Abstract group 3840.w: $C_2^3:\GL(2,5)$

• Label: 3840.w
• Order: \( 2^{8} \cdot 3 \cdot 5 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{4} \)
• $\card{\Aut(G)}$: \( 2^{13} \cdot 3 \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{9} \)
• Perm deg.: $30$
• Trans deg.: $192$
• Rank: $3$

Group information

Description:	$C_2^3:\GL(2,5)$
Order:	\(3840\)\(\medspace = 2^{8} \cdot 3 \cdot 5 \)
Exponent:	\(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \)
Automorphism group:	$C_9^2:F_9$, of order \(122880\)\(\medspace = 2^{13} \cdot 3 \cdot 5 \)
Composition factors:	$C_2$ x 6, $A_5$
Derived length:	$2$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	8	10	12	20	24
Elements	1	255	20	1216	24	300	320	360	320	384	640	3840
Conjugacy classes	1	17	1	34	1	11	8	15	12	16	16	132
Divisions	1	17	1	20	1	11	4	11	6	6	4	82
Autjugacy classes	1	8	1	8	1	5	1	5	3	3	1	37

Dimension	1	2	4	5	6	8	10	12	16	24
Irr. complex chars.	16	4	48	16	32	8	4	4	0	0	132
Irr. rational chars.	8	8	8	8	8	12	8	12	8	2	82

Minimal presentations

Permutation degree:	$30$
Transitive degree:	$192$
Rank:	$3$
Inequivalent generating triples:	not computed

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

Permutation group:	Degree $30$ $\langle(1,4)(2,8)(5,14)(6,16)(7,17)(9,10)(11,22)(15,20)(18,21)(19,24), (2,11,3,14) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 9 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 9 & 6 \\ 2 & 15 \end{array}\right), \left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 17 & 15 \\ 0 & 7 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/20\Z)$
Direct product:	$C_2$ $\, \times\, $ $(C_2^2:\GL(2,5))$
Semidirect product:	$C_2^3$ $\,\rtimes\,$ $\GL(2,5)$	$(C_2^3\times \SL(2,5))$ $\,\rtimes\,$ $C_4$	$(\SL(2,5):C_2^3)$ $\,\rtimes\,$ $C_4$ (2)	$\SL(2,5)$ $\,\rtimes\,$ $(C_2^3:C_4)$	all 10
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2^3\times C_4)$ . $S_5$	$C_4$ . $(C_2^3.S_5)$	$C_4$ . $(C_2^3:S_5)$	$C_2$ . $(C_2^4.S_5)$ (3)	all 23

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{20}\Z)$.

Homology

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

Subgroups

There are 40937 subgroups in 2342 conjugacy classes, 84 normal (22 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2\times C_4$	$G/Z \simeq$ $C_2\times S_5$
Commutator:	$G' \simeq$ $C_2\times \SL(2,5)$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $C_2^2\times S_5$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^3\times C_4$	$G/\operatorname{Fit} \simeq$ $S_5$
Radical:	$R \simeq$ $C_2^3\times C_4$	$G/R \simeq$ $S_5$
Socle:	$\operatorname{soc} \simeq$ $C_2^3$	$G/\operatorname{soc} \simeq$ $C_2^2.S_5$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^5.D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
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

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 3840: $C_2^3:\GL(2,5)$

Order 1920: $C_2^3.A_5:C_4$ x 4, $C_2^3.A_5:C_4$ x 2, $(C_2^3\times C_4).A_5$

Order 960: $\SL(2,5):C_2^3$ x 10, $C_2\times \GL(2,5)$ x 8, $C_2^3\times \SL(2,5)$

Order 768: $C_2\times C_4.\GL(2,\mathbb{Z}/4)$

Order 640: $C_2\times D_{10}:C_4^2$

Order 480: $\SL(2,5):C_2^2$ x 18, $C_2^2\times \SL(2,5)$ x 5, $\GL(2,5)$ x 4

Order 384: $C_4.\GL(2,\mathbb{Z}/4)$ x 4, $C_2^2\times \Unitary(2,3)$ x 2, $\SL(2,3):C_2^4$, $C_2\times D_6:\OD_{16}$

