C₂×#11921467 #72229080

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂³×C₃

Aliases

direct product

C₂×#11921467

Splittings

Fitting	C₂×H(3,𝔽₄) ⋊ C₃⋊D₃
derived	#5450⋊C₃ ⋊ C₂²

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#11921467
commutator, cosocle, hypocenter	#5450⋊C₃	C₂²
Frattini	C₂²	C₂×A₄⋊S₄
Fitting	C₂×H(3,𝔽₄)	C₃⋊D₃
socle	C₂³	A₄⋊S₄
LCS	G ▷ #5450⋊C₃
UCS	C₁ ◁ C₂

728 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 27 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) = 18
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 3 generators on 18 points:

(1,2)(3,5,17,4,9,18)(6,10)(7,13,11,8,12,16)(14,15);(3,4,8,7)(5,10,12,14)(6,9,15,13)(11,16,18,17);(3,7,4)(5,13,9)(6,14,10)(11,17,16)

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72215868, C₂×AΓL(2,𝔽₄), D₃×#11921467, …
maximal normal subgroup of	C₄×#11921467, C₂×C₃×#11921467
maximal quotient of	D₃×#11921467, C₄×#11921467, C₂×C₃×#11921467
automorphism group of	C₂×C₂²⋊SL(2,𝔽₃), #5948

GAP

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

gap> G := Group((1,2)(3,5,17,4,9,18)(6,10)(7,13,11,8,12,16)(14,15), (3,4,8,7)(5,10,12,14)(6,9,15,13)(11,16,18,17), (3,7,4)(5,13,9)(6,14,10)(11,17,16));