Abstract group 15360.dy: $C_2\wr D_6.F_5$

• Label: 15360.dy
• Order: \( 2^{10} \cdot 3 \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: 2
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3 \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $17$
• Trans deg.: not computed
• Rank: $4$

Group information

Description:	$C_2\wr D_6.F_5$
Order:	\(15360\)\(\medspace = 2^{10} \cdot 3 \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$C_2^2\times C_2^4.(C_6\times D_5).C_2^4$, of order \(61440\)\(\medspace = 2^{12} \cdot 3 \cdot 5 \)
Composition factors:	$C_2$ x 10, $C_3$, $C_5$
Derived length:	$3$

This group is nonabelian and solvable. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	5	6	10	12	15	20	30	60
Elements	1	863	32	7328	4	1120	572	2944	128	1472	640	256	15360
Conjugacy classes	1	35	1	96	1	7	17	12	1	15	3	1	190
Divisions	1	35	1	63	1	7	17	7	1	15	3	1	152
Autjugacy classes	1	31	1	72	1	7	15	10	1	12	3	1	155

Dimension	1	2	3	4	6	8	12	16	24	48
Irr. complex chars.	32	24	32	12	56	6	12	1	14	1	190
Irr. rational chars.	16	20	16	16	36	7	24	1	15	1	152

Minimal presentations

Permutation degree:	$17$
Transitive degree:	not computed
Rank:	$4$
Inequivalent generating quadruples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	24	24	24
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h \mid a^{4}=c^{6}=d^{4}=e^{2}=f^{2}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $17$ $\langle(1,2)(3,7)(4,8)(5,6)(10,11)(12,13)(15,17), (1,3)(2,7)(4,5)(6,8)(10,11)(12,13) \!\cdots\! \rangle$
Direct product:	not computed
Semidirect product:	$C_5$ $\,\rtimes\,$ $(C_2\wr D_6.C_4)$				more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_2\wr D_6)$ . $F_5$	$F_5$ . $(C_2\wr D_6)$ (2)	$D_5$ . $(C_2^7:D_6)$	$(F_5\times C_2^6)$ . $D_6$	all 157
Aut. group:	$\Aut(Q_8\times D_{10})$	$\Aut(Q_8:D_{10})$	$\Aut(C_2\times D_{20}.C_4)$	$\Aut(D_{10}.C_2^4)$

Elements of the group are displayed as permutations of degree 17.

Homology

Abelianization:	$C_{2}^{3} \times C_{4} $
Schur multiplier:	$C_{2}^{8}$
Commutator length:	$1$

Subgroups

There are 248 normal subgroups (224 characteristic).

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

Special subgroups

Center:	a subgroup isomorphic to $C_2$
Commutator:	a subgroup isomorphic to $C_2^4:C_{30}$
Frattini:	a subgroup isomorphic to $C_2^3$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^5.C_2^5$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$

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

All subgroups of index up to 20 (order at least 768) are shown, as well as all normal subgroups of any index.

Order 15360: $C_2\wr D_6.F_5$

Order 7680: $C_2^4.D_6:F_5.C_2$ x 2, $C_2^4.(C_6\times D_5).C_2^3$ x 2, $C_2^4.(S_3\times D_4\times D_5)$, $C_2^4.(S_3\times C_4:F_5)$, $C_2^4.(F_5\times D_{12})$, $C_2^4.(F_5\times C_3:D_4)$, $C_2^4.(D_4\times C_3:F_5)$, $C_2^4.(C_6\times F_5).C_2^2$, $C_2^4.(C_4\times S_3\times F_5)$, $(C_2^4\times C_{10}).C_6.C_2^3$, $C_2^4.(F_5\times S_4)$, $C_2\wr S_3\times F_5$, $C_2^4:S_4\times F_5$

