C₂×#13147076 #72268962

order	3456
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³×C₃

solvable

Aliases

direct product

C₂×#13147076

Subgroups

	subgroup	G/H
center	C₂²	#11722396
commutator, cosocle	C₃×#41⋊C₉	C₂³
Frattini	C₃×C₄	C₂²×C₃⋊S₄
Fitting	C₂×C₃²×H(5,𝔽₂)	D₃
socle	C₂²×C₃²	GL(2,ℤ/4)
hypercenter	C₂×Q₈	C₃⋊((C₂²⋊C₉)⋊C₂)
hypocenter	C₃×Q₈⋊C₉	C₂×D₄
LCS	G ▷ C₃×#41⋊C₉ ▷ C₃×Q₈⋊C₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₈

1948 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 1948 conjugacy classes of subgroups — too many to draw. Only its 157 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) = 38
transitive degree	μ_t(G) = 432

A minimal faithful permutation representation — 3 generators on 38 points:

(3,4,5)(15,23)(16,19)(17,21)(18,28)(20,24)(22,27)(26,30)(31,35)(34,38)(36,37);(1,2)(6,13,14,8,7,9,12,11,10)(15,18,26)(16,22,31)(17,25,35)(19,29,30)(20,27,38)(21,28,36)(23,33,34)(24,32,37);(4,5)(7,9)(8,12)(10,13)(11,14)(16,19)(17,23)(18,26)(20,24)(22,30)(25,34)(27,37)(28,36)(29,31)(32,38)(33,35)

Conjugacy classes & character table

186 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72245426, C₂×#72252958
maximal normal subgroup of	C₂×#72252958

GAP

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

gap> G := Group((3,4,5)(15,23)(16,19)(17,21)(18,28)(20,24)(22,27)(26,30)(31,35)(34,38)(36,37), (1,2)(6,13,14,8,7,9,12,11,10)(15,18,26)(16,22,31)(17,25,35)(19,29,30)(20,27,38)(21,28,36)(23,33,34)(24,32,37), (4,5)(7,9)(8,12)(10,13)(11,14)(16,19)(17,23)(18,26)(20,24)(22,30)(25,34)(27,37)(28,36)(29,31)(32,38)(33,35));