Finite Group Databasebeta

← Prev (#63771) Next (#63773) →

C₂×#3708 #63772

order	288
isoclinism	isoclinic class of #157
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆ · Schur multiplier (H₂) = C₂×C₃

Aliases

Splittings

C₃² ⋊ C₂×C₁₆

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₄	#157
commutator, hypocenter	C₃²	C₂×C₁₆
Frattini	C₄	C₂×#157
Fitting	C₂×C₃²×C₄	C₄
socle, prime radical	C₂²×C₃²	C₈
cosocle	C₃⋊(C₃⋊C₈)	C₂²
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂×C₄

50 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 2 generators on 24 points:

(3,4)(5,6,7,8)(9,10,11,14,12,15,17,20,13,16,18,21,19,22,23,24);(1,2)(3,7,5)

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10590061, #10590062, #10590063, …
maximal normal subgroup of	#10590061, #10590062, #10590063, …
maximal quotient of	C₄×#3708, C₄⋊#3708, C₂×#63540, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 420);