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C₂²×D₃×#41 #87563

order	384
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

C₂²×D₃×#41

Splittings

Fitting	C₂²×C₃×#41 ⋊ C₂
Sylow	C₃ ⋊ C₂³×#41

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃
commutator, cosocle	C₂×C₃	C₂⁶
Frattini	C₂	C₂⁵×D₃
Fitting	C₂²×C₃×#41	C₂
socle	C₂³×C₃	C₂⁴
hypercenter	C₂²×#41	D₃
hypocenter	C₃	C₂³×#41
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#41

6520 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 15 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#1780, #10794298, #10794302, …
maximal normal subgroup of	D₃×#1780, #10794298, #10794302, …
maximal quotient of	C₂²×D₃×#117, C₂²×C₄×D₃×D₄, C₂²×C₄×D₃×Q₈, …
automorphism group of	#4511, D₃×M₃₂, C₂×C₃×C₃⋊C₁₆, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 20139);