C₅₃⋊#17309 #27349656

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

unambiguous semidirect product

C₅₃⋊#17309

Splittings

Sylow

C₅₃ ⋊ #17309

Subgroups

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₅₃
commutator, socle, prime radical, cosocle	C₂⁴×C₅₃	C₂⁴
Fitting	C₅₃×#1446	C₂
hypercenter	#1446	D₅₃
hypocenter	C₅₃	#17309
LCS	G ▷ C₂⁴×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ #1446

1466 conjugacy classes of subgroups; only the 283 normal subgroups are shown.

normal subgroups only

This group has 1466 conjugacy classes of subgroups — too many to draw. Only its 283 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) = 85
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 4 generators on 85 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)(56,60)(58,61)(64,69)(67,68)(70,72,73,77)(71,74,75,76);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(55,59)(58,61)(62,64)(63,67)(65,69)(66,68)(78,80,81,85)(79,82,83,84);(54,56)(55,58)(57,60)(59,61)(63,66)(67,68)(78,79,81,83)(80,84,85,82);(54,55)(56,58)(57,59)(60,61)(62,63)(64,68)(65,66)(67,69)(70,71,73,75)(72,76,77,74)

Polycyclic presentation:

2^8,53|a^2=h,b^2=g,c^2=g,d^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,d^c=de,i^d=i^52

Conjugacy classes & character table

1190 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 135132);