D₅₃×C₃⋊D₂₁ #27211302

order	13356
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product	D₅₃×C₃⋊D₂₁
unambiguous semidirect product	(C₃×D₅₃)⋊D₂₁
unambiguous semidirect product	(C₇×D₅₃)⋊(C₃⋊D₃)

Splittings

Fitting, socle, derived, Hall	C₃²×C₇×C₅₃ ⋊ C₂²
Hall	C₃²×C₅₃ ⋊ C₂×D₇
Hall	C₇×C₅₃ ⋊ C₂×C₃⋊D₃
Hall	C₃²×C₇ ⋊ C₂×D₅₃
Sylow	C₅₃ ⋊ C₂×C₃⋊D₂₁
Sylow	C₃² ⋊ D₇×D₅₃
Sylow	C₇ ⋊ C₃⋊D₃×D₅₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃²×C₇×C₅₃	C₂²
LCS	G ▷ C₃²×C₇×C₅₃
UCS	C₁

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 66
transitive degree	μ_t(G) = 3339

A minimal faithful permutation representation — 3 generators on 66 points:

(2,3)(4,5)(6,7)(8,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53,55,57,59,60,58,56,54,52,50,48,46,44,42,40,38,36,34,32,30,28,26,24,22,20,18,16,14,12,10)(62,63)(65,66);(1,2,4,6,7,5,3)(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)(64,65,66);(61,62,63)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13356, 95);

D53×C3⋊D21 #27211302