#27270851

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1375 ⋊ C₂
Sylow	C₅₃ ⋊ #7109

Subgroups

	subgroup	G/H
center	C₄	#16810356
commutator	C₂×C₈×C₅₃	C₂×C₈
Frattini	C₂×#97	C₂×D₅₃
Fitting	C₅₃×#1375	C₂
socle	C₂×C₅₃	#1270
prime radical	C₂×#13133239	C₄
cosocle	C₂×#97×C₅₃	C₂²
hypercenter	#1375	D₅₃
hypocenter	C₅₃	#7109
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1375

228 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 63 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 — 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)(54,56)(55,60)(57,70)(58,66)(59,68)(61,76)(62,75)(63,79)(64,78)(65,73)(67,71)(69,80)(72,83)(74,77)(81,84)(82,85);(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)(56,64,66,80,68,81,67,76)(57,69,72,61)(58,63)(59,62)(60,65,74,70,75,82,79,83)(71,77)(73,84,85,78)

Polycyclic presentation:

2^8,53|a^2=d,c^2=fh,d^2=g,f^2=h,b^a=bc,c^a=ce,e^a=eh,i^a=i^52,c^b=cf,d^b=def,f^b=fh,g^b=gh,i^b=i^52,d^c=dh

Conjugacy classes & character table

1184 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 56327);