#27300493

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #12895

Subgroups

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator, socle, prime radical	C₂³×C₅₃	C₂²×C₈
Fitting	C₅₃×#1374	C₂
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1374	D₅₃
hypocenter	C₅₃	#12895
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

298 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 127 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) = 101
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 3 generators on 101 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)(55,59)(58,61)(62,64,65,69)(63,67,66,68)(72,80)(76,86)(81,92)(82,93)(87,96)(88,97)(94,100)(98,101);(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)(55,58,59,61)(63,66)(67,68)(70,72)(71,76)(73,81)(74,82)(75,80)(77,87)(78,88)(79,86)(83,94)(84,92)(85,93)(89,98)(90,96)(91,97)(95,100)(99,101);(54,55)(56,58)(57,59)(60,61)(62,63)(64,68)(65,66)(67,69)(70,71,73,77,74,78,83,89,75,79,84,90,85,91,95,99)(72,76,81,87,82,88,94,98,80,86,92,96,93,97,100,101)

Polycyclic presentation:

2^8,53|a^2=f,b^2=de,c^2=d,f^2=g,g^2=h,b^a=bd,c^a=ce,c^b=ch,i^c=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, 85969);