Order 320: $D_{10}:C_4^2$ x 8, $C_2^4:F_5$ x 2, $D_{10}:C_4^2$ x 2, $D_{10}.C_4^2$ x 2, $C_{20}:C_2^4$

Order 256: $C_2^5.D_4$

Order 240: $\SL(2,5):C_2$ x 6, $C_2\times \SL(2,5)$ x 5

Order 192: $\SL(2,3):C_2^3$ x 10, $C_2\times \Unitary(2,3)$ x 8, $D_6:\OD_{16}$ x 8, $C_2\times D_6:C_8$ x 2, $C_{24}:C_2^3$ x 2, $C_2^3:C_{24}$, $C_2\times C_{12}.D_4$, $C_{12}:C_2^4$, $C_2^3\times \SL(2,3)$

Order 160: $C_2^3:F_5$ x 24, $C_{20}:C_2^3$ x 20, $C_{10}:C_4^2$ x 16, $C_{10}.C_4^2$ x 8, $C_2^3\times F_5$ x 4, $C_2^3\times D_{10}$, $C_2^3\times C_{20}$, $C_{10}.C_2^4$

Order 128: $C_2^4.D_4$ x 8, $C_4^2:C_2^3$ x 4, $D_4:C_2^4$, $C_2^3:\OD_{16}$, $C_2^3:C_4^2$

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

Order 96: $C_{12}:C_2^3$ x 20, $\SL(2,3):C_2^2$ x 18, $C_8:D_6$ x 16, $D_6:C_8$ x 8, $C_2^2\times \SL(2,3)$ x 5, $\Unitary(2,3)$ x 4, $C_2^2:C_{24}$ x 4, $C_{12}.D_4$ x 4, $C_2^2\times C_{24}$ x 2, $C_{12}.C_2^3$ x 2, $C_2^3\times D_6$, $C_2^3\times C_{12}$, $C_6.C_2^4$

Order 80: $C_4\times D_{10}$ x 68, $C_2^2\times F_5$ x 40, $D_{10}:C_4$ x 32, $C_2^2\times D_{10}$ x 22, $C_4\times F_5$ x 16, $C_2^2\times C_{20}$ x 10, $C_2^2.D_{10}$ x 10, $C_2^3\times C_{10}$

Order 64: $C_4^2:C_2^2$ x 32, $D_4:C_2^3$ x 20, $C_2^2:\OD_{16}$ x 6, $C_2^2:C_4^2$ x 6, $C_2^3:C_8$ x 2, $C_2^2\times \OD_{16}$ x 2, $C_2^2\times C_4^2$ x 2, $C_2^2.C_4^2$, $Q_8\times C_2^3$, $D_4\times C_2^3$, $C_2^4\times C_4$, $C_2^4:C_4$

Order 48: $C_4\times D_6$ x 68, $C_2^2\times D_6$ x 18, $C_{24}:C_2$ x 16, $C_2^2\times C_{12}$ x 10, $C_6.C_2^3$ x 10, $C_6:C_8$ x 8, $C_2\times C_{24}$ x 8, $\SL(2,3):C_2$ x 6, $C_2\times \SL(2,3)$ x 5, $C_2^3\times C_6$

Order 40: $C_2\times D_{10}$ x 88, $C_2\times F_5$ x 64, $C_4\times D_5$ x 40, $C_2\times C_{20}$ x 18, $C_{10}:C_4$ x 18, $C_2^2\times C_{10}$ x 11

Order 32: $D_4:C_2^2$ x 100, $C_4\wr C_2$ x 32, $C_2^3\times C_4$ x 24, $C_2^2\times D_4$ x 17, $C_2\times \OD_{16}$ x 14, $C_2\times C_4^2$ x 14, $C_2^3:C_4$ x 12, $C_2^2\times Q_8$ x 10, $C_2^2:C_8$ x 8, $C_2.C_4^2$ x 4, $C_2^2\times C_8$ x 2, $C_2^5$

Order 24: $C_2\times D_6$ x 76, $C_4\times S_3$ x 40, $C_2\times C_{12}$ x 18, $C_6:C_4$ x 18, $C_2^2\times C_6$ x 11, $C_{24}$ x 4, $C_3:C_8$ x 4, $\SL(2,3)$

Order 20: $D_{10}$ x 76, $C_2\times C_{10}$ x 23, $F_5$ x 16, $C_{20}$ x 6, $C_5:C_4$ x 6

