D₆₇×D₇₅ #34117170

order	20100
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

direct product	D₆₇×D₇₅
unambiguous semidirect product	(C₃×D₆₇)⋊D₂₅
unambiguous semidirect product	(C₂₅×D₆₇)⋊D₃

Splittings

Fitting, derived, Hall	C₃×C₂₅×C₆₇ ⋊ C₂²
Hall	C₂₅×C₆₇ ⋊ C₂×D₃
Hall	C₃×C₆₇ ⋊ C₂×D₂₅
Hall	C₃×C₂₅ ⋊ C₂×D₆₇
Sylow	C₆₇ ⋊ C₂×D₇₅
Sylow	C₂₅ ⋊ D₃×D₆₇
Sylow	C₃ ⋊ D₂₅×D₆₇

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₃×C₂₅×C₆₇	C₂²
Frattini	C₅	D₁₅×D₆₇
socle	C₃×C₅×C₆₇	C₂×D₅
LCS	G ▷ C₃×C₂₅×C₆₇
UCS	C₁

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 95
transitive degree	μ_t(G) = 5025

A minimal faithful permutation representation — 2 generators on 95 points:

(2,4)(3,7)(5,10)(6,13)(8,16)(9,15)(11,12)(14,20)(17,21)(18,22)(19,24)(23,25)(27,28)(29,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31);(1,21,12,24,9,7,14,5,23,4,16,6,22,18,13,8,2,25,10,20,3,15,19,11,17)(26,27,28)(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)(90,91)(92,93)(94,95)

Conjugacy classes & character table

1365 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20100, 26);

D67×D75 #34117170