Abstract group 1600.10254: $F_5^2:C_2^2$

• Label: 1600.10254
• Order: \( 2^{6} \cdot 5^{2} \)
• Exponent: \( 2^{3} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 5^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $12$
• Trans deg.: $20$
• Rank: $3$

Group information

Description:	$F_5^2:C_2^2$
Order:	\(1600\)\(\medspace = 2^{6} \cdot 5^{2} \)
Exponent:	\(40\)\(\medspace = 2^{3} \cdot 5 \)
Automorphism group:	$F_5^2:C_2^3$, of order \(3200\)\(\medspace = 2^{7} \cdot 5^{2} \)
Composition factors:	$C_2$ x 6, $C_5$ x 2
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	4	5	8	10	20
Elements	1	111	640	24	400	264	160	1600
Conjugacy classes	1	7	16	2	4	6	4	40
Divisions	1	7	10	2	2	6	2	30
Autjugacy classes	1	6	11	2	2	5	2	29

Dimension	1	2	4	8	16
Irr. complex chars.	16	12	0	8	4	40
Irr. rational chars.	8	8	4	4	6	30

Minimal presentations

Permutation degree:	$12$
Transitive degree:	$20$
Rank:	$3$
Inequivalent generating triples:	$419328$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	8	8
Arbitrary	8	8	8

Constructions

Presentation:	${\langle a, b, c, d \mid a^{2}=b^{4}=c^{20}=d^{10}=1, b^{a}=b^{3}, c^{a}= \!\cdots\! \rangle}$
Permutation group:	Degree $12$ $\langle(1,2)(3,5)(4,7)(6,10)(8,9)(11,12), (3,6,7,8), (11,12), (3,7)(6,8), (3,8,7,6)(4,9,5,10), (3,7)(4,5)(6,8)(9,10), (1,3,6,8,7)(2,4,9,10,5), (2,4,9,10,5)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrr} 3 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 3 \end{array}\right), \left(\begin{array}{rrr} 1 & 2 & 0 \\ 0 & 1 & 0 \\ 0 & 2 & 1 \end{array}\right), \left(\begin{array}{rrr} 2 & 1 & 2 \\ 0 & 4 & 0 \\ 4 & 2 & 3 \end{array}\right), \left(\begin{array}{rrr} 4 & 2 & 1 \\ 0 & 1 & 0 \\ 3 & 1 & 1 \end{array}\right), \left(\begin{array}{rrr} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{array}\right), \left(\begin{array}{rrr} 1 & 2 & 4 \\ 0 & 1 & 0 \\ 0 & 1 & 4 \end{array}\right), \left(\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 3 & 1 \end{array}\right), \left(\begin{array}{rrr} 4 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}\right) \right\rangle \subseteq \GL_{3}(\F_{5})$
Transitive group:	20T212	20T215	40T1479	40T1485	all 12
Direct product:	$C_2$ $\, \times\, $ $(F_5\wr C_2)$
Semidirect product:	$F_5^2$ $\,\rtimes\,$ $C_2^2$ (2)	$(C_2\times F_5^2)$ $\,\rtimes\,$ $C_2$	$(D_5^2:C_2^2)$ $\,\rtimes\,$ $C_4$	$(D_5^2.C_2^2)$ $\,\rtimes\,$ $C_4$ (2)	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(D_5:F_5)$ . $D_4$ (2)	$(D_5\times D_{10})$ . $D_4$	$(C_{10}:F_5)$ . $D_4$	$D_5^2$ . $(C_2\times D_4)$	all 17
Aut. group:	$\Aut(C_5^2:C_8)$	$\Aut(C_{10}:F_5)$	$\Aut(C_{15}:F_5)$	$\Aut(C_3\times D_5^2)$	all 12

Elements of the group are displayed as matrices in $\GL_{3}(\F_{5})$.

Homology

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

Subgroups

There are 3970 subgroups in 251 conjugacy classes, 43 normal (27 characteristic).

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

Special subgroups

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

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 Click on a subgroup in the diagram to see information about it.

