#27278216

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1832 ⋊ C₂
Sylow	C₅₃ ⋊ #8877

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133185
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1832	C₂
socle	C₂²×C₅₃	#385
prime radical	#13133191	C₂×C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1832	D₅₃
hypocenter	C₅₃	#8877
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1832

360 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 93 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) = 85
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 85 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)(54,57,60,56)(55,62,65,61)(58,69,59,67)(63,68,64,66)(70,73,76,71)(72,81,84,77)(74,78,75,79)(80,83,85,82);(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)(54,56,60,57)(55,61,65,62)(58,69,59,67)(63,68,64,66)(70,72,74,83,76,84,75,82)(71,77,79,80,73,81,78,85);(54,55,60,65)(56,66,57,68)(58,64,59,63)(61,69,62,67)(70,71,74,79,76,73,75,78)(72,80,83,81,84,85,82,77)

Polycyclic presentation:

2^8,53|a^2=f,b^2=f,c^2=g,d^2=g,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dg,e^b=eh,i^b=i^52,d^c=dh,e^c=eg,i^c=i^52

Conjugacy classes & character table

1190 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 63692);