C₂×C₃²×D₇×C₅₃ #27211335

order	13356
isoclinism	isoclinic class of D₇
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃²×C₅₃ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product

C₂×C₃²×D₇×C₅₃

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃²×C₅₃	D₇
commutator, cosocle, hypocenter	C₇	C₂²×C₃²×C₅₃
Frattini	C₁	G
Fitting, socle	C₂×C₃²×C₇×C₅₃	C₂
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₃²×C₅₃

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 6678

A minimal faithful permutation representation — 2 generators on 68 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)(64,65,66);(1,2,4,6,7,5,3)(61,62,63)(67,68)

Conjugacy classes & character table

4770 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13356, 128);

C2×C32×D7×C53 #27211335