Order 5120: $F_5\times C_2^5:D_4$

Order 3840: $C_2^3:S_4\times F_5$ x 9, $C_2^4.D_6:F_5$ x 4, $F_5\times C_2^3:S_4$ x 4, $(C_2^3\times F_5):S_4$ x 3, $C_2^4.D_{15}:D_4$ x 2, $C_2^4.C_3:(C_4:F_5)$ x 2, $C_2^4.(D_5\times C_3:D_4)$ x 2, $C_2^4.(C_3:C_4\times F_5)$ x 2, $C_2^4.(C_2\times C_6):F_5$ x 2, $C_2\times C_2^4.(C_6\times D_5).C_2$ x 2, $(C_2^4\times C_{10}).C_{12}.C_2$ x 2, $(C_2^4\times C_{10}).C_6.C_2^2$ x 2, $C_2^4.D_{30}:C_2.C_2$, $C_2^4.D_{10}:S_3.C_2$, $C_2^4.C_{20}:D_6$, $C_2^4.C_{12}:F_5$, $C_2^4.(S_3\times D_{20})$, $C_2^4.(D_5\times D_{12})$, $C_2^4.(D_4\times D_{15})$, $C_2^4.(C_5\times S_3\times D_4)$, $C_2^4.(C_4\times S_3\times D_5)$, $C_2^4.(C_4\times C_3:F_5)$, $(C_5\times C_2^4:C_3).C_4^2$, $C_2^2\wr C_3\times F_5$, $(C_2^3\times D_{10}).S_4$, $D_4\times F_5\times S_4$, $C_2^4:A_4\times F_5$, $C_2^3:S_4\times D_{10}$

Order 3072: $C_2\wr D_6.C_4$

Order 2560: $C_5:(C_2^5.C_2^4)$ x 14, $C_5:(C_2^6.C_2^3)$ x 9, $C_2^4:D_4\times F_5$ x 4, $C_2^4:D_4\times F_5$ x 2, $C_2^4:D_4\times F_5$, $C_2^3\wr C_2\times F_5$

Order 1920: $F_5\times C_2^2:S_4$ x 12, $C_2^2:S_4\times D_{10}$ x 9, $(C_2^2\times D_{10}).S_4$ x 6, $(C_2^2\times F_5):S_4$ x 6, $C_2^3:A_4\times F_5$ x 4, $D_5\times C_2^3:S_4$ x 4, $C_2^2\times F_5\times S_4$ x 3, $F_5\times \GL(2,\mathbb{Z}/4)$ x 3, $\GL(2,\mathbb{Z}/4):F_5$ x 2, $D_{10}:C_4\times S_4$ x 2, $(C_2^2\times D_{10}):S_4$ x 2, $(C_2^3\times D_{10}):C_{12}$ x 2, $C_2^3:A_4\times F_5$ x 2, $(C_2^2\times D_{10}).S_4$ x 2, $C_2^4.C_5:D_{12}$ x 2, $C_2^2\times D_5\times C_2^4:C_3$, $C_2^2\times C_2^4:D_{15}$, $C_2^4.D_{30}.C_2$, $C_2^4.C_3:D_{20}$, $C_2^4.D_{10}:S_3$, $D_5\times C_2^2:(A_4:C_4)$, $\GL(2,\mathbb{Z}/4):D_{10}$, $D_4\times A_4\times F_5$, $D_4\times A_4:F_5$, $C_{20}:(C_4\times S_4)$, $C_{20}:C_4\times S_4$, $F_5\times C_4:S_4$, $C_4\times F_5\times S_4$, $C_2^3:A_4\times D_{10}$, $C_2^5:D_{30}$, $(C_2^4\times C_{10}).C_6.C_2$, $(C_2^4\times C_{10}).C_6.C_2$, $(C_2^4\times C_{10}).C_6.C_2$, $D_5\times C_2^5.C_6$, $C_2^4.D_{30}.C_2$, $C_2^4.D_{60}$, $C_2^4.D_{30}.C_2$, $C_2^4.C_3:D_{20}$, $C_2^4.D_{10}:S_3$, $C_2^4.D_{15}:C_4$, $C_2^4.(S_3\times C_5:C_4)$, $C_2\times C_{10}\times C_2^2:S_4$, $C_5\times C_2^4:C_3.D_4$, $C_2^3:S_4\times C_{10}$, $C_5\times C_2^5.(C_2\times C_6)$, $C_5\times C_2^4:C_3.D_4$, $C_5\times C_2^4.D_{12}$, $C_5\times C_2^4.(C_4\times S_3)$

