#27304726

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #13660

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₅₃
commutator, socle, prime radical	C₂²×C₅₃	C₂²×C₁₆
Fitting	C₅₃×C₄⋊C₃₂	C₂
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	C₄⋊C₃₂	D₅₃
hypocenter	C₅₃	#13660
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄⋊C₃₂

158 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 87 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) = 93
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 3 generators on 93 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)(62,64,65,74,66,75,77,89,67,72,73,86,79,87,88,93,68,76,78,82,80,90,91,92,81,63,69,70,83,71,84,85);(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)(54,56,57,61)(55,58,59,60);(54,55,57,59)(56,60,61,58)(62,63,65,70,66,71,77,85,67,64,73,74,79,75,88,89,68,72,78,86,80,87,91,93,81,76,69,82,83,90,84,92)

Polycyclic presentation:

2^8,53|a^2=e,b^2=d,c^2=de,e^2=f,f^2=g,g^2=h,b^a=bd,c^a=ch,i^b=i^52,i^c=i^52

Conjugacy classes & character table

2192 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 90202);