C₆₇×Dic₇₅ #34117152

order	20100
isoclinism	isoclinic class of D₇₅
automorphism	?
homology	Abelianization (H₁) = C₄×C₆₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₆₇×Dic₇₅
unambiguous semidirect product	C₃⋊(C₆₇×Dic₂₅)

Splittings

derived, Hall	C₃×C₂₅ ⋊ C₄×C₆₇
Hall	C₃×C₂₅×C₆₇ ⋊ C₄
Hall	C₂₅×C₆₇ ⋊ Dic₃
Hall	C₃×C₆₇ ⋊ Dic₂₅
Sylow	C₂₅ ⋊ Dic₃×C₆₇
Sylow	C₃ ⋊ C₆₇×Dic₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₆₇	D₇₅
commutator, hypocenter	C₃×C₂₅	C₄×C₆₇
Frattini	C₂×C₅	D₁₅×C₆₇
Fitting	C₂×C₃×C₂₅×C₆₇	C₂
socle, prime radical	C₂×C₃×C₅×C₆₇	D₅
cosocle	C₂×C₃×C₂₅	C₂×C₆₇
LCS	G ▷ C₃×C₂₅
UCS	C₁ ◁ C₂×C₆₇

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 99
transitive degree	μ_t(G) = 20100

A minimal faithful permutation representation — 2 generators on 99 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)(29,32,31,30)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99);(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)

Conjugacy classes & character table

5226 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20100, 8);

C67×Dic75 #34117152