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C₈×#261 #88099

order	416
isoclinism	isoclinic class of #261
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₄

Aliases

Splittings

Fitting	C₈×C₁₃ ⋊ C₄
derived, Sylow	C₁₃ ⋊ C₄×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₈	#261
commutator, hypocenter	C₁₃	C₄×C₈
Frattini	C₄	C₂×#261
Fitting	C₈×C₁₃	C₄
socle	C₂×C₁₃	C₄²
prime radical	C₂×D₁₃	C₂×C₄
cosocle	C₄×D₁₃	C₂²
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 104

A minimal faithful permutation representation — 2 generators on 21 points:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₁₆×#261, #11718216, C₂×C₈×#261, …
maximal normal subgroup of	C₁₆×#261, #11718216, C₂×C₈×#261, …
maximal quotient of	C₁₆×#261, #11718216, C₈×#1057, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(416, 66);