C₂×#11710898 #72227312

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#11710898

Splittings

Fitting, Sylow

C₂×#63091 ⋊ C₃

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊A₄
commutator, cosocle	#585	C₂³×C₃
Frattini	C₄	C₂³×C₂²⋊A₄
Fitting	C₂×#63091	C₃
hypercenter	C₂×D₈	C₂²⋊A₄
hypocenter	H(5,𝔽₂)	C₂×C₃×D₄
LCS	G ▷ #585 ▷ H(5,𝔽₂)
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

19279 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 19279 conjugacy classes of subgroups — too many to draw. Only its 155 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 34 points:

(3,4,10,21,8,13,5,9)(6,24,23,30,28,26,14,29,7,11,32,31,17,33,15,19,16,12,22,34,18,25,27,20);(1,2)(6,17)(11,25)(14,22)(16,28)(19,30)(24,33)(27,32)(29,34);(3,16)(4,29)(5,32)(6,7)(8,28)(9,24)(10,27)(11,20)(12,19)(13,34)(14,23)(15,22)(17,18)(21,33)(25,31)(26,30)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72228101, C₂×#72215685, #72342102, …
maximal normal subgroup of	C₂×#72228101, C₂×#72215685, #72342102, …
maximal quotient of	#72340745

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((3,4,10,21,8,13,5,9)(6,24,23,30,28,26,14,29,7,11,32,31,17,33,15,19,16,12,22,34,18,25,27,20), (1,2)(6,17)(11,25)(14,22)(16,28)(19,30)(24,33)(27,32)(29,34), (3,16)(4,29)(5,32)(6,7)(8,28)(9,24)(10,27)(11,20)(12,19)(13,34)(14,23)(15,22)(17,18)(21,33)(25,31)(26,30));