D₃²×C₇×C₅₃ #27211307

order	13356
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇×C₅₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	D₃²×C₇×C₅₃
unambiguous semidirect product	(C₃×D₃)⋊(C₂×C₇×C₅₃)
unambiguous semidirect product	(C₃⋊D₃)⋊(C₂×C₇×C₅₃)

Splittings

Fitting, socle, Hall	C₃²×C₇×C₅₃ ⋊ C₂²
derived, Sylow	C₃² ⋊ C₂²×C₇×C₅₃
Hall	C₃²×C₅₃ ⋊ C₂²×C₇
Hall	C₃²×C₇ ⋊ C₂²×C₅₃

Subgroups

	subgroup	G/H
center, hypercenter	C₇×C₅₃	D₃²
commutator, cosocle, hypocenter	C₃²	C₂²×C₇×C₅₃
Frattini	C₁	G
Fitting, socle	C₃²×C₇×C₅₃	C₂²
LCS	G ▷ C₃²
UCS	C₁ ◁ C₇×C₅₃

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 66
transitive degree	μ_t(G) = 2226

A minimal faithful permutation representation — 2 generators on 66 points:

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

Conjugacy classes & character table

3339 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13356, 100);

D32×C7×C53 #27211307