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C₈×#226 #73007

order	384
isoclinism	isoclinic class of #226
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

solvable

Aliases

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₈	S₄
commutator, hypocenter	SL(2,𝔽₃)	C₂×C₈
Frattini	C₂×C₄	C₂×S₄
Fitting	C₈×Q₈	D₃
socle	C₂²	C₄×S₄
prime radical	C₂×SL(2,𝔽₃)	C₂×C₄
cosocle	C₄×SL(2,𝔽₃)	C₂²
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₈

103 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 24 points:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₁₆×#226, #11711381, #226⋊C₁₆, …
maximal normal subgroup of	C₁₆×#226, #11711381, #226⋊C₁₆, …
maximal quotient of	C₈×#859, #11711086, #11711106, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5583);