C₃×C₇×A₄×C₅₃ #27211324

order	13356
isoclinism	isoclinic class of A₄
automorphism	?
homology	Abelianization (H₁) = C₃²×C₇×C₅₃ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product	C₃×C₇×A₄×C₅₃
unambiguous semidirect product	C₂²⋊(C₃²×C₇×C₅₃)
unambiguous semidirect product	(C₂²×C₃)⋊(C₃×C₇×C₅₃)
unambiguous semidirect product	(C₂²×C₅₃)⋊(C₃²×C₇)
unambiguous semidirect product	(C₂²×C₃×C₅₃)⋊(C₃×C₇)

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₇×C₅₃	A₄
commutator, cosocle, hypocenter	C₂²	C₃²×C₇×C₅₃
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₇×C₅₃	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₃×C₇×C₅₃

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 4452

A minimal faithful permutation representation — 2 generators on 67 points:

(2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57)(58,59,60,61,62,63,64);(1,4)(2,3)(65,66,67)

Conjugacy classes & character table

4452 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13356, 117);

C3×C7×A4×C53 #27211324