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C₄×D₃×B(4,𝔽₂) #72230833

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹¹

Aliases

C₄×D₃×B(4,𝔽₂)

Splittings

Fitting	C₃×C₄×B(4,𝔽₂) ⋊ C₂
Sylow	C₃ ⋊ C₂×C₄×B(4,𝔽₂)

Subgroups

	subgroup	G/H
center	C₂×C₄	D₃×#120
commutator	C₂³×C₃	C₂⁴×C₄
Frattini	C₂⁴	C₂⁴×D₃
Fitting	C₃×C₄×B(4,𝔽₂)	C₂
socle	C₂²×C₃	C₂²×#120
prime radical	C₂×D₃×B(4,𝔽₂)	C₂
cosocle	C₂⁴×C₃	C₂⁵
hypercenter	C₄×B(4,𝔽₂)	D₃
hypocenter	C₃	C₂×C₄×B(4,𝔽₂)
LCS	G ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₄×B(4,𝔽₂)

13388 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 5 generators on 15 points:

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

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×AGL(1,𝔽₅)×B(4,𝔽₂), D₃×D₄×B(4,𝔽₂), D₃×#6454041, …
maximal normal subgroup of	D₃×D₄×B(4,𝔽₂), D₃×#6454041, D₃×#6595772
maximal quotient of	D₃×AGL(1,𝔽₅)×B(4,𝔽₂)
automorphism group of	C₂×C₅×Dic₆, C₂×C₅×D₁₂

GAP

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

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