#27304643

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×C₄⋊C₃₂ ⋊ C₂
Sylow	C₅₃ ⋊ #13645

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₁₆
Fitting	C₅₃×C₄⋊C₃₂	C₂
prime radical	C₂×D₂₁₂	C₁₆
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	C₄⋊C₃₂	D₅₃
hypocenter	C₅₃	#13645
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄⋊C₃₂

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

normal subgroups only

This group has 204 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) = 89
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

2^8,53|a^2=e,c^2=e,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, 90119);