#27270757

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #7073

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133139
commutator	C₂×C₈×C₅₃	C₂×C₈
Frattini	C₈²	C₂×D₅₃
Fitting	C₅₃×#1312	C₂
socle, prime radical	C₂²×C₅₃	D₄⋊C₈
cosocle	C₈²×C₅₃	C₂²
hypercenter	#1312	D₅₃
hypocenter	C₅₃	#7073
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1312

132 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 55 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) = 13568

A minimal faithful permutation representation — 2 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)(55,59)(56,62)(61,66)(63,68)(70,71,74,77)(72,79,81,85)(73,82,80,76)(75,83,84,78);(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)(54,55,56,60,58,61,63,67,57,59,62,65,64,66,68,69)(71,75)(72,78)(73,80)(76,85)(77,84)(79,82)(81,83)

Polycyclic presentation:

2^8,53|a^2=d,b^2=g,c^2=egh,d^2=f,e^2=g,f^2=h,b^a=bc,c^a=ce,e^a=eg,i^a=i^52,c^b=ceh,d^b=dh,e^b=eg,i^b=i^52

Conjugacy classes & character table

2138 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 56233);