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Classes of subgroups up to conjugation

Order 1600: $F_5^2:C_2^2$

Order 800: $F_5\wr C_2$ x 4, $D_5^2.C_2^3$, $C_2\times C_5^2:\OD_{16}$, $C_2\times F_5^2$

Order 400: $D_5^2.C_2^2$ x 6, $C_5^2:\OD_{16}$ x 3, $F_5^2$ x 3, $D_5^2:C_2^2$ x 2, $D_{10}:F_5$ x 2, $C_2\times C_5^2:Q_8$, $D_{10}\times F_5$, $C_2\times C_5^2:C_8$

Order 200: $D_5:F_5$ x 8, $D_5\wr C_2$ x 7, $D_5\times F_5$ x 4, $C_5^2:Q_8$ x 3, $D_5\times D_{10}$ x 2, $C_{10}:F_5$ x 2, $C_5^2:C_8$ x 2, $C_{10}:F_5$, $C_{10}:F_5$, $C_{10}\times F_5$

Order 160: $C_{10}:C_4^2$

Order 100: $D_5^2$ x 6, $C_5:F_5$ x 4, $C_5\times F_5$ x 2, $C_5\times D_{10}$ x 2, $C_5:F_5$ x 2, $C_5:F_5$ x 2, $C_5:D_{10}$

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

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

Order 50: $C_5\times D_5$ x 4, $C_5:D_5$ x 2, $C_5\times C_{10}$

Order 40: $C_2\times F_5$ x 18, $C_4\times D_5$ x 4, $C_2\times D_{10}$ x 2, $C_2\times C_{20}$, $C_{10}:C_4$

Order 32: $C_4\wr C_2$ x 4, $D_4:C_2^2$, $C_2\times \OD_{16}$, $C_2\times C_4^2$

Order 25: $C_5^2$

Order 20: $D_{10}$ x 12, $F_5$ x 12, $C_2\times C_{10}$ x 2, $C_{20}$ x 2, $C_5:C_4$ x 2

Order 16: $D_4:C_2$ x 6, $\OD_{16}$ x 3, $C_4^2$ x 3, $C_2^2\times C_4$ x 3, $C_2\times D_4$ x 2, $C_2\times C_8$, $C_2\times Q_8$

Order 10: $D_5$ x 8, $C_{10}$ x 6

Order 8: $C_2\times C_4$ x 17, $D_4$ x 7, $Q_8$ x 3, $C_2^3$ x 2, $C_8$ x 2

Order 5: $C_5$ x 2

Order 4: $C_4$ x 10, $C_2^2$ x 9

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1600: $F_5^2:C_2^2$

Order 800: $D_5^2.C_2^3$, $C_2\times C_5^2:\OD_{16}$, $C_2\times F_5^2$, $F_5\wr C_2$

Order 400: $D_5^2.C_2^2$ x 4, $D_5^2:C_2^2$ x 2, $D_{10}:F_5$ x 2, $C_5^2:\OD_{16}$ x 2, $F_5^2$ x 2, $C_2\times C_5^2:Q_8$, $D_{10}\times F_5$, $C_2\times C_5^2:C_8$

Order 200: $D_5:F_5$ x 6, $D_5\wr C_2$ x 4, $D_5\times D_{10}$ x 2, $C_{10}:F_5$ x 2, $C_5^2:Q_8$ x 2, $D_5\times F_5$ x 2, $C_{10}:F_5$, $C_{10}:F_5$, $C_{10}\times F_5$, $C_5^2:C_8$

Order 160: $C_{10}:C_4^2$

Order 100: $D_5^2$ x 5, $C_5:F_5$ x 3, $C_5\times D_{10}$ x 2, $C_5:F_5$ x 2, $C_5\times F_5$, $C_5:D_{10}$, $C_5:F_5$

Order 80: $C_2^2\times F_5$ x 3, $C_4\times F_5$ x 2, $C_4\times D_{10}$

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

Order 50: $C_5\times D_5$ x 3, $C_5:D_5$ x 2, $C_5\times C_{10}$

Order 40: $C_2\times F_5$ x 14, $C_4\times D_5$ x 2, $C_2\times D_{10}$ x 2, $C_2\times C_{20}$, $C_{10}:C_4$

Order 32: $D_4:C_2^2$, $C_2\times \OD_{16}$, $C_2\times C_4^2$, $C_4\wr C_2$

Order 25: $C_5^2$

Order 20: $D_{10}$ x 10, $F_5$ x 9, $C_2\times C_{10}$ x 2, $C_{20}$, $C_5:C_4$

Order 16: $D_4:C_2$ x 4, $C_2^2\times C_4$ x 3, $\OD_{16}$ x 2, $C_4^2$ x 2, $C_2\times D_4$ x 2, $C_2\times C_8$, $C_2\times Q_8$

Order 10: $D_5$ x 7, $C_{10}$ x 5

Order 8: $C_2\times C_4$ x 13, $D_4$ x 4, $C_2^3$ x 2, $Q_8$ x 2, $C_8$

Order 5: $C_5$ x 2

Order 4: $C_2^2$ x 8, $C_4$ x 7

Order 2: $C_2$ x 6

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1600: $F_5^2:C_2^2$ ($C_1$)

