Abstract group 1280.1116308: $F_5\times C_2^6$

• Label: 1280.1116308
• Order: \( 2^{8} \cdot 5 \)
• Exponent: \( 2^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{8} \)
• $\card{Z(G)}$: \( 2^{6} \)
• $\card{\Aut(G)}$: \( 2^{23} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2} \cdot 31 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{21} \cdot 3^{4} \cdot 5 \cdot 7^{2} \cdot 31 \)
• Perm deg.: not computed
• Trans deg.: $320$
• Rank: $7$

Group information

Description:	$F_5\times C_2^6$
Order:	\(1280\)\(\medspace = 2^{8} \cdot 5 \)
Exponent:	\(20\)\(\medspace = 2^{2} \cdot 5 \)
Automorphism group:	$C_2^6.\GL(6,2)\times F_5$, of order \(25803148492800\)\(\medspace = 2^{23} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2} \cdot 31 \)
Composition factors:	$C_2$ x 8, $C_5$
Derived length:	$2$

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

Group statistics

Order	1	2	4	5	10
Elements	1	383	640	4	252	1280
Conjugacy classes	1	127	128	1	63	320
Divisions	1	127	64	1	63	256
Autjugacy classes	1	3	2	1	1	8

Dimension	1	2	4
Irr. complex chars.	256	0	64	320
Irr. rational chars.	128	64	64	256

Minimal presentations

Permutation degree:	not computed
Transitive degree:	$320$
Rank:	$7$
Inequivalent generating 7-tuples:	$63495936$

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:	${\langle a, b, c, d, e, f, g \mid a^{4}=b^{10}=c^{2}=d^{2}=e^{2}=f^{2}=g^{2}= \!\cdots\! \rangle}$
Matrix group:	$\left\langle \left(\begin{array}{rr} 3 & 0 \\ 30 & 1 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 39 & 0 \\ 0 & 39 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 39 & 20 \\ 30 & 9 \end{array}\right), \left(\begin{array}{rr} 21 & 0 \\ 0 & 21 \end{array}\right), \left(\begin{array}{rr} 11 & 20 \\ 20 & 31 \end{array}\right), \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/40\Z)$
Direct product:	$C_2$ ${}^6$ $\, \times\, $ $F_5$
Semidirect product:	$D_5$ $\,\rtimes\,$ $(C_2^5\times C_4)$	$C_5$ $\,\rtimes\,$ $(C_2^6\times C_4)$	$(C_2^4\times D_{10})$ $\,\rtimes\,$ $C_4$	$D_{10}$ $\,\rtimes\,$ $(C_2^4\times C_4)$	all 13
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_5$ . $C_2^7$	$D_{10}$ . $C_2^6$	$(C_2^6\times D_5)$ . $C_2$	$(C_2\times D_{10})$ . $C_2^5$	all 7
Aut. group:	$\Aut(C_2^3.D_{10})$	$\Aut(C_2^3.D_{10})$	$\Aut(C_2^3.D_{10})$	$\Aut(D_{10}.D_4)$	all 25

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{40}\Z)$.

Homology

Abelianization:	$C_{2}^{6} \times C_{4} $
Schur multiplier:	$C_{2}^{21}$
Commutator length:	$1$

Subgroups

There are 322294 subgroups in 111198 conjugacy classes, 58424 normal (7 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^6$	$G/Z \simeq$ $F_5$
Commutator:	$G' \simeq$ $C_5$	$G/G' \simeq$ $C_2^6\times C_4$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $F_5\times C_2^6$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^5\times C_{10}$	$G/\operatorname{Fit} \simeq$ $C_4$
Radical:	$R \simeq$ $F_5\times C_2^6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^5\times C_{10}$	$G/\operatorname{soc} \simeq$ $C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^6\times C_4$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

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 1280: $F_5\times C_2^6$

Order 640: $F_5\times C_2^5$ x 126, $C_2^6\times D_5$

Order 320: $C_2^4\times F_5$ x 2604, $C_2^4\times D_{10}$ x 126, $C_2^5\times C_{10}$

Order 256: $C_2^6\times C_4$

Order 160: $C_2^3\times F_5$ x 11160, $C_2^3\times D_{10}$ x 2604, $C_2^4\times C_{10}$ x 63

Order 128: $C_2^5\times C_4$ x 126, $C_2^7$

Order 80: $C_2^2\times D_{10}$ x 11160, $C_2^2\times F_5$ x 10416, $C_2^3\times C_{10}$ x 651

Order 64: $C_2^4\times C_4$ x 2604, $C_2^6$ x 127

Order 40: $C_2\times D_{10}$ x 10416, $C_2\times F_5$ x 2016, $C_2^2\times C_{10}$ x 1395

Order 32: $C_2^3\times C_4$ x 11160, $C_2^5$ x 2667

Order 20: $D_{10}$ x 2016, $C_2\times C_{10}$ x 651, $F_5$ x 64

Order 16: $C_2^4$ x 11811, $C_2^2\times C_4$ x 10416

Order 10: $D_5$ x 64, $C_{10}$ x 63

Order 8: $C_2^3$ x 11811, $C_2\times C_4$ x 2016

Order 5: $C_5$

Order 4: $C_2^2$ x 2667, $C_4$ x 64

Order 2: $C_2$ x 127

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1280: $F_5\times C_2^6$

Order 640: $C_2^6\times D_5$, $F_5\times C_2^5$

Order 320: $C_2^4\times D_{10}$ x 2, $C_2^5\times C_{10}$, $C_2^4\times F_5$

Order 256: $C_2^6\times C_4$

Order 160: $C_2^3\times D_{10}$ x 2, $C_2^4\times C_{10}$, $C_2^3\times F_5$

Order 128: $C_2^7$, $C_2^5\times C_4$

Order 80: $C_2^2\times D_{10}$ x 2, $C_2^3\times C_{10}$, $C_2^2\times F_5$

Order 64: $C_2^6$ x 3, $C_2^4\times C_4$

Order 40: $C_2\times D_{10}$ x 2, $C_2^2\times C_{10}$, $C_2\times F_5$

Order 32: $C_2^5$ x 3, $C_2^3\times C_4$

Order 20: $D_{10}$ x 2, $C_2\times C_{10}$, $F_5$

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

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

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

Order 5: $C_5$

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1280: $F_5\times C_2^6$ ($C_1$)

