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C₂³×#215 #87240

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

Aliases

C₂³×#215

Splittings

Fitting	C₂³×C₃×Q₈ ⋊ C₂
Sylow	C₃ ⋊ C₂³×SD₁₆

Subgroups

	subgroup	G/H
center	C₂⁴	#67
commutator, cosocle	C₃×C₄	C₂⁵
Frattini	C₄	C₂⁴×D₃
Fitting	C₂³×C₃×Q₈	C₂
socle	C₂⁴×C₃	D₄
prime radical	C₂³×D₁₂	C₂
hypercenter	C₂³×Q₈	D₃
hypocenter	C₃	C₂³×SD₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×Q₈

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

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10701988, #10701992, #10702015, …
maximal normal subgroup of	#10701988, #10701992, #10702015, …
maximal quotient of	C₂³×#809, C₂³×#835, C₂³×#808, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 19816);