C₃²×D₁₁₅ #14523397

order	2070
isoclinism	isoclinic class of D₁₁₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃

Aliases

direct product	C₃²×D₁₁₅
unambiguous semidirect product	C₅⋊(C₃²×D₂₃)
unambiguous semidirect product	C₂₃⋊(C₃²×D₅)

Splittings

Fitting, socle, Hall	C₃²×C₅×C₂₃ ⋊ C₂
derived, Hall	C₅×C₂₃ ⋊ C₂×C₃²
Hall	C₃²×C₂₃ ⋊ D₅
Hall	C₃²×C₅ ⋊ D₂₃
Sylow	C₂₃ ⋊ C₃²×D₅
Sylow	C₅ ⋊ C₃²×D₂₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃²	D₁₁₅
commutator, cosocle, hypocenter	C₅×C₂₃	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₂₃	C₂
LCS	G ▷ C₅×C₂₃
UCS	C₁ ◁ C₃²

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 1035

A minimal faithful permutation representation — 2 generators on 34 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃²×D₅×D₂₃, C₃×D₃×D₁₁₅, C₃×D₁₅⋊D₂₃, …
maximal normal subgroup of	C₃²×D₅×D₂₃, C₃×D₃×D₁₁₅, C₃×D₁₅⋊D₂₃, …
maximal quotient of	C₃²×Dic₁₁₅, C₂×C₃²×D₁₁₅, C₃×C₅⋊(ΓL(1,𝔽₈)⋊D₂₃), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2070, 12);

C32×D115 #14523397