Order 1536: $C_2^7.D_6$, $C_2^7.D_6$, $C_2^7:D_6$, $C_4\times C_2^4:S_4$, $C_2^6.C_2.D_6$, $C_4\times C_2^4:C_3.D_4$, $C_2^6.C_2^2.S_3$, $C_2^6.C_2^2.S_3$, $C_2^6.C_2^2.S_3$, $C_2^6.C_2^2.S_3$, $C_2^6.C_2.D_6$, $C_2^7.D_6$, $C_4\times C_2^4.D_{12}$, $C_4\times C_2^4.(C_4\times S_3)$, $C_2^4.C_2^5.C_3$

Order 1280: $C_2^3:D_4\times F_5$ x 37, $C_2\wr C_2^2\times F_5$ x 16, $C_2^5:(C_2\times F_5)$ x 8, $C_2^4:C_4\times F_5$ x 8, $C_5:(C_2^6.C_2^2)$ x 8, $C_5:(C_2^6.C_2^2)$ x 8, $C_2^4:C_4\times F_5$ x 8, $C_5:(C_2^5.C_2^3)$ x 8, $C_5:(C_2^6.C_2^2)$ x 8, $(D_4\times D_{10}).D_4$ x 6, $C_5:(C_2^4.C_2^4)$ x 4, $C_2^4.C_2^3\times D_5$ x 4, $C_5:(C_2^3.C_2^5)$ x 4, $C_5:(C_2^3.C_2^5)$ x 4, $C_5:(C_2^3.C_2^5)$ x 4, $F_5\times D_4^2$ x 4, $C_5:(C_2^5.C_2^3)$ x 4, $C_5:(C_2^6.C_2^2)$ x 4, $C_5:(C_2^5.C_2^3)$ x 4, $C_5:(C_2^5.C_2^3)$ x 4, $C_5:(C_2^5.C_2^3)$ x 4, $C_5:(C_2^5.C_2^3)$ x 4, $C_5:(C_2^5.C_2^3)$ x 4, $C_5:(C_2^5.C_2^3)$ x 4, $C_2^2:C_4:C_4\times F_5$ x 4, $C_5:(C_2^5.C_2^3)$ x 3, $C_5:(C_2^6.C_2^2)$ x 3, $F_5\times C_2^4:C_2^2$ x 3, $C_2^6.C_2\times D_5$ x 2, $C_5:(C_2^4.C_2^4)$ x 2, $C_5:(C_2^4.C_2^4)$ x 2, $C_5:(C_2^4.C_2^4)$ x 2, $C_5:(C_2^4.C_2^4)$ x 2, $C_2^5.C_2^2\times D_5$ x 2, $C_2^4.C_2^3\times D_5$ x 2, $D_{10}.C_2^6$ x 2, $C_2^2\times C_2^3:(C_2\times F_5)$ x 2, $C_2^2\times D_5.C_2^2\wr C_2$ x 2, $C_5:(C_2^3.C_2^5)$ x 2, $C_5:(C_2\times C_2^5.C_2^2)$ x 2, $(C_2^3\times F_5):C_2^3$ x 2, $C_5:(C_2^7.C_2)$ x 2, $C_5:(C_2^5.C_2^3)$ x 2, $C_5:(C_2^5.C_2^3)$ x 2, $C_5:(C_2^6.C_2^2)$ x 2, $C_5:(C_2^6.C_2^2)$ x 2, $C_5:(C_2^6.C_2^2)$ x 2, $C_2\wr C_2^2\times D_{10}$ x 2, $C_5:(C_2^4.C_2^4)$, $C_5:(C_2^4.C_2^4)$, $C_5:(C_2^4.C_2^4)$, $C_5\times C_2^4.C_2^4$, $C_2^5.C_2^2\times D_5$, $F_5\times C_2^6$, $D_{10}.C_2^6$, $C_5:(C_2^4.C_2^4)$, $C_5:(C_2^5.C_2^3)$, $C_5:(C_2^5.C_2^3)$, $C_5:(C_2^4.C_2^4)$, $C_5:(C_2^6.C_2^2)$, $C_5:(C_2^3.C_2^5)$, $F_5\times C_4^2:C_2^2$, $C_2^2\times (C_2^3\times C_{10}):C_4$, $C_5:(C_2^3.C_2^5)$, $C_5:(C_2\times C_2^5.C_2^2)$, $C_2^4.C_2^3\times D_5$, $C_2^2\times C_2^2\wr C_2\times D_5$

