C₃×#88322 #11929653

order	1272
isoclinism	isoclinic class of D₅₃
automorphism	?
homology	Abelianization (H₁) = C₃×C₈ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₃×#88322

Splittings

derived, Sylow	C₅₃ ⋊ C₃×C₈
Hall	C₃×C₅₃ ⋊ C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₄	D₅₃
commutator, hypocenter	C₅₃	C₃×C₈
Frattini	C₄	C₃×D₅₃
Fitting	C₃×C₄×C₅₃	C₂
socle, prime radical	C₂×C₃×C₅₃	C₄
cosocle	C₄×C₅₃	C₂×C₃
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₃×C₄

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 1272

A minimal faithful permutation representation — 2 generators on 64 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)(57,58,59,61,60,62,63,64);(4,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53,55,56,54,52,50,48,46,44,42,40,38,36,34,32,30,28,26,24,22,20,18,16,14,12,10,8,6)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1272, 2);

C3×#88322 #11929653