Order 640: $F_5\times C_2^5$ ($C_2$) x 126, $C_2^6\times D_5$ ($C_2$)

Order 320: $C_2^4\times F_5$ ($C_2^2$) x 2604, $C_2^4\times D_{10}$ ($C_2^2$) x 63, $C_2^4\times D_{10}$ ($C_4$) x 63, $C_2^5\times C_{10}$ ($C_4$)

Order 160: $C_2^3\times F_5$ ($C_2^3$) x 11160, $C_2^3\times D_{10}$ ($C_2\times C_4$) x 1953, $C_2^3\times D_{10}$ ($C_2^3$) x 651, $C_2^4\times C_{10}$ ($C_2\times C_4$) x 63

Order 80: $C_2^2\times F_5$ ($C_2^4$) x 10416, $C_2^2\times D_{10}$ ($C_2^2\times C_4$) x 9765, $C_2^2\times D_{10}$ ($C_2^4$) x 1395, $C_2^3\times C_{10}$ ($C_2^2\times C_4$) x 651

Order 64: $C_2^6$ ($F_5$)

Order 40: $C_2\times D_{10}$ ($C_2^3\times C_4$) x 9765, $C_2\times F_5$ ($C_2^5$) x 2016, $C_2^2\times C_{10}$ ($C_2^3\times C_4$) x 1395, $C_2\times D_{10}$ ($C_2^5$) x 651

Order 32: $C_2^5$ ($C_2\times F_5$) x 63

Order 20: $D_{10}$ ($C_2^4\times C_4$) x 1953, $C_2\times C_{10}$ ($C_2^4\times C_4$) x 651, $F_5$ ($C_2^6$) x 64, $D_{10}$ ($C_2^6$) x 63

Order 16: $C_2^4$ ($C_2^2\times F_5$) x 651

Order 10: $C_{10}$ ($C_2^5\times C_4$) x 63, $D_5$ ($C_2^5\times C_4$) x 63, $D_5$ ($C_2^7$)

Order 8: $C_2^3$ ($C_2^3\times F_5$) x 1395

Order 5: $C_5$ ($C_2^6\times C_4$)

Order 4: $C_2^2$ ($C_2^4\times F_5$) x 651

Order 2: $C_2$ ($F_5\times C_2^5$) x 63

Order 1: $C_1$ ($F_5\times C_2^6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1280: $F_5\times C_2^6$ ($C_1$)

Order 640: $C_2^6\times D_5$ ($C_2$), $F_5\times C_2^5$ ($C_2$)

Order 320: $C_2^5\times C_{10}$ ($C_4$), $C_2^4\times D_{10}$ ($C_2^2$), $C_2^4\times D_{10}$ ($C_4$), $C_2^4\times F_5$ ($C_2^2$)

Order 160: $C_2^4\times C_{10}$ ($C_2\times C_4$), $C_2^3\times D_{10}$ ($C_2^3$), $C_2^3\times D_{10}$ ($C_2\times C_4$), $C_2^3\times F_5$ ($C_2^3$)

Order 80: $C_2^3\times C_{10}$ ($C_2^2\times C_4$), $C_2^2\times D_{10}$ ($C_2^4$), $C_2^2\times D_{10}$ ($C_2^2\times C_4$), $C_2^2\times F_5$ ($C_2^4$)

Order 64: $C_2^6$ ($F_5$)

Order 40: $C_2^2\times C_{10}$ ($C_2^3\times C_4$), $C_2\times D_{10}$ ($C_2^5$), $C_2\times D_{10}$ ($C_2^3\times C_4$), $C_2\times F_5$ ($C_2^5$)

Order 32: $C_2^5$ ($C_2\times F_5$)

Order 20: $C_2\times C_{10}$ ($C_2^4\times C_4$), $D_{10}$ ($C_2^6$), $D_{10}$ ($C_2^4\times C_4$), $F_5$ ($C_2^6$)

Order 16: $C_2^4$ ($C_2^2\times F_5$)

Order 10: $C_{10}$ ($C_2^5\times C_4$), $D_5$ ($C_2^7$), $D_5$ ($C_2^5\times C_4$)

Order 8: $C_2^3$ ($C_2^3\times F_5$)

Order 5: $C_5$ ($C_2^6\times C_4$)

Order 4: $C_2^2$ ($C_2^4\times F_5$)

Order 2: $C_2$ ($F_5\times C_2^5$)

Order 1: $C_1$ ($F_5\times C_2^6$)

Series

Derived series

$F_5\times C_2^6$

$\rhd$

$C_5$

$\rhd$

$C_1$

Chief series

$F_5\times C_2^6$

$\rhd$

$F_5\times C_2^5$

$\rhd$

$C_2^4\times D_{10}$

$\rhd$

$C_2^4\times C_{10}$

$\rhd$

$C_2^5$

$\rhd$

$C_2^4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$F_5\times C_2^6$

$\rhd$

$C_5$

Upper central series

$C_1$

$\lhd$

$C_2^6$

Supergroups

This group is a maximal subgroup of 3 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $320 \times 320$ 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 $256 \times 256$ rational character table (warning: may be slow to load).