Order 1024: $C_2^5.C_2^5$

Order 960: $C_2\times F_5\times S_4$ x 29, $D_5\times C_2^2:S_4$ x 12, $C_2^4:D_{30}$ x 5, $C_2^2:S_4\times C_{10}$ x 5, $(C_2\times D_{10}).S_4$ x 5, $D_5.\GL(2,\mathbb{Z}/4)$ x 5, $(C_2\times D_{10}):S_4$ x 4, $C_2^2:A_4\times D_{10}$ x 4, $C_2^2\times A_4\times F_5$ x 4, $C_2\times D_{10}\times S_4$ x 3, $C_2^2:A_4\times F_5$ x 3, $C_2^2\times A_4:F_5$ x 3, $D_5\times \GL(2,\mathbb{Z}/4)$ x 3, $D_5.\GL(2,\mathbb{Z}/4)$ x 3, $D_5.\GL(2,\mathbb{Z}/4)$ x 3, $A_4:C_4\times F_5$ x 3, $D_5\times C_2^3:A_4$ x 2, $C_2^4:D_{30}$ x 2, $C_5\times C_2^3:S_4$ x 2, $(C_2^3\times C_{10}):C_{12}$ x 2, $\GL(2,\mathbb{Z}/4):D_5$ x 2, $C_5:D_4\times S_4$ x 2, $A_4\times D_{10}:C_4$ x 2, $C_2^5:C_{30}$, $(C_2^3\times C_{10}):C_{12}$, $(C_2^2\times D_{10}):A_4$, $C_2^4:C_{60}$, $C_2^5:C_{30}$, $D_4\times D_5\times A_4$, $C_2^5.D_{15}$, $C_5\times C_2^3.S_4$, $C_2^4:D_{30}$, $\GL(2,\mathbb{Z}/4):C_{10}$, $S_3\times D_4\times F_5$, $D_{20}:S_4$, $S_4\times D_{20}$, $D_5\times C_4:S_4$, $C_4\times D_5\times S_4$, $A_4\times C_{20}:C_4$, $C_4\times A_4\times F_5$, $(A_4\times C_{20}):C_4$, $C_4\times A_4:F_5$

Order 768: $C_2^6.D_6$ x 9, $C_2\wr D_6$ x 8, $C_4\times C_2^3:S_4$ x 4, $C_2^6.D_6$ x 4, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 3, $C_2^4:C_3.C_4^2$ x 2, $C_2^5:S_4$, $C_2^3:\GL(2,\mathbb{Z}/4)$, $C_2^2\times C_2^2:(A_4:C_4)$, $C_2^3.\GL(2,\mathbb{Z}/4)$, $C_2^4.(C_6.Q_8)$, $C_2^3.\GL(2,\mathbb{Z}/4)$, $C_4\times C_2^2:(A_4:C_4)$, $C_2^5:S_4$, $C_4\times D_4\times S_4$, $C_2^5:D_{12}$, $C_2^5:S_4$, $C_2^5.S_4$, $C_2^4.C_{12}:C_4$, $C_2^4.(C_6.Q_8)$, $C_2^3.\GL(2,\mathbb{Z}/4)$, $C_2^6:C_{12}$, $C_2^7:C_6$, $C_4\times C_2^4:A_4$, $C_2^5.(C_2\times C_{12})$, $C_2^6.C_2^2.C_3$, $C_2^6:C_{12}$, $C_2^4.C_2^4.C_3$

