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C₂³×C₃⋊C₁₆ #83722

order	384
isoclinism	isoclinic class of D₃
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂⁶

Aliases

C₂³×C₃⋊C₁₆

Splittings

C₃ ⋊ C₂³×C₁₆

Subgroups

	subgroup	G/H
center, hypercenter	C₂³×C₈	D₃
commutator, hypocenter	C₃	C₂³×C₁₆
Frattini	C₈	C₂³×D₃
Fitting	C₂³×C₃×C₈	C₂
socle, prime radical	C₂⁴×C₃	C₈
cosocle	C₃×C₈	C₂⁴
LCS	G ▷ C₃
UCS	C₁ ◁ C₂³×C₈

440 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 25 points:

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

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#67804, C₂×#67657, #10710030, …
maximal normal subgroup of	C₂×#67804, C₂×#67657, #10710030, …
maximal quotient of	C₂²×C₄×C₃⋊C₁₆, C₂²×#4549, C₂²×C₄⋊(C₃⋊C₁₆), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 16298);