#27300619

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1388 ⋊ C₂
Sylow	C₅₃ ⋊ #12924

Subgroups

	subgroup	G/H
center	C₄	#16810758
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1388	C₂
socle	C₂×C₅₃	#1852
prime radical	C₂³×D₅₃	Q₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1388	D₅₃
hypocenter	C₅₃	#12924
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1388

386 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 109 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 — 3 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)(57,71)(62,80)(65,81)(66,83)(69,84)(70,85)(72,73)(79,82);(54,56,58,68,60,67,59,75)(55,61,63,77,64,76,74,78)(57,70,72,83,71,85,73,66)(62,79,69,65,80,82,84,81);(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)(54,55,58,63,60,64,59,74)(56,65,68,82,67,81,75,79)(57,69,72,80,71,84,73,62)(61,70,77,83,76,85,78,66)

Polycyclic presentation:

2^8,53|a^2=f,b^2=f,d^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,i^a=i^52,d^b=dg,e^b=eh,g^b=gh,d^c=dh,i^c=i^52

Conjugacy classes & character table

872 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 86095);