Order 16: $C_2^2\times C_4$ x 108, $D_4:C_2$ x 104, $C_2\times D_4$ x 62, $C_2\times Q_8$ x 27, $C_2^4$ x 17, $C_2^2:C_4$ x 16, $\OD_{16}$ x 12, $C_4^2$ x 12, $C_2\times C_8$ x 8

Order 12: $D_6$ x 64, $C_2\times C_6$ x 23, $C_{12}$ x 6, $C_3:C_4$ x 6

Order 10: $D_5$ x 12, $C_{10}$ x 11

Order 8: $C_2\times C_4$ x 108, $C_2^3$ x 65, $D_4$ x 36, $Q_8$ x 15, $C_8$ x 4

Order 6: $C_6$ x 11, $S_3$ x 8

Order 5: $C_5$

Order 4: $C_2^2$ x 65, $C_4$ x 20

Order 3: $C_3$

Order 2: $C_2$ x 17

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 3840: $C_2^3:\GL(2,5)$

Order 1920: $(C_2^3\times C_4).A_5$, $C_2^3.A_5:C_4$, $C_2^3.A_5:C_4$

Order 960: $\SL(2,5):C_2^3$ x 5, $C_2\times \GL(2,5)$ x 2, $C_2^3\times \SL(2,5)$

Order 768: $C_2\times C_4.\GL(2,\mathbb{Z}/4)$

Order 640: $C_2\times D_{10}:C_4^2$

Order 480: $\SL(2,5):C_2^2$ x 7, $C_2^2\times \SL(2,5)$ x 3, $\GL(2,5)$

Order 384: $C_4.\GL(2,\mathbb{Z}/4)$, $\SL(2,3):C_2^4$, $C_2^2\times \Unitary(2,3)$, $C_2\times D_6:\OD_{16}$

Order 320: $D_{10}:C_4^2$ x 2, $C_{20}:C_2^4$, $C_2^4:F_5$, $D_{10}:C_4^2$, $D_{10}.C_4^2$

Order 256: $C_2^5.D_4$

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

Order 192: $\SL(2,3):C_2^3$ x 5, $C_2\times \Unitary(2,3)$ x 2, $D_6:\OD_{16}$ x 2, $C_2^3:C_{24}$, $C_2\times C_{12}.D_4$, $C_2\times D_6:C_8$, $C_{12}:C_2^4$, $C_2^3\times \SL(2,3)$, $C_{24}:C_2^3$

Order 160: $C_{20}:C_2^3$ x 10, $C_{10}:C_4^2$ x 4, $C_2^3:F_5$ x 2, $C_{10}.C_4^2$, $C_2^3\times D_{10}$, $C_2^3\times F_5$, $C_2^3\times C_{20}$, $C_{10}.C_2^4$

Order 128: $C_2^4.D_4$ x 2, $C_4^2:C_2^3$ x 2, $D_4:C_2^4$, $C_2^3:\OD_{16}$, $C_2^3:C_4^2$

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

Order 96: $C_{12}:C_2^3$ x 10, $\SL(2,3):C_2^2$ x 7, $C_8:D_6$ x 4, $C_2^2\times \SL(2,3)$ x 3, $\Unitary(2,3)$, $C_2^2:C_{24}$, $C_{12}.D_4$, $D_6:C_8$, $C_2^3\times D_6$, $C_2^3\times C_{12}$, $C_6.C_2^4$, $C_2^2\times C_{24}$, $C_{12}.C_2^3$

Order 80: $C_4\times D_{10}$ x 22, $C_2^2\times D_{10}$ x 7, $C_2^2.D_{10}$ x 6, $C_2^2\times C_{20}$ x 5, $C_2^2\times F_5$ x 4, $C_4\times F_5$ x 3, $C_2^3\times C_{10}$, $D_{10}:C_4$

Order 64: $D_4:C_2^3$ x 10, $C_4^2:C_2^2$ x 6, $C_2^2:\OD_{16}$ x 2, $C_2^3:C_8$ x 2, $C_2^2:C_4^2$ x 2, $C_2^2.C_4^2$, $Q_8\times C_2^3$, $D_4\times C_2^3$, $C_2^4\times C_4$, $C_2^2\times \OD_{16}$, $C_2^4:C_4$, $C_2^2\times C_4^2$

