#27415622

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1794 ⋊ C₂
Sylow	C₅₃ ⋊ #23893

Subgroups

	subgroup	G/H
center	C₂	#20278028
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1794	C₂
socle	C₂×C₅₃	#2403
hypercenter	#1794	D₅₃
hypocenter	C₅₃	#23893
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1794

1610 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)(54,58,61,82)(55,64,66,93)(56,69,72,100)(57,75,78,108)(59,81,60,80)(62,85,87,111)(63,89,91,105)(65,92,83,79)(67,95,86,110)(68,97,98,117)(70,101,71,99)(73,103,90,113)(74,104,102,116)(76,107,77,106)(84,109,96,115)(88,112,94,114);(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)(56,68)(58,75)(59,76)(60,77)(61,78)(62,84)(64,89)(65,90)(66,91)(67,94)(69,97)(70,74)(71,102)(72,98)(73,83)(79,103)(80,106)(81,107)(82,108)(85,109)(86,88)(87,96)(92,113)(93,105)(95,114)(99,116)(100,117)(101,104)(110,112)(111,115);(54,56)(55,62)(57,74)(58,69)(59,71)(60,70)(61,72)(63,88)(64,85)(65,86)(66,87)(67,83)(68,76)(73,84)(75,104)(77,98)(78,102)(79,95)(80,101)(81,99)(82,100)(89,112)(90,96)(91,94)(92,110)(93,111)(97,107)(103,109)(105,114)(106,117)(108,116)(113,115);(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)(56,67)(57,73)(58,79)(59,83)(60,65)(61,66)(62,70)(63,77)(64,80)(68,96)(69,85)(71,87)(72,86)(74,94)(75,105)(76,91)(78,90)(81,93)(82,92)(84,98)(88,102)(89,108)(95,99)(97,114)(100,111)(101,110)(103,106)(104,109)(107,113)(112,117)(115,116)

Polycyclic presentation:

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

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, 201098);