C₃×C₇⋊D₄₉₇ #35229477

order	20874
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₇

Aliases

direct product	C₃×C₇⋊D₄₉₇
unambiguous semidirect product	C₇₁⋊(C₃×C₇⋊D₇)

Splittings

Fitting, socle, Hall	C₃×C₇²×C₇₁ ⋊ C₂
derived, Hall	C₇²×C₇₁ ⋊ C₂×C₃
Hall	C₃×C₇₁ ⋊ C₇⋊D₇
Hall	C₃×C₇² ⋊ D₇₁
Sylow	C₇₁ ⋊ C₃×C₇⋊D₇
Sylow	C₇² ⋊ C₃×D₇₁

Subgroups

	subgroup	G/H
center, hypercenter	C₃	C₇⋊D₄₉₇
commutator, cosocle, hypocenter	C₇²×C₇₁	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₃×C₇²×C₇₁	C₂
LCS	G ▷ C₇²×C₇₁
UCS	C₁ ◁ C₃

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 88
transitive degree	μ_t(G) = 10437

A minimal faithful permutation representation — 3 generators on 88 points:

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

Conjugacy classes & character table

5223 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20874, 56);

C3×C7⋊D497 #35229477