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C₂×#3622 #63222

order	272
isoclinism	isoclinic class of D₁₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

Splittings

C₁₇ ⋊ C₂×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₄	D₁₇
commutator, hypocenter	C₁₇	C₂×C₈
Frattini	C₄	C₂×D₁₇
Fitting	C₂×C₄×C₁₇	C₂
socle, prime radical	C₂²×C₁₇	C₄
cosocle	C₄×C₁₇	C₂²
LCS	G ▷ C₁₇
UCS	C₁ ◁ C₂×C₄

22 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 272

A minimal faithful permutation representation — 2 generators on 27 points:

(1,2)(3,4,5,7,6,8,9,10)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27);(1,2)(11,12,14,16,18,20,22,24,26,27,25,23,21,19,17,15,13)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#3622, #10587474, C₄⋊#3622, …
maximal normal subgroup of	C₄×#3622, #10587474, C₄⋊#3622, …
maximal quotient of	C₄×#3622, C₄⋊#3622, C₂×#63214, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(272, 9);