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C₂²×#874 #85678

order	384
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

C₂²×#874

Splittings

Fitting	C₂²×C₃×C₄² ⋊ C₂
Sylow	C₃ ⋊ C₂²×#127

Subgroups

	subgroup	G/H
center	C₂⁴	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂⁵
Frattini	C₂²	C₂⁴×D₃
Fitting	C₂²×C₃×C₄²	C₂
socle	C₂⁴×C₃	C₂³
hypercenter	C₂²×C₄²	D₃
hypocenter	C₃	C₂²×#127
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄²

3000 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 192

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

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10686670, #10701724, #10701751, …
maximal normal subgroup of	#10686670, #10701724, #10701751, …
maximal quotient of	C₂²×#4935, C₂²×#4939, C₂×C₄×#874, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18254);