C₂×C₃×D₁₀₇₅ #27166367

order	12900
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₃×D₁₀₇₅
unambiguous semidirect product	(C₂×C₂₅)⋊(C₃×D₄₃)

Splittings

Fitting	C₂×C₃×C₂₅×C₄₃ ⋊ C₂
derived, Hall	C₂₅×C₄₃ ⋊ C₂²×C₃
Hall	C₃×C₂₅×C₄₃ ⋊ C₂²
Hall	C₃×C₄₃ ⋊ C₂×D₂₅
Hall	C₃×C₂₅ ⋊ C₂×D₄₃
Sylow	C₄₃ ⋊ C₂×C₃×D₂₅
Sylow	C₂₅ ⋊ C₂×C₃×D₄₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	D₁₀₇₅
commutator, cosocle, hypocenter	C₂₅×C₄₃	C₂²×C₃
Frattini	C₅	C₂×C₃×D₂₁₅
Fitting	C₂×C₃×C₂₅×C₄₃	C₂
socle	C₂×C₃×C₅×C₄₃	D₅
LCS	G ▷ C₂₅×C₄₃
UCS	C₁ ◁ C₂×C₃

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 6450

A minimal faithful permutation representation — 2 generators on 73 points:

(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)(36,37)(38,39)(40,41)(42,43)(45,47)(46,50)(48,53)(49,56)(51,59)(52,58)(54,55)(57,63)(60,64)(61,65)(62,67)(66,68)(69,70,71);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(44,58,68,61,47,50,64,62,53,56,59,57,55,54,63,51,49,48,67,60,46,45,65,66,52)(72,73)

Conjugacy classes & character table

3234 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12900, 36);

C2×C3×D1075 #27166367