C₂₅×D₂₆₇ #27211162

order	13350
automorphism	?
homology	Abelianization (H₁) = C₂×C₂₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₂₅×D₂₆₇
unambiguous semidirect product	C₃⋊(C₂₅×D₈₉)
unambiguous semidirect product	C₈₉⋊(D₃×C₂₅)

Splittings

Fitting, Hall	C₃×C₂₅×C₈₉ ⋊ C₂
derived, Hall	C₃×C₈₉ ⋊ C₂×C₂₅
Hall	C₂₅×C₈₉ ⋊ D₃
Sylow	C₈₉ ⋊ D₃×C₂₅
Hall	C₃×C₂₅ ⋊ D₈₉
Sylow	C₃ ⋊ C₂₅×D₈₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂₅	D₂₆₇
commutator, hypocenter	C₃×C₈₉	C₂×C₂₅
Frattini	C₅	C₅×D₂₆₇
Fitting	C₃×C₂₅×C₈₉	C₂
socle, cosocle	C₃×C₅×C₈₉	C₂×C₅
prime radical	C₅×D₂₆₇	C₅
LCS	G ▷ C₃×C₈₉
UCS	C₁ ◁ C₂₅

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)(91,92)(93,94,96,99,103,95,97,100,104,108,98,101,105,109,112,102,106,110,113,115,107,111,114,116,117);(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)(90,91,92)

Conjugacy classes & character table

3375 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃×C₂₅×D₈₉, D₂₅×D₂₆₇, D₇₅⋊D₈₉, …
maximal normal subgroup of	D₃×C₂₅×D₈₉, D₂₅×D₂₆₇, D₇₅⋊D₈₉, …
maximal quotient of	C₂₅×Dic₂₆₇, C₂×C₂₅×D₂₆₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13350, 6);

C25×D267 #27211162