Order 640: $F_5\times C_2^5$ x 2, $C_2^5:F_5$ x 2, $C_5:(C_2^6.C_2)$, $C_2^5.D_{10}$, $C_2^6\times D_5$, $D_{10}.C_2^5$, $C_2^5:D_{10}$, $C_5\times C_2^6.C_2$, $C_5:(C_2^6.C_2)$, $C_5:(C_2^6.C_2)$

Order 480: $D_5\times C_2^2:A_4$ x 2, $C_2^4:D_{15}$ x 2, $C_5\times C_2^2:S_4$ x 2, $C_2^4:C_{30}$

Order 384: $C_2\wr C_6$, $C_2^4:S_4$, $C_2^4:D_{12}$, $C_2^5.D_6$, $C_2^2\wr S_3$, $C_2\wr S_3$, $C_2^4:S_4$

Order 320: $C_2^4\times F_5$ x 4, $C_2^4\times D_{10}$ x 2, $C_2^4:F_5$ x 2, $C_2^4:C_{20}$, $C_2^4.D_{10}$, $C_2^5\times C_{10}$, $C_{20}:C_2^4$, $D_{10}.C_2^4$, $D_{10}:C_4^2$, $C_2^5:C_{10}$, $C_2^4:D_{10}$

Order 240: $C_2^4:C_{15}$

Order 192: $C_2^3:S_4$ x 3, $C_2^4:C_{12}$, $C_2^2\wr C_3$, $C_2^3.S_4$, $C_2^4:A_4$

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

Order 128: $C_2^3\wr C_2$

Order 96: $C_2^2:S_4$ x 2, $C_2^3:A_4$

Order 80: $C_2^3\times C_{10}$ x 3, $C_2^2\times D_{10}$ x 3, $C_2^2\times F_5$ x 3, $C_2^2\times C_{20}$, $C_2^2.D_{10}$

Order 64: $C_2^4:C_4$, $C_2^6$, $C_2^3:D_4$

Order 48: $C_2^2:A_4$

Order 40: $C_2\times D_{10}$ x 3, $C_2\times F_5$ x 2, $C_2^2\times C_{10}$

Order 32: $C_2^5$, $C_2^2\times D_4$

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

Order 16: $C_2^4$ x 3, $C_2^2\times C_4$

Order 10: $D_5$ x 2, $C_{10}$

Order 8: $C_2^3$

Order 5: $C_5$

Order 4: $C_2^2$ x 2

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 15360: $C_2\wr D_6.F_5$ ($C_1$)

Order 7680: $C_2^4.D_6:F_5.C_2$ ($C_2$) x 2, $C_2^4.(C_6\times D_5).C_2^3$ ($C_2$) x 2, $C_2^4.(S_3\times D_4\times D_5)$ ($C_2$), $C_2^4.(S_3\times C_4:F_5)$ ($C_2$), $C_2^4.(F_5\times D_{12})$ ($C_2$), $C_2^4.(F_5\times C_3:D_4)$ ($C_2$), $C_2^4.(D_4\times C_3:F_5)$ ($C_2$), $C_2^4.(C_6\times F_5).C_2^2$ ($C_2$), $C_2^4.(C_4\times S_3\times F_5)$ ($C_2$), $(C_2^4\times C_{10}).C_6.C_2^3$ ($C_2$), $C_2^4.(F_5\times S_4)$ ($C_2$), $C_2\wr S_3\times F_5$ ($C_2$), $C_2^4:S_4\times F_5$ ($C_2$)

