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D₃×#525 #86034

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

Aliases

Splittings

Fitting	C₃×#525 ⋊ C₂
Sylow	C₃ ⋊ C₂×#525

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₃
commutator, socle, cosocle	C₂²×C₃	C₂⁵
Frattini	C₂²	C₂⁴×D₃
Fitting	C₃×#525	C₂
hypercenter	#525	D₃
hypocenter	C₃	C₂×#525
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ #525

1880 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 48

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

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#1501, D₃×#1504, #10762347, …
maximal normal subgroup of	D₃×#1501, D₃×#1504, #10762347, …
maximal quotient of	D₃×#2280, D₃×#2290, D₃×#2296, …
automorphism group of	#4512, #4514

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18610);