#27300441

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #12880

Subgroups

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator, socle, prime radical	C₂³×C₅₃	C₂²×C₈
Fitting	C₂×C₅₃×C₄⋊C₁₆	C₂
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₅₃
hypocenter	C₅₃	#12880
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

322 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 151 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) = 97
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 3 generators on 97 points:

(1,2)(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)(54,56,57,55)(59,63)(62,65)(68,76)(72,82)(77,88)(78,89)(83,92)(84,93)(90,96)(94,97);(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)(54,55,57,56)(58,60,61,64)(59,62,63,65)(66,68)(67,72)(69,77)(70,78)(71,76)(73,83)(74,84)(75,82)(79,90)(80,88)(81,89)(85,94)(86,92)(87,93)(91,96)(95,97);(55,56)(58,59)(60,62)(61,63)(64,65)(66,67,69,73,70,74,79,85,71,75,80,86,81,87,91,95)(68,72,77,83,78,84,90,94,76,82,88,92,89,93,96,97)

Polycyclic presentation:

2^8,53|a^2=f,b^2=e,c^2=d,f^2=g,g^2=h,b^a=bd,c^a=ce,c^b=ch,i^b=i^52,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, 85917);