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C₄×D₅₁ #87989

order	408
isoclinism	isoclinic class of D₅₁
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

C₄×D₅₁

Splittings

Fitting	C₃×C₄×C₁₇ ⋊ C₂
derived, Hall	C₃×C₁₇ ⋊ C₂×C₄
Sylow	C₁₇ ⋊ C₄×D₃
Sylow	C₃ ⋊ C₄×D₁₇

Subgroups

	subgroup	G/H
center, hypercenter	C₄	D₅₁
commutator, hypocenter	C₃×C₁₇	C₂×C₄
Frattini	C₂	C₂×D₅₁
Fitting	C₃×C₄×C₁₇	C₂
socle, cosocle	C₂×C₃×C₁₇	C₂²
prime radical	C₂×D₅₁	C₂
LCS	G ▷ C₃×C₁₇
UCS	C₁ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 204

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

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

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11717739, #11717742, C₈×D₅₁, …
maximal normal subgroup of	#11717739, #11717742, C₈×D₅₁, …
maximal quotient of	C₈×D₅₁, #11717796, C₄×Dic₅₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(408, 26);