Order 48: $C_4\times D_6$ x 22, $C_2^2\times D_6$ x 7, $C_6.C_2^3$ x 6, $C_2^2\times C_{12}$ x 5, $C_{24}:C_2$ x 3, $\SL(2,3):C_2$ x 3, $C_2\times \SL(2,3)$ x 3, $C_6:C_8$ x 2, $C_2\times C_{24}$ x 2, $C_2^3\times C_6$

Order 40: $C_2\times D_{10}$ x 16, $C_4\times D_5$ x 13, $C_{10}:C_4$ x 9, $C_2\times C_{20}$ x 7, $C_2^2\times C_{10}$ x 5, $C_2\times F_5$ x 4

Order 32: $D_4:C_2^2$ x 29, $C_2^3\times C_4$ x 13, $C_2^2\times D_4$ x 8, $C_2^2\times Q_8$ x 6, $C_2\times \OD_{16}$ x 4, $C_2\times C_4^2$ x 4, $C_4\wr C_2$ x 4, $C_2^2:C_8$ x 2, $C_2^3:C_4$ x 2, $C_2^5$, $C_2^2\times C_8$, $C_2.C_4^2$

Order 24: $C_2\times D_6$ x 16, $C_4\times S_3$ x 13, $C_6:C_4$ x 9, $C_2\times C_{12}$ x 7, $C_2^2\times C_6$ x 5, $\SL(2,3)$, $C_{24}$, $C_3:C_8$

Order 20: $D_{10}$ x 13, $C_2\times C_{10}$ x 8, $C_5:C_4$ x 4, $C_{20}$ x 3, $F_5$

Order 16: $C_2^2\times C_4$ x 37, $D_4:C_2$ x 23, $C_2\times D_4$ x 17, $C_2\times Q_8$ x 12, $C_2^4$ x 8, $\OD_{16}$ x 3, $C_4^2$ x 3, $C_2\times C_8$ x 2, $C_2^2:C_4$

Order 12: $D_6$ x 13, $C_2\times C_6$ x 8, $C_3:C_4$ x 4, $C_{12}$ x 3

Order 10: $C_{10}$ x 5, $D_5$ x 3

Order 8: $C_2\times C_4$ x 33, $C_2^3$ x 21, $D_4$ x 10, $Q_8$ x 7, $C_8$

Order 6: $C_6$ x 5, $S_3$ x 3

Order 5: $C_5$

Order 4: $C_2^2$ x 21, $C_4$ x 8

Order 3: $C_3$

Order 2: $C_2$ x 8

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 3840: $C_2^3:\GL(2,5)$ ($C_1$)

Order 1920: $C_2^3.A_5:C_4$ ($C_2$) x 4, $C_2^3.A_5:C_4$ ($C_2$) x 2, $(C_2^3\times C_4).A_5$ ($C_2$)

Order 960: $C_2\times \GL(2,5)$ ($C_2^2$) x 4, $\SL(2,5):C_2^3$ ($C_2^2$) x 3, $\SL(2,5):C_2^3$ ($C_4$) x 3, $C_2^3\times \SL(2,5)$ ($C_4$)

Order 480: $\SL(2,5):C_2^2$ ($D_4$) x 4, $C_2^2\times \SL(2,5)$ ($C_2\times C_4$) x 3, $\SL(2,5):C_2^2$ ($C_2\times C_4$) x 3, $\SL(2,5):C_2^2$ ($C_2^3$)

Order 240: $C_2\times \SL(2,5)$ ($C_2^2:C_4$) x 2, $\SL(2,5):C_2$ ($C_2^2:C_4$) x 2, $\SL(2,5):C_2$ ($C_2\times D_4$) x 2, $C_2\times \SL(2,5)$ ($C_2^2\times C_4$)

Order 120: $\SL(2,5)$ ($C_2^3:C_4$)

Order 32: $C_2^3\times C_4$ ($S_5$)

Order 16: $C_2^2\times C_4$ ($A_5:C_4$) x 3, $C_2^2\times C_4$ ($C_2\times S_5$) x 3, $C_2^4$ ($A_5:C_4$)

