C₃×Dic₁₀₇₅ #27166339

order	12900
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₃×Dic₁₀₇₅

Splittings

derived, Hall	C₂₅×C₄₃ ⋊ C₃×C₄
Hall	C₃×C₂₅×C₄₃ ⋊ C₄
Hall	C₃×C₄₃ ⋊ Dic₂₅
Hall	C₃×C₂₅ ⋊ Dic₄₃
Sylow	C₄₃ ⋊ C₃×Dic₂₅
Sylow	C₂₅ ⋊ C₃×Dic₄₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	D₁₀₇₅
commutator, hypocenter	C₂₅×C₄₃	C₃×C₄
Frattini	C₂×C₅	C₃×D₂₁₅
Fitting	C₂×C₃×C₂₅×C₄₃	C₂
socle, prime radical	C₂×C₃×C₅×C₄₃	D₅
cosocle	C₂×C₂₅×C₄₃	C₂×C₃
LCS	G ▷ C₂₅×C₄₃
UCS	C₁ ◁ C₂×C₃

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 12900

A minimal faithful permutation representation — 2 generators on 75 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)(69,72,71,70)(73,74,75);(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)

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, 8);

C3×Dic1075 #27166339