#27498593

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2047 ⋊ C₂
Sylow	C₅₃ ⋊ #30440

Subgroups

	subgroup	G/H
center	C₂	#20280922
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2047	C₂
socle	C₂×C₅₃	#2668
prime radical	#20284082	C₂
hypercenter	#2047	D₅₃
hypocenter	C₅₃	#30440
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2047

850 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)(54,58,61,81)(55,64,67,93)(56,70,73,101)(57,75,77,107)(59,80,60,79)(62,84,86,109)(63,88,89,114)(65,92,66,91)(68,96,85,90)(69,97,99,116)(71,100,72,78)(74,103,94,104)(76,108,82,106)(83,110,87,112)(95,113,111,115)(98,117,102,105);(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)(55,63)(56,69)(58,75)(59,76)(60,82)(61,77)(62,83)(64,88)(65,74)(66,94)(67,89)(68,95)(70,97)(71,98)(72,102)(73,99)(78,105)(79,106)(80,108)(81,107)(84,110)(85,111)(86,87)(90,115)(91,104)(92,103)(93,114)(96,113)(100,117)(101,116)(109,112);(54,56,61,73)(55,62,67,86)(57,69,77,99)(58,78,81,100)(59,72,60,71)(63,87,89,83)(64,90,93,96)(65,85,66,68)(70,80,101,79)(74,95,94,111)(75,105,107,117)(76,102,82,98)(84,92,109,91)(88,113,114,115)(97,108,116,106)(103,110,104,112);(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)(54,55)(56,68)(57,74)(58,64)(59,66)(60,65)(61,67)(62,71)(63,76)(69,87)(70,96)(72,86)(73,85)(75,104)(77,94)(78,109)(79,92)(80,91)(81,93)(82,89)(83,99)(84,100)(88,106)(90,101)(95,98)(97,110)(102,111)(103,107)(105,113)(108,114)(112,116)(115,117)

Polycyclic presentation:

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

Conjugacy classes & character table

710 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 284069);