C₃×C₅×Dic₂₃ #13063398

order	1380
isoclinism	isoclinic class of D₂₃
automorphism	?
homology	Abelianization (H₁) = C₃×C₄×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×C₅×Dic₂₃
unambiguous semidirect product	C₂₃⋊(C₃×C₄×C₅)

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃×C₅	D₂₃
commutator, hypocenter	C₂₃	C₃×C₄×C₅
Frattini	C₂	C₃×C₅×D₂₃
Fitting, socle, prime radical	C₂×C₃×C₅×C₂₃	C₂
cosocle	C₂×C₂₃	C₂×C₃×C₅
LCS	G ▷ C₂₃
UCS	C₁ ◁ C₂×C₃×C₅

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);(9,11,10,12)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₅×C₉×Dic₂₃, C₃²×C₅×Dic₂₃, C₃×C₅×Dic₆₉, …
maximal normal subgroup of	C₅×C₉×Dic₂₃, C₃²×C₅×Dic₂₃, C₃×C₅×C₁₁×Dic₂₃, …
maximal quotient of	C₅×C₉×Dic₂₃, C₃²×C₅×Dic₂₃, C₃×C₅×Dic₆₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1380, 1);

C3×C5×Dic23 #13063398