C₅₃×#535 #16810290

order	3392
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅₃ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₅₃×#535

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₅₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ G

334 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 65
transitive degree	μ_t(G) = 848

A minimal faithful permutation representation — 4 generators on 65 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)(55,57)(59,62)(60,64);(58,59,61,62)(60,63,64,65);(54,55)(56,57)(58,59)(60,63)(61,62)(64,65);(58,60)(59,63)(61,64)(62,65)

Conjugacy classes & character table

1166 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	#28302264, #28302615, #28302628, …
maximal subgroup of	C₅₃×#1910, C₅₃×#1911, C₅₃×#1914, …
maximal normal subgroup of	C₅₃×#1910, C₅₃×#1911, C₅₃×#1914, …
maximal quotient of	C₅₃×#2309, C₅₃×#2311, C₅₃×#2317, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3392, 216);