#27271945

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₈

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1410 ⋊ C₂
Sylow	C₅₃ ⋊ #7527

Subgroups

	subgroup	G/H
center	C₂	#20272877
commutator	C₄×C₈×C₅₃	C₂×C₄
Frattini	#492	C₂×D₅₃
Fitting	C₅₃×#1410	C₂
socle	C₂×C₅₃	#1408
prime radical	#20274379	C₂
cosocle	C₅₃×#492	C₂²
hypercenter	#1410	D₅₃
hypocenter	C₅₃	#7527
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #492 ◁ #1410

164 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 25 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) = 69
transitive degree	μ_t(G) = 848

A minimal faithful permutation representation — 2 generators on 69 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)(56,64)(57,65)(60,68)(61,69)(62,63)(66,67);(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)(56,63,66,59)(57,68,65,62)(58,61)(60,64,69,67)

Polycyclic presentation:

2^8,53|a^2=d,c^2=fgh,e^2=gh,f^2=h,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh,i^a=i^52,c^b=cfg,d^b=defh,e^b=egh,f^b=fh,i^b=i^52,d^c=dfg,e^d=egh,f^d=fh

Conjugacy classes & character table

461 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 57421);