C₂²×C₃×ΓL(1,𝔽₈)×C₅₃ #27211325

order	13356
isoclinism	isoclinic class of ΓL(1,𝔽₈)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃²×C₅₃ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product

C₂²×C₃×ΓL(1,𝔽₈)×C₅₃

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₂²×C₃×C₅₃	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₂²×C₃²×C₅₃
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₇×C₅₃	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂²×C₃×C₅₃

120 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)(5,7,6)(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)(66,67);(1,2,3,5,4,6,7)(61,62,63)(64,65)

Conjugacy classes & character table

3180 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13356, 118);

C22×C3×ΓL(1,𝔽8)×C53 #27211325