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C₄×#1004 #85750

order	384
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

Splittings

Fitting	C₂×C₃×C₄×D₄ ⋊ C₂
Sylow	C₃ ⋊ C₄×H(5,𝔽₂)

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₃
commutator	C₂×C₃	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₃
Fitting	C₂×C₃×C₄×D₄	C₂
socle, cosocle	C₂²×C₃	C₂⁵
prime radical	C₂×#1004	C₂
hypercenter	C₂×C₄×D₄	D₃
hypocenter	C₃	C₄×H(5,𝔽₂)
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄

1660 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)(9,13)(12,15);(10,14)(12,15);(8,10)(9,12)(11,14)(13,15);(2,3)(8,9)(10,12)(11,13)(14,15);(4,5,6,7)

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10686578, #10686579, #10686585, …
maximal normal subgroup of	#10686578, #10686579, #10686585, …
maximal quotient of	C₄×#5486, C₄×#5545, C₄×#5798, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18326);