#27274540

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #8139

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄²×D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1275	C₂
socle	C₂²×C₅₃	#376
prime radical	C₂³×C₅₃	#117
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1275	D₅₃
hypocenter	C₅₃	#8139
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #1275

330 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 131 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)(55,58)(59,61)(62,65,66,76,67,77,78,89)(63,69,70,81,71,82,83,91)(64,73,74,86,75,87,88,93)(68,79,72,84,80,90,85,92);(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)(62,64)(63,68,71,80)(65,73)(66,74)(67,75)(69,79,82,90)(70,72,83,85)(76,86)(77,87)(78,88)(81,84,91,92)(89,93);(54,55,56,59,57,58,60,61)(62,63)(64,72,75,85)(65,69)(66,70)(67,71)(68,74,80,88)(73,84,87,92)(76,81)(77,82)(78,83)(79,86,90,93)(89,91)

Polycyclic presentation:

2^8,53|a^2=e,b^2=f,c^2=d,d^2=g,e^2=h,b^a=bd,c^a=ch,f^a=fg,e^b=eg,i^b=i^52,i^c=i^52

Conjugacy classes & character table

1208 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 60016);