C₇×C₃⋊D₃×D₅₃ #27211300

order	13356
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇ · Schur multiplier (H₂) = C₂×C₃

Aliases

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

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₇	C₃⋊D₃×D₅₃
commutator, cosocle, hypocenter	C₃²×C₅₃	C₂²×C₇
Frattini	C₁	G
Fitting, socle	C₃²×C₇×C₅₃	C₂²
LCS	G ▷ C₃²×C₅₃
UCS	C₁ ◁ C₇

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

Conjugacy classes & character table

1176 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13356, 93);

C7×C3⋊D3×D53 #27211300