Abstract group 2400.bt: $\SL(2,5):D_{10}$

• Label: 2400.bt
• Order: \( 2^{5} \cdot 3 \cdot 5^{2} \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3 \cdot 5^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $31$
• Trans deg.: $240$
• Rank: $2$

Group information

Description:	$\SL(2,5):D_{10}$
Order:	\(2400\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{2} \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$D_4\times F_5\times S_5$, of order \(19200\)\(\medspace = 2^{8} \cdot 3 \cdot 5^{2} \)
Composition factors:	$C_2$ x 3, $C_5$, $A_5$
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	303	20	80	124	60	372	400	80	720	240	2400
Conjugacy classes	1	5	1	6	8	3	24	4	2	12	6	72
Divisions	1	5	1	4	4	3	12	2	1	4	3	40
Autjugacy classes	1	3	1	3	3	2	6	1	1	3	2	26

Dimension	1	2	3	4	5	6	8	10	12	16	20	24
Irr. complex chars.	4	12	8	16	4	12	8	4	4	0	0	0	72
Irr. rational chars.	4	0	0	6	4	4	8	0	6	4	2	2	40

Minimal presentations

Permutation degree:	$31$
Transitive degree:	$240$
Rank:	$2$
Inequivalent generating pairs:	$57$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	4	5	9

Constructions

Permutation group:	Degree $31$ $\langle(1,2,5,3,6,9)(4,7,10,8,12,13)(11,16,20,14,18,23)(15,19,24,17,22,21)(25,26,27,28,29) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 0 & 4 & 0 & 0 \\ 0 & 0 & 4 & 0 \\ 0 & 0 & 0 & 4 \end{array}\right), \left(\begin{array}{rrrr} 2 & 4 & 2 & 0 \\ 3 & 3 & 4 & 0 \\ 0 & 1 & 1 & 4 \\ 1 & 3 & 2 & 2 \end{array}\right), \left(\begin{array}{rrrr} 3 & 1 & 4 & 0 \\ 4 & 3 & 1 & 2 \\ 3 & 0 & 4 & 2 \\ 4 & 1 & 1 & 4 \end{array}\right), \left(\begin{array}{rrrr} 1 & 1 & 3 & 4 \\ 1 & 0 & 4 & 0 \\ 4 & 0 & 0 & 1 \\ 1 & 1 & 4 & 4 \end{array}\right), \left(\begin{array}{rrrr} 1 & 3 & 0 & 2 \\ 1 & 0 & 4 & 0 \\ 1 & 2 & 0 & 2 \\ 1 & 1 & 4 & 4 \end{array}\right), \left(\begin{array}{rrrr} 0 & 0 & 4 & 4 \\ 1 & 4 & 4 & 4 \\ 0 & 0 & 4 & 0 \\ 1 & 0 & 4 & 3 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{5})$
Direct product:	$C_2$ $\, \times\, $ $(\SL(2,5):D_5)$
Semidirect product:	$\SL(2,5)$ $\,\rtimes\,$ $D_{10}$	$(C_2\times \SL(2,5))$ $\,\rtimes\,$ $D_5$	$(C_{10}\times \SL(2,5))$ $\,\rtimes\,$ $C_2$	$C_{10}$ $\,\rtimes\,$ $(\SL(2,5):C_2)$ (2)	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2$ . $(D_{10}\times A_5)$	$(C_{10}:C_4)$ . $A_5$	$C_2^2$ . $(D_5\times A_5)$	$C_{10}$ . $(C_2^2\times A_5)$	all 6

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

Homology

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

Subgroups

There are 5980 subgroups in 232 conjugacy classes, 20 normal (12 characteristic).

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

Special subgroups

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

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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 2400: $\SL(2,5):D_{10}$

Order 1200: $\SL(2,5):D_5$ x 2, $C_{10}\times \SL(2,5)$

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

Order 480: $\SL(2,5):C_2^2$, $\SL(2,3):D_{10}$

Order 400: $C_{10}^2.C_2^2$

Order 240: $\SL(2,5):C_2$ x 2, $\SL(2,3):D_5$ x 2, $C_2\times \SL(2,5)$, $C_{10}\times \SL(2,3)$, $D_{30}:C_4$

Order 200: $C_{10}.D_{10}$ x 4, $C_{10}:C_{20}$ x 2, $C_{10}:D_{10}$

Order 160: $Q_8:D_{10}$

Order 120: $D_{15}:C_4$ x 4, $\SL(2,5)$, $C_2\times D_{30}$, $C_6:C_{20}$, $C_{10}:C_{12}$, $C_5\times \SL(2,3)$

Order 100: $C_5:D_{10}$ x 6, $C_5:C_{20}$ x 4, $C_{10}^2$

Order 96: $\SL(2,3):C_2^2$

Order 80: $D_{20}:C_2$ x 4, $C_4\times D_{10}$ x 2, $Q_8\times C_{10}$, $C_2\times D_{20}$

Order 60: $D_{30}$ x 6, $C_5:C_{12}$ x 2, $C_3:C_{20}$ x 2, $C_2\times C_{30}$

Order 50: $C_5:D_5$ x 4, $C_5\times C_{10}$ x 3

Order 48: $\SL(2,3):C_2$ x 2, $C_4\times D_6$, $C_2\times \SL(2,3)$

Order 40: $C_4\times D_5$ x 8, $D_{20}$ x 4, $C_2\times D_{10}$ x 4, $C_2\times C_{20}$ x 2, $C_{10}:C_4$ x 2, $C_5\times Q_8$ x 2

