#27421508

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1868 ⋊ C₂
Sylow	C₅₃ ⋊ #24411

Subgroups

	subgroup	G/H
center	C₂	#20278087
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1868	C₂
socle	C₂×C₅₃	#2410
prime radical	#20289503	C₂
hypercenter	#1868	D₅₃
hypocenter	C₅₃	#24411
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1868

1294 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 117
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 117 points:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(54,58)(55,63)(56,68)(57,73)(59,78)(60,77)(61,83)(62,86)(64,76)(65,88)(66,93)(67,94)(69,97)(70,96)(71,100)(72,101)(74,102)(75,103)(79,107)(80,105)(81,106)(82,108)(84,109)(85,110)(87,111)(89,104)(90,112)(91,113)(92,114)(95,115)(98,116)(99,117);(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)(54,57)(55,62)(56,67)(58,73)(59,74)(60,81)(61,82)(63,86)(64,71)(65,91)(66,92)(68,94)(69,98)(70,95)(72,80)(75,79)(76,100)(77,106)(78,102)(83,108)(84,99)(85,90)(87,89)(88,113)(93,114)(96,115)(97,116)(101,105)(103,107)(104,111)(109,117)(110,112);(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)(54,56,59,70)(55,61,64,66)(57,72,74,98)(58,68,78,96)(60,80,79,69)(62,85,71,99)(63,83,76,93)(65,90,89,84)(67,81,95,75)(73,101,102,116)(77,105,107,97)(82,91,92,87)(86,110,100,117)(88,112,104,109)(94,106,115,103)(108,113,114,111);(54,55)(56,66,70,61)(57,71)(58,76,78,63)(59,64)(60,65)(62,74)(67,92,95,82)(68,83)(69,90,80,84)(72,99,98,85)(73,100,102,86)(75,91)(77,104,107,88)(79,89)(81,87)(93,96)(94,114)(97,109)(101,117)(103,113,106,111)(105,112)(108,115)(110,116)

Polycyclic presentation:

2^8,53|a^2=fh,b^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^b=fh,g^b=gh,i^b=i^52,e^c=eh,g^c=gh,i^c=i^52,f^d=fh

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, 206984);