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C₂×#4675 #73373

order	384
isoclinism	isoclinic class of #4656
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

Splittings

Fitting	C₂×C₃×#95 ⋊ C₂
Sylow	C₃ ⋊ C₂×#397

Subgroups

	subgroup	G/H
center	C₂⁴	C₂²×D₃
commutator, cosocle	C₂³×C₃	C₂⁴
Frattini	C₂³	C₂³×D₃
Fitting	C₂×C₃×#95	C₂
socle	C₂⁴×C₃	C₂³
prime radical	C₂⁴×D₃	C₂²
hypercenter	C₂×#95	D₃
hypocenter	C₃	C₂×#397
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴ ◁ C₂×#95

820 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 21 points:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10717004, C₄×#4675, #10720542, …
maximal normal subgroup of	#10717004, C₄×#4675, #10720542, …
maximal quotient of	C₂×#69428, C₂×#69429, C₂×#69414, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5949);