#27270866

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1273 ⋊ C₂
Sylow	C₅₃ ⋊ #7114

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133139
commutator	C₂×C₈×C₅₃	C₂×C₈
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₅₃×#1273	C₂
socle	C₂²×C₅₃	D₄⋊C₈
prime radical	C₂×#13133239	C₄
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	#1273	D₅₃
hypocenter	C₅₃	#7114
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ #1273

258 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 67 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) = 77
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 2 generators on 77 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)(55,58)(57,61)(59,60)(62,64)(63,68)(65,74)(66,71)(67,75)(69,77)(70,72)(73,76);(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)(56,60,61,58)(57,59)(62,63,65,70,67,71,77,73,68,64,72,74,66,75,76,69)

Polycyclic presentation:

2^8,53|a^2=d,c^2=fh,d^2=g,e^2=h,f^2=h,g^2=h,b^a=bc,c^a=ce,e^a=eh,f^a=fh,i^a=i^52,c^b=cf,d^b=defh,e^b=eh,f^b=fh,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, 56342);