Finite Group Databasebeta

← Prev (#63628) Next (#63630) →

C₄×C₃⋊(C₃⋊C₈) #63629

order	288
isoclinism	isoclinic class of C₃⋊D₃
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₃×C₄

Aliases

C₄×C₃⋊(C₃⋊C₈)

Splittings

C₃² ⋊ C₄×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₄²	C₃⋊D₃
commutator, hypocenter	C₃²	C₄×C₈
Frattini	C₂×C₄	C₂×C₃⋊D₃
Fitting	C₃²×C₄²	C₂
socle, prime radical	C₂²×C₃²	C₂×C₄
cosocle	C₂×C₃²×C₄	C₂²
LCS	G ▷ C₃²
UCS	C₁ ◁ C₄²

132 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:

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

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃⋊C₈², #10589078, #10589079, …
maximal normal subgroup of	C₃⋊C₈², #10589078, #10589079, …
maximal quotient of	C₈×C₃⋊(C₃⋊C₈), #10589321, C₄×C₃⋊(C₃⋊C₁₆), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 277);