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C₂³×#214 #87226

order	384
isoclinism	isoclinic class of #213
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

C₂³×#214

Splittings

C₃ ⋊ C₂³×SD₁₆

Subgroups

	subgroup	G/H
center	C₂⁴	#67
commutator, cosocle	C₃×C₄	C₂⁵
Frattini	C₄	C₂⁴×D₃
Fitting, prime radical	C₂³×C₃×D₄	C₂
socle	C₂⁴×C₃	D₄
hypercenter	C₂³×D₄	D₃
hypocenter	C₃	C₂³×SD₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×D₄

1720 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 17 points:

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

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10701989, #10701993, #10702016, …
maximal normal subgroup of	#10701989, #10701993, #10702016, …
maximal quotient of	C₂³×#832, C₂³×#810, C₂³×#808, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 19802);