), $C_2\times C_4$ ($C_5:D_4$), $C_2\times C_4$ ($C_2\times D_{10}$)

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

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

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

Order 1: $C_1$ ($C_4.(C_4\times D_{10})$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 320: $C_4.(C_4\times D_{10})$ ($C_1$)

Order 160: $C_4^2:C_{10}$ ($C_2$), $C_{20}.Q_8$ ($C_2$), $C_2^3.D_{10}$ ($C_2$), $C_{20}.C_2^3$ ($C_2$), $C_{10}.D_8$ ($C_2$)

Order 80: $C_{10}:C_8$ ($C_2^2$) x 2, $C_{10}:C_8$ ($C_4$), $C_2^2\times C_{20}$ ($C_2^2$), $C_4:C_{20}$ ($C_2^2$), $C_{20}:C_4$ ($C_2^2$)

Order 40: $C_5:C_8$ ($C_2\times C_4$) x 2, $C_2\times C_{20}$ ($C_2^3$), $C_2\times C_{20}$ ($Q_8$), $C_2\times C_{20}$ ($D_4$), $C_2\times C_{20}$ ($C_2\times C_4$), $C_2^2\times C_{10}$ ($D_4$)

Order 32: $C_4^2:C_2$ ($D_5$)

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

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

Order 10: $C_{10}$ ($D_8:C_2$), $C_{10}$ ($C_2^2.D_4$)

Order 8: $C_2^3$ ($C_5:D_4$), $C_2\times C_4$ ($C_5:D_4$), $C_2\times C_4$ ($C_4\times D_5$), $C_2\times C_4$ ($C_5:Q_8$), $C_2\times C_4$ ($C_2\times D_{10}$)

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

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

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

Order 1: $C_1$ ($C_4.(C_4\times D_{10})$)

Series

Derived series

$C_4.(C_4\times D_{10})$

$\rhd$

$C_{20}$

$\rhd$

$C_1$

Chief series

$C_4.(C_4\times D_{10})$

$\rhd$

$C_{10}.D_8$

$\rhd$

$C_4:C_{20}$

$\rhd$

$C_2\times C_{20}$

$\rhd$

$C_2\times C_{10}$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_4.(C_4\times D_{10})$

$\rhd$

$C_{20}$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

Upper central series

$C_1$

$\lhd$

$C_2\times C_4$

$\lhd$

$C_2^2\times C_4$

$\lhd$

$C_4^2:C_2$

Supergroups

This group is a maximal subgroup of 14 larger groups in the database.

This group is a maximal quotient of 14 larger groups in the database.

Character theory

Complex character table

See the $68 \times 68$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $30 \times 30$ rational character table.