Abstract group 2400.bp: $C_5.\GL(2,5)$

• Label: 2400.bp
• Order: \( 2^{5} \cdot 3 \cdot 5^{2} \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3 \cdot 5^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $29$
• Trans deg.: $120$
• Rank: $2$

Group information

Description:	$C_5.\GL(2,5)$
Order:	\(2400\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{2} \)
Exponent:	\(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \)
Automorphism group:	$C_2\times F_5\times S_5$, of order \(4800\)\(\medspace = 2^{6} \cdot 3 \cdot 5^{2} \)
Composition factors:	$C_2$ x 3, $C_5$, $A_5$
Derived length:	$2$

This group is nonabelian and nonsolvable. Whether it is almost simple has not been computed.

Group statistics

Order	1	2	3	4	5	6	8	10	12	15	20	24	30
Elements	1	151	20	640	124	20	200	124	200	80	360	400	80	2400
Conjugacy classes	1	2	1	7	4	1	2	4	2	1	3	4	1	33
Divisions	1	2	1	4	4	1	1	4	1	1	2	1	1	24
Autjugacy classes	1	2	1	5	3	1	2	3	2	1	3	2	1	27

Dimension	1	2	4	5	6	8	10	12	16	20	24
Irr. complex chars.	4	0	11	4	6	2	0	2	2	1	1	33
Irr. rational chars.	2	1	3	2	2	4	1	4	3	1	1	24

Minimal presentations

Permutation degree:	$29$
Transitive degree:	$120$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

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

Constructions

Permutation group:	Degree $29$ $\langle(1,2,4,7)(3,6,9,14)(5,8,11,16)(10,15,18,22)(12,13,17,20)(19,23,24,21)(25,26) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 2 & 1 & 2 & 4 \\ 1 & 1 & 1 & 0 \\ 0 & 4 & 1 & 1 \\ 3 & 2 & 0 & 2 \end{array}\right), \left(\begin{array}{rrrr} 0 & 0 & 1 & 1 \\ 4 & 1 & 1 & 1 \\ 0 & 0 & 1 & 0 \\ 4 & 0 & 1 & 2 \end{array}\right), \left(\begin{array}{rrrr} 3 & 4 & 3 & 4 \\ 3 & 3 & 4 & 0 \\ 1 & 0 & 3 & 4 \\ 1 & 4 & 1 & 1 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 1 & 2 \\ 0 & 4 & 2 & 4 \\ 4 & 4 & 4 & 2 \\ 3 & 1 & 1 & 4 \end{array}\right), \left(\begin{array}{rrrr} 0 & 4 & 4 & 0 \\ 1 & 1 & 1 & 2 \\ 2 & 0 & 0 & 3 \\ 2 & 4 & 0 & 3 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{5})$
Direct product:	not computed
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$\SL(2,5)$ . $F_5$	$C_5$ . $\GL(2,5)$	$(C_5:C_4)$ . $S_5$	$C_2$ . $(A_5:F_5)$	all 7

Elements of the group are displayed as matrices in $\GL_{4}(\F_{5})$.

Homology

Abelianization:	$C_{4} $
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 3260 subgroups in 112 conjugacy classes, 9 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $A_5:F_5$
Commutator:	$G' \simeq$ $C_5\times \SL(2,5)$	$G/G' \simeq$ $C_4$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $A_5:F_5$
Fitting:	$\operatorname{Fit} \simeq$ $C_{10}$	$G/\operatorname{Fit} \simeq$ $A_5:C_4$
Radical:	$R \simeq$ $C_5:C_4$	$G/R \simeq$ $S_5$
Socle:	$\operatorname{soc} \simeq$ $C_{10}$	$G/\operatorname{soc} \simeq$ $A_5:C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4\wr C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

Order 2400: $C_5.\GL(2,5)$

Order 1200: $\SL(2,5):D_5$

Order 600: $C_5\times \SL(2,5)$

Order 480: $\SL(2,3):F_5$, $\GL(2,5)$

Order 400: $C_5^2:C_4^2$

Order 240: $\SL(2,5):C_2$, $\SL(2,3):D_5$, $C_{15}:\OD_{16}$

Order 200: $C_{10}:F_5$, $C_{10}:F_5$, $C_{10}.D_{10}$

Order 160: $D_{20}:C_4$

Order 120: $C_{15}:C_8$, $C_5:C_{24}$, $\SL(2,5)$, $C_5\times \SL(2,3)$, $D_{15}:C_4$

Order 100: $C_5:C_{20}$ x 2, $C_5:F_5$ x 2, $C_5:F_5$ x 2, $C_5:D_{10}$

Order 96: $\Unitary(2,3)$

Order 80: $C_4\times F_5$ x 2, $D_{20}:C_2$, $C_5:\OD_{16}$

Order 60: $C_5:C_{12}$, $D_{30}$, $C_3:C_{20}$

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

Order 48: $C_{24}:C_2$, $\SL(2,3):C_2$

Order 40: $C_2\times F_5$ x 5, $C_4\times D_5$ x 2, $D_{20}$, $C_5:C_8$, $C_5\times Q_8$

Order 32: $C_4\wr C_2$

Order 30: $C_{30}$, $D_{15}$

Order 25: $C_5^2$

Order 24: $C_4\times S_3$, $\SL(2,3)$, $C_{24}$, $C_3:C_8$

Order 20: $F_5$ x 10, $D_{10}$ x 4, $C_{20}$ x 2, $C_5:C_4$ x 2

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

Order 15: $C_{15}$

Order 12: $D_6$, $C_{12}$, $C_3:C_4$

Order 10: $D_5$ x 7, $C_{10}$ x 4

Order 8: $C_2\times C_4$ x 2, $Q_8$, $D_4$, $C_8$

Order 6: $C_6$, $S_3$

Order 5: $C_5$ x 4

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

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 2400: $C_5.\GL(2,5)$

Order 1200: $\SL(2,5):D_5$

Order 600: $C_5\times \SL(2,5)$

Order 480: $\GL(2,5)$

Order 400: $C_5^2:C_4^2$

Order 160: $D_{20}:C_4$

Order 96: $\Unitary(2,3)$

Order 32: $C_4\wr C_2$

Order 25: $C_5^2$

Order 15: $C_{15}$

Order 3: $C_3$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 2400: $C_5.\GL(2,5)$ ($C_1$)

Order 1200: $\SL(2,5):D_5$ ($C_2$)

Order 600: $C_5\times \SL(2,5)$ ($C_4$)

Order 120: $\SL(2,5)$ ($F_5$)

Order 20: $C_5:C_4$ ($S_5$)

Order 10: $C_{10}$ ($A_5:C_4$)

Order 5: $C_5$ ($\GL(2,5)$)

Order 2: $C_2$ ($A_5:F_5$)

Order 1: $C_1$ ($C_5.\GL(2,5)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2400: $C_5.\GL(2,5)$ ($C_1$)

Order 1200: $\SL(2,5):D_5$ ($C_2$)

Order 600: $C_5\times \SL(2,5)$ ($C_4$)

Order 120: $\SL(2,5)$ ($F_5$)

Order 1: $C_1$ ($C_5.\GL(2,5)$)

Series

Derived series

$C_5.\GL(2,5)$

$\rhd$

$C_5\times \SL(2,5)$

$\rhd$

$\SL(2,5)$

Chief series

$C_5.\GL(2,5)$

$\rhd$

$\SL(2,5):D_5$

$\rhd$

$C_5:C_4$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_5.\GL(2,5)$

$\rhd$

$C_5\times \SL(2,5)$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

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

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

See the $33 \times 33$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $24 \times 24$ rational character table.