Order 8: $C_2^3$ ($\GL(2,5)$) x 4, $C_2\times C_4$ ($C_2^2:S_5$) x 4, $C_2^3$ ($C_2^2.S_5$) x 3, $C_2\times C_4$ ($C_2^2.S_5$) x 3, $C_2\times C_4$ ($C_2^2\times S_5$)

Order 4: $C_2^2$ ($C_2\times \GL(2,5)$) x 6, $C_2^2$ ($C_2^3.S_5$) x 2, $C_2^2$ ($\GL(2,5):C_2$) x 2, $C_4$ ($C_2^3.S_5$) x 2, $C_4$ ($C_2^3:S_5$) x 2, $C_2^2$ ($C_2^3.S_5$)

Order 2: $C_2$ ($C_2^3.A_5:C_4$) x 4, $C_2$ ($(C_2^2\times C_4).S_5$), $C_2$ ($C_2^4.S_5$), $C_2$ ($(C_2^2\times C_4).S_5$)

Order 1: $C_1$ ($C_2^3:\GL(2,5)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 3840: $C_2^3:\GL(2,5)$ ($C_1$)

Order 1920: $(C_2^3\times C_4).A_5$ ($C_2$), $C_2^3.A_5:C_4$ ($C_2$), $C_2^3.A_5:C_4$ ($C_2$)

Order 960: $\SL(2,5):C_2^3$ ($C_2^2$) x 2, $\SL(2,5):C_2^3$ ($C_4$) x 2, $C_2\times \GL(2,5)$ ($C_2^2$), $C_2^3\times \SL(2,5)$ ($C_4$)

Order 480: $C_2^2\times \SL(2,5)$ ($C_2\times C_4$) x 2, $\SL(2,5):C_2^2$ ($C_2\times C_4$) x 2, $\SL(2,5):C_2^2$ ($C_2^3$), $\SL(2,5):C_2^2$ ($D_4$)

Order 240: $C_2\times \SL(2,5)$ ($C_2^2:C_4$), $C_2\times \SL(2,5)$ ($C_2^2\times C_4$), $\SL(2,5):C_2$ ($C_2^2:C_4$), $\SL(2,5):C_2$ ($C_2\times D_4$)

Order 120: $\SL(2,5)$ ($C_2^3:C_4$)

Order 32: $C_2^3\times C_4$ ($S_5$)

Order 16: $C_2^2\times C_4$ ($A_5:C_4$) x 2, $C_2^2\times C_4$ ($C_2\times S_5$) x 2, $C_2^4$ ($A_5:C_4$)

Order 8: $C_2^3$ ($C_2^2.S_5$) x 2, $C_2\times C_4$ ($C_2^2.S_5$) x 2, $C_2^3$ ($\GL(2,5)$), $C_2\times C_4$ ($C_2^2:S_5$), $C_2\times C_4$ ($C_2^2\times S_5$)

Order 4: $C_2^2$ ($C_2\times \GL(2,5)$) x 2, $C_2^2$ ($C_2^3.S_5$), $C_2^2$ ($\GL(2,5):C_2$), $C_2^2$ ($C_2^3.S_5$), $C_4$ ($C_2^3.S_5$), $C_4$ ($C_2^3:S_5$)

Order 2: $C_2$ ($C_2^3.A_5:C_4$), $C_2$ ($(C_2^2\times C_4).S_5$), $C_2$ ($C_2^4.S_5$), $C_2$ ($(C_2^2\times C_4).S_5$)

Order 1: $C_1$ ($C_2^3:\GL(2,5)$)

Series

Derived series

$C_2^3:\GL(2,5)$

$\rhd$

$C_2\times \SL(2,5)$

$\rhd$

$\SL(2,5)$

Chief series

$C_2^3:\GL(2,5)$

$\rhd$

$(C_2^3\times C_4).A_5$

$\rhd$

$C_2^3\times C_4$

$\rhd$

$C_2^4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^3:\GL(2,5)$

$\rhd$

$C_2\times \SL(2,5)$

$\rhd$

$\SL(2,5)$

Upper central series

$C_1$

$\lhd$

$C_2^2\times C_4$

$\lhd$

$C_2^3\times C_4$

Supergroups

This group is a maximal subgroup of 6 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

See the $132 \times 132$ 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 $82 \times 82$ rational character table.