C₂×D₄×#4169 #72276009

order	2592
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁶×C₃

Aliases

direct product

C₂×D₄×#4169

Splittings

Fitting	C₂×D₄×C₃²⋊C₉ ⋊ C₂
Sylow	C₃²⋊C₉ ⋊ C₂²×D₄

Subgroups

	subgroup	G/H
center	C₂²	C₂²×#4169
commutator, cosocle	C₂×C₃×C₉	C₂⁴×C₃
Frattini	C₂×C₃²	C₂³×C₃×D₃
Fitting	C₂×D₄×C₃²⋊C₉	C₂
socle	C₂²×C₃²	C₂²×C₃×D₃
hypercenter	C₂×D₄	#4169
hypocenter	C₃×C₉	C₂²×C₃×D₄
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ C₂×D₄

1878 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 216

A minimal faithful permutation representation — 4 generators on 24 points:

(3,4,5,6)(7,8,14)(9,11,13)(10,12,15)(16,23,24,18,17,19,22,21,20);(8,10)(9,12)(11,13)(14,15)(17,19)(18,22)(20,23)(21,24);(1,2);(4,6)

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

C₂×D₄×#92299

GAP

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

gap> G := Group((3,4,5,6)(7,8,14)(9,11,13)(10,12,15)(16,23,24,18,17,19,22,21,20), (8,10)(9,12)(11,13)(14,15)(17,19)(18,22)(20,23)(21,24), (1,2), (4,6));

C2×D4×#4169 #72276009