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C₂×#5232 #84071

order	384
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

Splittings

C₃ ⋊ C₂×#417

Subgroups

	subgroup	G/H
center	C₂³	C₂×#67
commutator	C₃×C₄	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃
Fitting, prime radical	C₂²×C₃×#41	C₂
socle	C₂³×C₃	C₂×D₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₂²×#41	D₃
hypocenter	C₃	C₂×#417
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#41

816 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 17 points:

(3,4,5)(6,7)(8,9)(10,11)(12,17)(13,15)(14,16);(1,2)(11,15)(14,17);(4,5)(6,7,8,9)(11,14)(12,16)(15,17);(1,2)

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10700661, #10700663, #10700704, …
maximal normal subgroup of	#10700661, #10700663, #10700704, …
maximal quotient of	C₂×#71639, C₂×#71641, #10761648, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 16647);