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C₂²×#817 #83729

order	384
isoclinism	isoclinic class of #204
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

C₂²×#817

Splittings

C₃ ⋊ C₂²×#106

Subgroups

	subgroup	G/H
center	C₂⁴	D₁₂
commutator	C₃×C₄	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂³×C₃×C₈	C₂
socle, prime radical	C₂⁴×C₃	D₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₂³×C₈	D₃
hypocenter	C₃	C₂²×#106
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈

600 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 4 generators on 19 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10701081, #10701083, #10701091, …
maximal normal subgroup of	#10701081, #10701083, #10701091, …
maximal quotient of	C₂²×#4457, C₂×#69730, C₂×#70749, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 16305);