Finite Group Databasebeta

← Prev (#87235) Next (#87237) →

C₂×#5804 #87236

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

Aliases

Splittings

Fitting	C₂³×C₃×D₄ ⋊ C₂
Sylow	C₃ ⋊ C₂×#534

Subgroups

	subgroup	G/H
center	C₂³	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁵
Frattini	C₂²	C₂⁴×D₃
Fitting	C₂³×C₃×D₄	C₂
socle	C₂³×C₃	C₂⁴
hypercenter	C₂³×D₄	D₃
hypocenter	C₃	C₂×#534
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂³×D₄

1984 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 5 generators on 17 points:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10701707, #10701708, #10701711, …
maximal normal subgroup of	#10701707, #10701708, #10701711, …
maximal quotient of	C₂×#82121, C₂×#84043, C₂×#82095, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 19812);