Order 3840: $C_2^3:S_4\times F_5$ ($C_2^2$) x 5, $C_2^4.D_6:F_5$ ($C_2^2$) x 4, $(C_2^3\times F_5):S_4$ ($C_2^2$) x 3, $C_2^4.D_{15}:D_4$ ($C_4$) x 2, $C_2^4.C_3:(C_4:F_5)$ ($C_2^2$) x 2, $C_2^4.(D_5\times C_3:D_4)$ ($C_2^2$) x 2, $C_2^4.(C_3:C_4\times F_5)$ ($C_2^2$) x 2, $C_2^4.(C_2\times C_6):F_5$ ($C_2^2$) x 2, $C_2\times C_2^4.(C_6\times D_5).C_2$ ($C_2^2$) x 2, $(C_2^4\times C_{10}).C_{12}.C_2$ ($C_2^2$) x 2, $(C_2^4\times C_{10}).C_6.C_2^2$ ($C_2^2$) x 2, $C_2^4.D_{30}:C_2.C_2$ ($C_4$), $C_2^4.D_{10}:S_3.C_2$ ($C_4$), $C_2^4.C_{20}:D_6$ ($C_4$), $C_2^4.C_{12}:F_5$ ($C_2^2$), $C_2^4.(S_3\times D_{20})$ ($C_4$), $C_2^4.(D_5\times D_{12})$ ($C_2^2$), $C_2^4.(D_4\times D_{15})$ ($C_4$), $C_2^4.(C_5\times S_3\times D_4)$ ($C_4$), $C_2^4.(C_4\times S_3\times D_5)$ ($C_2^2$), $C_2^4.(C_4\times C_3:F_5)$ ($C_2^2$), $(C_5\times C_2^4:C_3).C_4^2$ ($C_2^2$), $C_2^2\wr C_3\times F_5$ ($C_2^2$), $(C_2^3\times D_{10}).S_4$ ($C_2^2$), $C_2^4:A_4\times F_5$ ($C_2^2$), $C_2^3:S_4\times D_{10}$ ($C_2^2$)

Order 2560: $C_2^3\wr C_2\times F_5$ ($S_3$)

Order 1920: $(C_2^2\times D_{10}).S_4$ ($C_2^3$) x 4, $F_5\times C_2^2:S_4$ ($D_4$) x 4, $C_2^2:S_4\times D_{10}$ ($C_2^3$) x 3, $C_2^2:S_4\times D_{10}$ ($C_2\times C_4$) x 2, $C_2^3:A_4\times F_5$ ($C_2^3$) x 2, $(C_2^2\times D_{10}):S_4$ ($C_2\times C_4$) x 2, $(C_2^3\times D_{10}):C_{12}$ ($C_2^3$) x 2, $C_2^4.C_5:D_{12}$ ($C_2\times C_4$) x 2, $C_2^2\times D_5\times C_2^4:C_3$ ($C_2^3$), $C_2^2\times C_2^4:D_{15}$ ($C_2\times C_4$), $C_2^4.D_{30}.C_2$ ($C_2\times C_4$), $C_2^4.C_3:D_{20}$ ($C_2\times C_4$), $C_2^4.D_{10}:S_3$ ($C_2\times C_4$), $D_5\times C_2^2:(A_4:C_4)$ ($C_2^3$), $C_2^3:A_4\times D_{10}$ ($C_2^3$), $C_2^5:D_{30}$ ($C_2\times C_4$), $(C_2^4\times C_{10}).C_6.C_2$ ($C_2\times C_4$), $(C_2^4\times C_{10}).C_6.C_2$ ($C_2\times C_4$), $(C_2^4\times C_{10}).C_6.C_2$ ($C_2\times C_4$), $D_5\times C_2^5.C_6$ ($C_2^3$), $C_2^4.D_{30}.C_2$ ($C_2\times C_4$), $C_2^4.D_{60}$ ($C_2\times C_4$), $C_2^4.D_{30}.C_2$ ($C_2\times C_4$), $C_2^4.C_3:D_{20}$ ($C_2\times C_4$), $C_2^4.D_{10}:S_3$ ($C_2\times C_4$), $C_2^4.D_{15}:C_4$ ($C_2\times C_4$), $C_2^4.(S_3\times C_5:C_4)$ ($C_2\times C_4$), $C_2\times C_{10}\times C_2^2:S_4$ ($C_2\times C_4$), $C_5\times C_2^4:C_3.D_4$ ($C_2\times C_4$), $C_2^3:S_4\times C_{10}$ ($C_2\times C_4$), $C_5\times C_2^5.(C_2\times C_6)$ ($C_2\times C_4$), $C_5\times C_2^4:C_3.D_4$ ($C_2\times C_4$), $C_5\times C_2^4.D_{12}$ ($C_2\times C_4$), $C_5\times C_2^4.(C_4\times S_3)$ ($C_2\times C_4$)

