#27287875

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #10471

Subgroups

	subgroup	G/H
center	C₂²	#16810538
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1805	C₂
socle	C₂²×C₅₃	#394
prime radical	C₂³×C₅₃	#121
hypercenter	#1805	D₅₃
hypocenter	C₅₃	#10471
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1805

366 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 93 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) = 93
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 93 points:

(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)(57,61)(58,60)(62,65,66,80,68,81,67,82)(63,70,71,87,72,78,83,92)(64,74,75,90,77,91,76,79)(69,84,73,86,85,93,88,89);(55,58)(59,60)(62,64,68,77)(63,69,72,85)(65,79,81,90)(66,76,67,75)(70,86,78,89)(71,88,83,73)(74,80,91,82)(84,92,93,87);(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)(54,55,56,58)(57,59,61,60)(62,63)(64,73)(65,78)(66,83)(67,71)(68,72)(69,75)(70,81)(74,89)(76,85)(77,88)(79,84)(80,87)(82,92)(86,91)(90,93)

Polycyclic presentation:

2^8,53|a^2=h,b^2=g,c^2=dh,d^2=g,e^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,i^a=i^52,c^b=cf,d^b=dg,f^b=fg,e^c=egh

Conjugacy classes & character table

1025 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 73351);