Abstract group 1200.481: $\SL(2,5):D_5$

• Label: 1200.481
• Order: \( 2^{4} \cdot 3 \cdot 5^{2} \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3 \cdot 5^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $29$
• Trans deg.: $120$
• Rank: $2$

Group information

Description:	$\SL(2,5):D_5$
Order:	\(1200\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
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 2, $C_5$, $A_5$
Derived length:	$2$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	10	12	15	20	30
Elements	1	151	20	40	124	20	124	200	80	360	80	1200
Conjugacy classes	1	2	1	3	8	1	8	2	2	6	2	36
Divisions	1	2	1	2	4	1	4	1	1	2	1	20
Autjugacy classes	1	2	1	2	3	1	3	1	1	2	1	18

Dimension	1	2	3	4	5	6	8	10	12	16	20	24
Irr. complex chars.	2	6	4	8	2	6	4	2	2	0	0	0	36
Irr. rational chars.	2	0	0	3	2	2	4	0	3	2	1	1	20

Minimal presentations

Permutation degree:	$29$
Transitive degree:	$120$
Rank:	$2$
Inequivalent generating pairs:	$114$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	4	8
Arbitrary	4	4	8

Constructions

Permutation group:	Degree $29$ $\langle(1,2,7,6,14,17,8,4,13,5,10,3)(9,18,12,23,20,16,11,21,24,15,22,19)(25,28) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 0 & 4 & 0 & 0 \\ 0 & 0 & 4 & 0 \\ 0 & 0 & 0 & 4 \end{array}\right), \left(\begin{array}{rrrr} 3 & 1 & 2 & 1 \\ 1 & 2 & 0 & 4 \\ 3 & 4 & 4 & 2 \\ 2 & 0 & 2 & 2 \end{array}\right), \left(\begin{array}{rrrr} 3 & 0 & 0 & 0 \\ 3 & 2 & 0 & 0 \\ 2 & 2 & 1 & 2 \\ 3 & 2 & 4 & 4 \end{array}\right), \left(\begin{array}{rrrr} 2 & 4 & 4 & 1 \\ 1 & 1 & 0 & 3 \\ 2 & 1 & 3 & 0 \\ 1 & 1 & 2 & 4 \end{array}\right), \left(\begin{array}{rrrr} 0 & 2 & 4 & 2 \\ 1 & 3 & 4 & 4 \\ 3 & 0 & 0 & 2 \\ 0 & 2 & 1 & 2 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{5})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$\SL(2,5)$ $\,\rtimes\,$ $D_5$	$(C_5\times \SL(2,5))$ $\,\rtimes\,$ $C_2$	$C_5$ $\,\rtimes\,$ $(\SL(2,5):C_2)$		more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2$ . $(D_5\times A_5)$	$(C_5:C_4)$ . $A_5$	$C_{10}$ . $(C_2\times A_5)$		more information

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

Homology

Abelianization:	$C_{2} $
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 1562 subgroups in 70 conjugacy classes, 8 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $D_5\times A_5$
Commutator:	$G' \simeq$ $C_5\times \SL(2,5)$	$G/G' \simeq$ $C_2$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $D_5\times A_5$
Fitting:	$\operatorname{Fit} \simeq$ $C_{10}$	$G/\operatorname{Fit} \simeq$ $C_2\times A_5$
Radical:	$R \simeq$ $C_5:C_4$	$G/R \simeq$ $A_5$
Socle:	$\operatorname{soc} \simeq$ $C_{10}$	$G/\operatorname{soc} \simeq$ $C_2\times A_5$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4:C_2$
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

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

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

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

Order 240: $\SL(2,5):C_2$, $\SL(2,3):D_5$

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

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

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

Order 80: $D_{20}:C_2$

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: $\SL(2,3):C_2$

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

Order 30: $D_{15}$ x 2, $C_{30}$

Order 25: $C_5^2$

Order 24: $C_4\times S_3$, $\SL(2,3)$

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

Order 16: $D_4:C_2$

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: $Q_8$, $D_4$, $C_2\times C_4$

Order 6: $S_3$ x 2, $C_6$

Order 5: $C_5$ x 4

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

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

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

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

Order 240: $\SL(2,5):C_2$, $\SL(2,3):D_5$

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

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

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

Order 80: $D_{20}:C_2$

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

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

Order 48: $\SL(2,3):C_2$

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

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

Order 25: $C_5^2$

Order 24: $C_4\times S_3$, $\SL(2,3)$

Order 20: $D_{10}$ x 3, $C_{20}$ x 2, $C_5:C_4$ x 2

Order 16: $D_4:C_2$

Order 15: $C_{15}$

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

Order 10: $C_{10}$ x 3, $D_5$ x 3

Order 8: $Q_8$, $D_4$, $C_2\times C_4$

Order 6: $C_6$, $S_3$

Order 5: $C_5$ x 3

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

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1200: $\SL(2,5):D_5$ ($C_1$)

Order 600: $C_5\times \SL(2,5)$ ($C_2$)

Order 120: $\SL(2,5)$ ($D_5$)

Order 20: $C_5:C_4$ ($A_5$)

Order 10: $C_{10}$ ($C_2\times A_5$)

Order 5: $C_5$ ($\SL(2,5):C_2$)

Order 2: $C_2$ ($D_5\times A_5$)

Order 1: $C_1$ ($\SL(2,5):D_5$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1200: $\SL(2,5):D_5$ ($C_1$)

Order 600: $C_5\times \SL(2,5)$ ($C_2$)

Order 120: $\SL(2,5)$ ($D_5$)

Order 20: $C_5:C_4$ ($A_5$)

Order 10: $C_{10}$ ($C_2\times A_5$)

Order 5: $C_5$ ($\SL(2,5):C_2$)

Order 2: $C_2$ ($D_5\times A_5$)

Order 1: $C_1$ ($\SL(2,5):D_5$)

Series

Derived series	$\SL(2,5):D_5$	$\rhd$	$C_5\times \SL(2,5)$	$\rhd$	$\SL(2,5)$
Chief series	$\SL(2,5):D_5$	$\rhd$	$C_5:C_4$	$\rhd$	$C_{10}$	$\rhd$	$C_5$	$\rhd$	$C_1$
Lower central series	$\SL(2,5):D_5$	$\rhd$	$C_5\times \SL(2,5)$
Upper central series	$C_1$	$\lhd$	$C_2$

Supergroups

This group is a maximal subgroup of 7 larger groups in the database.

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

Character theory

Complex character table

See the $36 \times 36$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $20 \times 20$ rational character table.