Order 1280: $C_2^6.C_2\times D_5$ ($D_6$), $F_5\times C_2^6$ ($D_6$), $C_2^3:D_4\times F_5$ ($D_6$), $C_5:(C_2^6.C_2^2)$ ($D_6$), $C_5:(C_2^6.C_2^2)$ ($D_6$), $C_5:(C_2^7.C_2)$ ($D_6$), $C_2^4:C_4\times F_5$ ($D_6$)

Order 960: $C_2^4:D_{30}$ ($C_2^2\times C_4$) x 3, $C_2^2:S_4\times C_{10}$ ($C_2^2\times C_4$) x 3, $D_5\times C_2^2:S_4$ ($D_4:C_2$) x 2, $D_5\times C_2^2:S_4$ ($C_2\times D_4$) x 2, $C_2^2:A_4\times F_5$ ($C_2\times D_4$) x 2, $(C_2\times D_{10}).S_4$ ($C_2\times D_4$) x 2, $C_2^5:C_{30}$ ($C_2^2\times C_4$), $(C_2^3\times C_{10}):C_{12}$ ($C_2^2\times C_4$), $(C_2^2\times D_{10}):A_4$ ($C_2^2\times C_4$), $C_2^4:C_{60}$ ($C_2^2\times C_4$), $C_2^5:C_{30}$ ($C_2^2\times C_4$), $C_2^2:A_4\times D_{10}$ ($C_2^4$), $C_2^2:A_4\times D_{10}$ ($C_2^2\times C_4$), $C_2^5.D_{15}$ ($C_2^2\times C_4$), $C_5\times C_2^3.S_4$ ($C_2^2\times C_4$)

Order 768: $C_2\wr D_6$ ($F_5$)

Order 640: $F_5\times C_2^5$ ($C_2\times D_6$) x 2, $C_2^5:F_5$ ($C_2\times D_6$) x 2, $C_5:(C_2^6.C_2)$ ($C_4\times S_3$), $C_2^5.D_{10}$ ($C_2\times D_6$), $C_2^6\times D_5$ ($C_2\times D_6$), $D_{10}.C_2^5$ ($S_4$), $C_2^5:D_{10}$ ($C_2\times D_6$), $C_5\times C_2^6.C_2$ ($C_4\times S_3$), $C_5:(C_2^6.C_2)$ ($C_4\times S_3$), $C_5:(C_2^6.C_2)$ ($C_4\times S_3$)

Order 480: $C_2^4:D_{15}$ ($C_4\times D_4$) x 2, $C_5\times C_2^2:S_4$ ($C_4\times D_4$) x 2, $C_2^4:C_{30}$ ($C_2^3\times C_4$), $D_5\times C_2^2:A_4$ ($D_4:C_2^2$), $D_5\times C_2^2:A_4$ ($C_2^2\times D_4$)

Order 384: $C_2\wr C_6$ ($C_2\times F_5$), $C_2^4:S_4$ ($C_2\times F_5$), $C_2^4:D_{12}$ ($C_2\times F_5$), $C_2^5.D_6$ ($C_2\times F_5$), $C_2^2\wr S_3$ ($C_2\times F_5$), $C_2\wr S_3$ ($C_2\times F_5$), $C_2^4:S_4$ ($C_2\times F_5$)

Order 320: $C_2^4\times F_5$ ($C_2\times S_4$) x 2, $C_2^4\times F_5$ ($S_3\times D_4$) x 2, $C_2^4:F_5$ ($C_2\times S_4$) x 2, $C_2^4:C_{20}$ ($C_4\times D_6$), $C_2^4.D_{10}$ ($C_4\times D_6$), $C_2^5\times C_{10}$ ($C_4\times D_6$), $C_2^4\times D_{10}$ ($C_2^2\times D_6$), $C_2^4\times D_{10}$ ($C_4\times D_6$), $C_{20}:C_2^4$ ($C_2\times S_4$), $D_{10}.C_2^4$ ($C_2\times S_4$), $D_{10}:C_4^2$ ($C_2\times S_4$), $C_2^5:C_{10}$ ($C_4\times D_6$), $C_2^4:D_{10}$ ($C_4\times D_6$)

