C₅₃×#538 #16810293

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

Aliases

direct product

C₅₃×#538

Subgroups

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

250 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 69
transitive degree	μ_t(G) = 848

A minimal faithful permutation representation — 4 generators on 69 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)(56,62)(57,61)(58,60)(59,63)(64,66)(65,68)(67,69);(54,56,58,64)(55,59,60,67)(57,63,65,69)(61,62,68,66);(55,60)(56,63)(59,66)(61,68)(62,67)(64,69);(55,61)(59,62)(60,68)(66,67)

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	#28302339, #28302348, #28302355, …
maximal subgroup of	C₅₃×#1600, C₅₃×#1606, C₅₃×#1612, …
maximal normal subgroup of	C₅₃×#1600, C₅₃×#1606, C₅₃×#1612, …
maximal quotient of	C₅₃×#2312, C₅₃×#2318, C₅₃×#2321, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3392, 219);