#27284429

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1789 ⋊ C₂
Sylow	C₅₃ ⋊ #9934

Subgroups

	subgroup	G/H
center	C₂²	#16810532
commutator, cosocle	C₂³×C₄×C₅₃	C₂³
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#1789	C₂
socle	C₂²×C₅₃	#392
prime radical	#20280439	C₂
hypercenter	#1789	D₅₃
hypocenter	C₅₃	#9934
LCS	G ▷ C₂³×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1789

808 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 808 conjugacy classes of subgroups — too many to draw. Only its 83 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 — 3 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)(57,60)(59,61)(63,69)(65,66)(68,72)(70,71)(73,75)(76,77);(58,61)(62,64)(63,68)(65,75)(66,73)(67,74)(69,77)(70,72)(71,76);(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)(56,58)(57,59)(60,61)(62,63,67,71)(64,72,74,77)(65,70,66,69)(68,75,76,73)

Polycyclic presentation:

2^8,53|a^2=g,d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fh,i^a=i^52,c^b=cf,d^b=dg,d^c=dh,e^c=eg,i^c=i^52

Conjugacy classes & character table

1019 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 69905);