Order 240: $C_2^4:C_{15}$ ($C_4^2:C_2^2$)

Order 192: $C_2^3:S_4$ ($C_2^2\times F_5$) x 3, $C_2^4:C_{12}$ ($C_2^2\times F_5$), $C_2^2\wr C_3$ ($C_2^2\times F_5$), $C_2^3.S_4$ ($C_2^2\times F_5$), $C_2^4:A_4$ ($C_2^2\times F_5$)

Order 160: $C_2^3\times F_5$ ($C_2^2\times S_4$) x 4, $C_2^3\times D_{10}$ ($C_2^2\times S_4$) x 2, $C_2^3:D_{10}$ ($C_4\times S_4$) x 2, $C_2^4\times C_{10}$ ($C_{12}:C_2^3$), $C_2^3\times D_{10}$ ($D_4:D_6$), $C_2^3\times D_{10}$ ($D_4\times D_6$), $C_{20}:C_2^3$ ($C_4\times S_4$), $C_2^2\times D_{20}$ ($C_4\times S_4$), $C_{20}:C_2^3$ ($C_2^2\times S_4$)

Order 128: $C_2^3\wr C_2$ ($S_3\times F_5$)

Order 96: $C_2^2:S_4$ ($D_4\times F_5$) x 2, $C_2^3:A_4$ ($C_2^3\times F_5$)

Order 80: $C_2^3\times C_{10}$ ($C_2^4.D_6$) x 2, $C_2^2\times D_{10}$ ($C_2^4.D_6$) x 2, $C_2^2\times F_5$ ($D_4\times S_4$) x 2, $C_2^3\times C_{10}$ ($C_4^2:D_6$), $C_2^2\times D_{10}$ ($C_2^3\times S_4$), $C_2^2\times F_5$ ($C_2^3:S_4$), $C_2^2\times C_{20}$ ($C_2^4.D_6$), $C_2^2.D_{10}$ ($C_2^4.D_6$)

Order 64: $C_2^4:C_4$ ($D_6\times F_5$), $C_2^6$ ($D_6\times F_5$), $C_2^3:D_4$ ($D_6\times F_5$)

Order 48: $C_2^2:A_4$ ($D_{10}.C_2^4$)

Order 40: $C_2\times F_5$ ($C_2^4:S_4$) x 2, $C_2^2\times C_{10}$ ($C_2^5.D_6$), $C_2\times D_{10}$ ($\GL(2,\mathbb{Z}/4):C_2^2$), $C_2\times D_{10}$ ($\GL(2,\mathbb{Z}/4):C_2^2$), $C_2\times D_{10}$ ($C_2^4:S_4$)

Order 32: $C_2^5$ ($C_2\times D_6\times F_5$), $C_2^2\times D_4$ ($F_5\times S_4$)

Order 20: $F_5$ ($C_2\wr D_6$) x 2, $C_2\times C_{10}$ ($C_4\times D_4\times S_4$), $C_2\times C_{10}$ ($C_4\times C_2^3:S_4$), $D_{10}$ ($C_2^5:S_4$), $D_{10}$ ($C_4\times C_2^3:S_4$)

Order 16: $C_2^4$ ($C_2\times F_5\times S_4$) x 2, $C_2^4$ ($S_3\times D_4\times F_5$), $C_2^2\times C_4$ ($C_2\times F_5\times S_4$)

Order 10: $C_{10}$ ($C_4\times C_2^4:S_4$), $D_5$ ($C_2\wr D_6:C_2$), $D_5$ ($C_2^7:D_6$)

Order 8: $C_2^3$ ($C_2^2\times F_5\times S_4$)

Order 5: $C_5$ ($C_2\wr D_6.C_4$)

Order 4: $C_2^2$ ($D_4\times F_5\times S_4$), $C_2^2$ ($F_5\times C_2^3:S_4$)

Order 2: $C_2$ ($C_2^4:S_4\times F_5$)

Order 1: $C_1$ ($C_2\wr D_6.F_5$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

Supergroups

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

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $190 \times 190$ 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 $152 \times 152$ rational character table (warning: may be slow to load).