D₇×C₃⋊D₃×C₅₃ #27211303

order	13356
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅₃ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product	D₇×C₃⋊D₃×C₅₃
unambiguous semidirect product	(C₃×D₇)⋊(D₃×C₅₃)

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₅₃	D₇×C₃⋊D₃
commutator, cosocle, hypocenter	C₃²×C₇	C₂²×C₅₃
Frattini	C₁	G
Fitting, socle	C₃²×C₇×C₅₃	C₂²
LCS	G ▷ C₃²×C₇
UCS	C₁ ◁ C₅₃

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:

(1,2,4,6,7,5,3)(9,10)(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);(2,3)(4,5)(6,7)(11,12,13);(8,9,10)

Conjugacy classes & character table

1590 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13356, 96);

D7×C3⋊D3×C53 #27211303