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C₄×#803 #69387

order	384
isoclinism	isoclinic class of Dic₆
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂×C₄²

Aliases

Splittings

C₃ ⋊ C₄×#97

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂×D₃
commutator	C₂×C₃	C₄³
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×C₄³	C₂
socle, prime radical	C₂³×C₃	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₄³	D₃
hypocenter	C₃	C₄×#97
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄³

360 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 19 points:

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

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10681722, #10681751, #10681753, …
maximal normal subgroup of	#10681722, #10681751, #10681753, …
maximal quotient of	C₄×#4454, C₈×#803, #10684547, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 1963);