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C₂²×#869 #85670

order	384
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

C₂²×#869

Splittings

C₃ ⋊ C₂²×C₄⋊Q₈

Subgroups

	subgroup	G/H
center	C₂⁴	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂⁵
Frattini	C₂²	C₂⁴×D₃
Fitting	C₂²×C₃×C₄²	C₂
socle, prime radical	C₂⁴×C₃	C₂³
hypercenter	C₂²×C₄²	D₃
hypocenter	C₃	C₂²×C₄⋊Q₈
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄²

1624 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 19 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10686673, #10686675, #10686682, …
maximal normal subgroup of	#10686673, #10686675, #10686682, …
maximal quotient of	C₂²×#4935, C₂²×#4928, C₂²×#4929, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18246);