Order 800: $F_5\wr C_2$ ($C_2$) x 4, $D_5^2.C_2^3$ ($C_2$), $C_2\times C_5^2:\OD_{16}$ ($C_2$), $C_2\times F_5^2$ ($C_2$)

Order 400: $D_5^2.C_2^2$ ($C_2^2$) x 2, $D_5^2.C_2^2$ ($C_4$) x 2, $C_5^2:\OD_{16}$ ($C_2^2$) x 2, $F_5^2$ ($C_2^2$) x 2, $C_2\times C_5^2:Q_8$ ($C_4$), $D_5^2:C_2^2$ ($C_4$), $D_{10}:F_5$ ($C_2^2$)

Order 200: $C_5^2:Q_8$ ($C_2\times C_4$) x 2, $D_5\wr C_2$ ($C_2\times C_4$) x 2, $D_5:F_5$ ($D_4$) x 2, $D_5\times D_{10}$ ($D_4$), $C_{10}:F_5$ ($C_2\times C_4$), $C_{10}:F_5$ ($D_4$), $D_5:F_5$ ($C_2^3$), $D_5:F_5$ ($C_2\times C_4$)

Order 100: $C_5:D_{10}$ ($C_2^2:C_4$), $D_5^2$ ($C_2^2:C_4$), $D_5^2$ ($C_2\times D_4$), $C_5:F_5$ ($C_2^2:C_4$), $C_5:F_5$ ($C_2^2\times C_4$), $C_5:F_5$ ($C_2^2:C_4$), $C_5:F_5$ ($C_2\times D_4$)

Order 50: $C_5\times C_{10}$ ($C_4\wr C_2$), $C_5:D_5$ ($C_2^3:C_4$), $C_5:D_5$ ($C_4\wr C_2$)

Order 25: $C_5^2$ ($C_4^2:C_2^2$)

Order 2: $C_2$ ($F_5\wr C_2$)

Order 1: $C_1$ ($F_5^2:C_2^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1600: $F_5^2:C_2^2$ ($C_1$)

Order 800: $D_5^2.C_2^3$ ($C_2$), $C_2\times C_5^2:\OD_{16}$ ($C_2$), $C_2\times F_5^2$ ($C_2$), $F_5\wr C_2$ ($C_2$)

Order 400: $C_2\times C_5^2:Q_8$ ($C_4$), $D_5^2:C_2^2$ ($C_4$), $D_{10}:F_5$ ($C_2^2$), $D_5^2.C_2^2$ ($C_2^2$), $D_5^2.C_2^2$ ($C_4$), $C_5^2:\OD_{16}$ ($C_2^2$), $F_5^2$ ($C_2^2$)

Order 200: $D_5:F_5$ ($D_4$) x 2, $D_5\times D_{10}$ ($D_4$), $C_{10}:F_5$ ($C_2\times C_4$), $C_{10}:F_5$ ($D_4$), $C_5^2:Q_8$ ($C_2\times C_4$), $D_5\wr C_2$ ($C_2\times C_4$), $D_5:F_5$ ($C_2^3$), $D_5:F_5$ ($C_2\times C_4$)

Order 100: $C_5:D_{10}$ ($C_2^2:C_4$), $D_5^2$ ($C_2^2:C_4$), $D_5^2$ ($C_2\times D_4$), $C_5:F_5$ ($C_2^2:C_4$), $C_5:F_5$ ($C_2^2\times C_4$), $C_5:F_5$ ($C_2^2:C_4$), $C_5:F_5$ ($C_2\times D_4$)

Order 50: $C_5\times C_{10}$ ($C_4\wr C_2$), $C_5:D_5$ ($C_2^3:C_4$), $C_5:D_5$ ($C_4\wr C_2$)

Order 25: $C_5^2$ ($C_4^2:C_2^2$)

Order 2: $C_2$ ($F_5\wr C_2$)

Order 1: $C_1$ ($F_5^2:C_2^2$)

Series

Derived series

$F_5^2:C_2^2$

$\rhd$

$C_5:F_5$

$\rhd$

$C_5^2$

$\rhd$

$C_1$

Chief series

$F_5^2:C_2^2$

$\rhd$

$F_5\wr C_2$

$\rhd$

$D_5^2.C_2^2$

$\rhd$

$D_5:F_5$

$\rhd$

$C_5:F_5$

$\rhd$

$C_5:D_5$

$\rhd$

$C_5^2$

$\rhd$

$C_1$

Lower central series

$F_5^2:C_2^2$

$\rhd$

$C_5:F_5$

$\rhd$

$C_5:D_5$

$\rhd$

$C_5^2$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 32 larger groups in the database.

This group is a maximal quotient of 40 larger groups in the database.

Character theory

Complex character table

See the $40 \times 40$ 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.