#27304273

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₆ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1397 ⋊ C₂
Sylow	C₅₃ ⋊ #13549

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₅₃
commutator	C₂×C₅₃	C₂×C₄×C₁₆
Fitting	C₅₃×#1397	C₂
socle	C₂²×C₅₃	C₂²×C₁₆
prime radical	C₂²×D₅₃	C₂×C₁₆
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	#1397	D₅₃
hypocenter	C₅₃	#13549
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ #1397

238 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 121
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 3 generators on 121 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)(60,70)(65,80)(71,90)(72,91)(73,92)(81,100)(82,101)(83,102)(93,110)(94,111)(95,112)(103,115)(104,116)(105,117)(113,120)(118,121);(54,55,56,57)(58,60)(59,65)(61,71)(62,72)(63,73)(64,70)(66,81)(67,82)(68,83)(69,80)(74,93)(75,94)(76,90)(77,95)(78,91)(79,92)(84,103)(85,104)(86,100)(87,105)(88,101)(89,102)(96,113)(97,110)(98,111)(99,112)(106,118)(107,115)(108,116)(109,117)(114,120)(119,121);(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)(58,59,61,66,62,67,74,84,63,68,75,85,77,87,96,106,64,69,76,86,78,88,97,107,79,89,98,108,99,109,114,119)(60,65,71,81,72,82,93,103,73,83,94,104,95,105,113,118,70,80,90,100,91,101,110,115,92,102,111,116,112,117,120,121)

Polycyclic presentation:

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

Conjugacy classes & character table

2240 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 89749);