#28297993

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×H(7,𝔽₂) ⋊ C₂
Sylow	C₅₃ ⋊ #63103

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂⁶×D₅₃
commutator, socle, cosocle	C₂×C₅₃	C₂⁷
Fitting	C₅₃×H(7,𝔽₂)	C₂
hypercenter	H(7,𝔽₂)	D₅₃
hypocenter	C₅₃	#63103
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ H(7,𝔽₂)

59452 conjugacy classes of subgroups; only the 32039 normal subgroups are shown.

normal subgroups only

This group has 59452 conjugacy classes of subgroups — too many to draw. Only its 32039 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) = 1696

A minimal faithful permutation representation — 7 generators on 85 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)(54,58,59,70)(55,63,62,75)(56,67,66,80)(57,69,68,81)(60,72,64,77)(61,74,73,83)(65,79,78,85)(71,82,76,84);(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)(55,62)(60,64)(61,73)(63,75)(71,76)(72,77)(74,83)(82,84);(56,66)(60,64)(65,78)(67,80)(71,76)(72,77)(79,85)(82,84);(57,68)(61,73)(65,78)(69,81)(71,76)(74,83)(79,85)(82,84);(54,57)(55,61)(56,65)(58,69)(59,68)(60,71)(62,73)(63,74)(64,76)(66,78)(67,79)(70,81)(72,82)(75,83)(77,84)(80,85);(54,56)(55,60)(57,65)(58,67)(59,66)(61,71)(62,64)(63,72)(68,78)(69,79)(70,80)(73,76)(74,82)(75,77)(81,85)(83,84);(54,55)(56,64)(57,61)(58,63)(59,62)(60,66)(65,76)(67,77)(68,73)(69,74)(70,75)(71,78)(72,80)(79,84)(81,83)(82,85)

Polycyclic presentation:

2^8,53|g^2=h,b^a=bh,f^a=fh,e^b=eh,d^c=dh,i^g=i^52

Conjugacy classes & character table

1820 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 1083469);