Finite Group Databasebeta

← Prev (#86067) Next (#86069) →

C₂×D₃×#124 #86068

order	384
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

C₂×D₃×#124

Splittings

Fitting	C₂×C₃×#124 ⋊ C₂
Sylow	C₃ ⋊ C₂²×#124

Subgroups

	subgroup	G/H
center	C₂³	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁵
Frattini	C₂²	C₂⁴×D₃
Fitting	C₂×C₃×#124	C₂
socle	C₂³×C₃	C₂⁴
prime radical	C₂²×D₃×D₄	C₂
hypercenter	C₂×#124	D₃
hypocenter	C₃	C₂²×#124
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×#124

1880 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:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10686652, #10710674, #10710691, …
maximal normal subgroup of	#10686652, #10710674, #10710691, …
maximal quotient of	C₂×D₃×#382, C₂×D₃×#388, C₂×D₃×#390, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18644);