Order 32: $D_4:C_2^2$

Order 30: $D_{15}$ x 4, $C_{30}$ x 3

Order 25: $C_5^2$

Order 24: $C_4\times S_3$ x 4, $C_2\times C_{12}$, $C_6:C_4$, $\SL(2,3)$, $C_2\times D_6$

Order 20: $D_{10}$ x 22, $C_2\times C_{10}$ x 4, $C_{20}$ x 4, $C_5:C_4$ x 4

Order 16: $D_4:C_2$ x 4, $C_2\times Q_8$, $C_2\times D_4$, $C_2^2\times C_4$

Order 15: $C_{15}$

Order 12: $D_6$ x 6, $C_{12}$ x 2, $C_3:C_4$ x 2, $C_2\times C_6$

Order 10: $D_5$ x 14, $C_{10}$ x 12

Order 8: $C_2\times C_4$ x 6, $D_4$ x 4, $Q_8$ x 2, $C_2^3$

Order 6: $S_3$ x 4, $C_6$ x 3

Order 5: $C_5$ x 4

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

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 2400: $\SL(2,5):D_{10}$

Order 1200: $C_{10}\times \SL(2,5)$, $\SL(2,5):D_5$

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

Order 480: $\SL(2,5):C_2^2$, $\SL(2,3):D_{10}$

Order 400: $C_{10}^2.C_2^2$

Order 240: $C_2\times \SL(2,5)$, $\SL(2,5):C_2$, $C_{10}\times \SL(2,3)$, $D_{30}:C_4$, $\SL(2,3):D_5$

Order 200: $C_{10}:C_{20}$ x 2, $C_{10}.D_{10}$ x 2, $C_{10}:D_{10}$

Order 160: $Q_8:D_{10}$

Order 120: $D_{15}:C_4$ x 2, $\SL(2,5)$, $C_2\times D_{30}$, $C_6:C_{20}$, $C_{10}:C_{12}$, $C_5\times \SL(2,3)$

Order 100: $C_5:C_{20}$ x 3, $C_5:D_{10}$ x 2, $C_{10}^2$

Order 96: $\SL(2,3):C_2^2$

Order 80: $D_{20}:C_2$ x 2, $C_4\times D_{10}$ x 2, $Q_8\times C_{10}$, $C_2\times D_{20}$

Order 60: $D_{30}$ x 2, $C_3:C_{20}$ x 2, $C_5:C_{12}$, $C_2\times C_{30}$

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

Order 48: $C_4\times D_6$, $\SL(2,3):C_2$, $C_2\times \SL(2,3)$

Order 40: $C_4\times D_5$ x 4, $C_2\times D_{10}$ x 3, $C_2\times C_{20}$ x 2, $C_{10}:C_4$ x 2, $D_{20}$ x 2, $C_5\times Q_8$ x 2

Order 32: $D_4:C_2^2$

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

Order 25: $C_5^2$

Order 24: $C_4\times S_3$ x 2, $C_2\times C_{12}$, $C_6:C_4$, $\SL(2,3)$, $C_2\times D_6$

Order 20: $D_{10}$ x 6, $C_2\times C_{10}$ x 3, $C_{20}$ x 3, $C_5:C_4$ x 3

Order 16: $D_4:C_2$ x 2, $C_2\times Q_8$, $C_2\times D_4$, $C_2^2\times C_4$

Order 15: $C_{15}$

Order 12: $D_6$ x 2, $C_3:C_4$ x 2, $C_2\times C_6$, $C_{12}$

Order 10: $C_{10}$ x 6, $D_5$ x 3

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

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

Order 5: $C_5$ x 3

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 2400: $\SL(2,5):D_{10}$ ($C_1$)

Order 1200: $\SL(2,5):D_5$ ($C_2$) x 2, $C_{10}\times \SL(2,5)$ ($C_2$)

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

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

Order 120: $\SL(2,5)$ ($D_{10}$)

Order 40: $C_{10}:C_4$ ($A_5$)

Order 20: $C_5:C_4$ ($C_2\times A_5$) x 2, $C_2\times C_{10}$ ($C_2\times A_5$)

Order 10: $C_{10}$ ($\SL(2,5):C_2$) x 2, $C_{10}$ ($C_2^2\times A_5$)

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

Order 4: $C_2^2$ ($D_5\times A_5$)

Order 2: $C_2$ ($\SL(2,5):D_5$) x 2, $C_2$ ($D_{10}\times A_5$)

Order 1: $C_1$ ($\SL(2,5):D_{10}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2400: $\SL(2,5):D_{10}$ ($C_1$)

Order 1200: $C_{10}\times \SL(2,5)$ ($C_2$), $\SL(2,5):D_5$ ($C_2$)

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

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

Order 120: $\SL(2,5)$ ($D_{10}$)

Order 40: $C_{10}:C_4$ ($A_5$)

Order 20: $C_2\times C_{10}$ ($C_2\times A_5$), $C_5:C_4$ ($C_2\times A_5$)

Order 10: $C_{10}$ ($\SL(2,5):C_2$), $C_{10}$ ($C_2^2\times A_5$)

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

Order 4: $C_2^2$ ($D_5\times A_5$)

Order 2: $C_2$ ($D_{10}\times A_5$), $C_2$ ($\SL(2,5):D_5$)

Order 1: $C_1$ ($\SL(2,5):D_{10}$)

Series

Derived series

$\SL(2,5):D_{10}$

$\rhd$

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

$\rhd$

$\SL(2,5)$

Chief series

$\SL(2,5):D_{10}$

$\rhd$

$C_{10}:C_4$

$\rhd$

$C_2\times C_{10}$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$\SL(2,5):D_{10}$

$\rhd$

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

Upper central series

$C_1$

$\lhd$

$C_2^2$

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 $72 \times 72$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $40 \times 40$ rational character table.