C₃×D₂₂₂₅ #27211160

order	13350
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×D₂₂₂₅
unambiguous semidirect product	C₂₅⋊(C₃×D₈₉)
unambiguous semidirect product	C₈₉⋊(C₃×D₂₅)

Splittings

Fitting, Hall	C₃×C₂₅×C₈₉ ⋊ C₂
derived, Hall	C₂₅×C₈₉ ⋊ C₂×C₃
Hall	C₃×C₈₉ ⋊ D₂₅
Sylow	C₈₉ ⋊ C₃×D₂₅
Hall	C₃×C₂₅ ⋊ D₈₉
Sylow	C₂₅ ⋊ C₃×D₈₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃	D₂₂₂₅
commutator, cosocle, hypocenter	C₂₅×C₈₉	C₂×C₃
Frattini	C₅	C₃×D₄₄₅
Fitting	C₃×C₂₅×C₈₉	C₂
socle	C₃×C₅×C₈₉	D₅
LCS	G ▷ C₂₅×C₈₉
UCS	C₁ ◁ C₃

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 117
transitive degree	μ_t(G) = 6675

A minimal faithful permutation representation — 2 generators on 117 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)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)(86,87)(88,89)(91,93)(92,96)(94,99)(95,102)(97,105)(98,104)(100,101)(103,109)(106,110)(107,111)(108,113)(112,114)(115,116,117);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(90,104,114,107,93,96,110,108,99,102,105,103,101,100,109,97,95,94,113,106,92,91,111,112,98)

Conjugacy classes & character table

3342 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×#22581808, C₃×#22581807, C₃×D₂₅×D₈₉, …
maximal normal subgroup of	C₃×#22581808, C₃×#22581807, C₃×D₂₅×D₈₉, …
maximal quotient of	C₃×Dic₂₂₂₅, C₂×C₃×D₂₂₂₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13350, 4);

C3×D2225 #27211160