Finite Group Databasebeta

← Prev (#64076) Next (#64078) →

C₄×#634 #64077

order	288
isoclinism	isoclinic class of #634
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₃

Aliases

Splittings

C₃² ⋊ C₄×Q₈

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×C₃⋊D₃
commutator	C₂×C₃²	C₂²×C₄
Frattini	C₂²	C₂²×C₃⋊D₃
Fitting	C₃²×C₄²	C₂
socle, prime radical, cosocle	C₂²×C₃²	C₂³
hypercenter	C₄²	C₃⋊D₃
hypocenter	C₃²	C₄×Q₈
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

210 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 3 generators on 18 points:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10589085, #10589322, #10589350, …
maximal normal subgroup of	#10589085, #10589322, #10589350, …
maximal quotient of	#10592345, #10592346, #10592353, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 725);