#27293336

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1323 ⋊ C₂
Sylow	C₅₃ ⋊ #11470

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1323	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	#16811135	C₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1323	D₅₃
hypocenter	C₅₃	#11470
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1323

392 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 89 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)(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)(62,65,68,82,69,67,66,81)(63,71,74,89,75,73,72,88)(64,77,78,84,80,79,83,93)(70,85,76,91,87,86,90,92);(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)(54,56)(55,58)(57,61)(59,60)(62,64)(63,70)(65,79)(66,83)(67,77)(68,78)(69,80)(71,86)(72,90)(73,85)(74,76)(75,87)(81,84)(82,93)(88,91)(89,92);(54,55)(56,60,61,58)(57,59)(62,63,67,73,68,74,81,88,69,75,65,71,66,72,82,89)(64,76,79,92,78,87,93,85,80,90,77,91,83,70,84,86)

Polycyclic presentation:

2^8,53|a^2=e,c^2=g,d^2=h,e^2=g,g^2=h,b^a=bd,d^a=df,c^b=ch,d^b=dh,e^b=efh,i^b=i^52,i^c=i^52

Conjugacy classes & character table

1052 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 78812);