#27304239

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #13542

Subgroups

	subgroup	G/H
center	C₂²	#16810912
commutator, cosocle	C₂²×C₈×C₅₃	C₂³
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1383	C₂
socle	C₂²×C₅₃	#476
prime radical	C₂³×C₅₃	AGL(1,ℤ/8)
hypercenter	#1383	D₅₃
hypocenter	C₅₃	#13542
LCS	G ▷ C₂²×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1383

252 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 87 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)(58,61)(62,64,65,75,66,76,78,77)(63,69,70,85,71,73,68,82)(67,81,79,74,80,91,93,92)(72,84,87,86,88,89,83,90);(54,56,57,60)(55,58,59,61)(63,68)(64,74,76,92)(65,78)(67,79)(69,84,73,89)(70,71)(75,81,77,91)(80,93)(82,86,85,90)(83,87);(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)(60,61)(62,63,65,70,66,71,78,68)(64,73,75,82,76,69,77,85)(67,72,79,87,80,88,93,83)(74,90,91,84,92,86,81,89)

Polycyclic presentation:

2^8,53|a^2=d,b^2=deg,c^2=d,d^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,i^a=i^52,c^b=cf,d^b=dh,f^b=fg,g^b=gh,f^c=fg,g^c=gh

Conjugacy classes & character table

872 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 89715);