#27274527

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #8135

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₃
commutator, socle, prime radical	C₂³×C₅₃	C₂×C₄²
Fitting	C₅₃×#1744	C₂
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1744	D₅₃
hypocenter	C₅₃	#8135
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ #1744

382 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 179 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) = 13568

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

Polycyclic presentation:

2^8,53|a^2=e,b^2=f,c^2=d,e^2=g,f^2=h,b^a=bd,c^a=cgh,c^b=cg,i^b=i^52

Conjugacy classes & character table

2168 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 60003);