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C₃×Dic₅×C₂₃ #13063399

order	1380
isoclinism	isoclinic class of D₅
automorphism	?
homology	Abelianization (H₁) = C₃×C₄×C₂₃ · Schur multiplier (H₂) = C₁

Aliases

C₃×Dic₅×C₂₃

Splittings

derived, Sylow	C₅ ⋊ C₃×C₄×C₂₃
Hall	C₃×C₅×C₂₃ ⋊ C₄
Hall	C₅×C₂₃ ⋊ C₃×C₄
Hall	C₃×C₅ ⋊ C₄×C₂₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃×C₂₃	D₅
commutator, hypocenter	C₅	C₃×C₄×C₂₃
Frattini	C₂	C₃×D₅×C₂₃
Fitting, socle, prime radical	C₂×C₃×C₅×C₂₃	C₂
cosocle	C₂×C₅	C₂×C₃×C₂₃
LCS	G ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₂₃

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 1380

A minimal faithful permutation representation — 2 generators on 35 points:

(1,2,3)(4,5)(6,7)(8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35);(4,6,5,7)(9,12)(10,11)

Conjugacy classes & character table

552 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₉×Dic₅×C₂₃, C₃²×Dic₅×C₂₃, C₃×C₂₃×Dic₁₅, …
maximal normal subgroup of	C₉×Dic₅×C₂₃, C₃²×Dic₅×C₂₃, C₃×C₁₁×Dic₅×C₂₃, …
maximal quotient of	C₉×Dic₅×C₂₃, C₃²×Dic₅×C₂₃, C₃×C₂₃×Dic₁₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1380, 2);