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C₂³×#216 #87242

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

Aliases

C₂³×#216

Splittings

C₃ ⋊ C₂³×Q₁₆

Subgroups

	subgroup	G/H
center	C₂⁴	#67
commutator, cosocle	C₃×C₄	C₂⁵
Frattini	C₄	C₂⁴×D₃
Fitting	C₂³×C₃×Q₈	C₂
socle, prime radical	C₂⁴×C₃	D₄
hypercenter	C₂³×Q₈	D₃
hypocenter	C₃	C₂³×Q₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×Q₈

1464 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 5 generators on 25 points:

(7,8,9)(10,12,14,20)(11,15,17,24)(13,22,23,19)(16,25,21,18);(8,9)(10,11,14,17)(12,18,20,25)(13,21,23,16)(15,22,24,19);(5,6);(3,4);(1,2)

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10701995, #10701996, #10702022, …
maximal normal subgroup of	#10701995, #10701996, #10702022, …
maximal quotient of	C₂³×#810, C₂³×#835, C₂³×#807, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 19818);