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C₄×D₅₇ #90715

order	456
isoclinism	isoclinic class of D₅₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

C₄×D₅₇

Splittings

Fitting	C₃×C₄×C₁₉ ⋊ C₂
derived, Hall	C₃×C₁₉ ⋊ C₂×C₄
Sylow	C₁₉ ⋊ C₄×D₃
Sylow	C₃ ⋊ C₄×D₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₄	D₅₇
commutator, hypocenter	C₃×C₁₉	C₂×C₄
Frattini	C₂	C₂×D₅₇
Fitting	C₃×C₄×C₁₉	C₂
socle, cosocle	C₂×C₃×C₁₉	C₂²
prime radical	C₂×D₅₇	C₂
LCS	G ▷ C₃×C₁₉
UCS	C₁ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 228

A minimal faithful permutation representation — 2 generators on 26 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₅₇⋊C₈, #11744817, C₈×D₅₇, …
maximal normal subgroup of	D₅₇⋊C₈, #11744817, C₈×D₅₇, …
maximal quotient of	C₈×D₅₇, #11744880, C₄×Dic₅₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(456, 35);