#27421640

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1858 ⋊ C₂
Sylow	C₅₃ ⋊ #24423

Subgroups

	subgroup	G/H
center	C₂	#20278087
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1858	C₂
socle	C₂×C₅₃	#2410
prime radical	#20289503	C₂
hypercenter	#1858	D₅₃
hypocenter	C₅₃	#24423
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1858

1284 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)(55,64)(56,69)(57,74)(59,79)(60,80)(61,78)(62,85)(63,88)(65,77)(66,91)(67,95)(68,97)(70,100)(71,101)(72,103)(73,104)(75,105)(76,107)(81,99)(82,106)(83,90)(84,109)(86,111)(87,112)(89,108)(92,110)(93,113)(94,114)(96,115)(98,116)(102,117);(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)(55,63)(56,68)(58,74)(59,82)(60,75)(61,76)(62,84)(64,88)(65,89)(66,93)(67,94)(69,97)(70,102)(71,98)(72,83)(73,81)(77,108)(78,107)(79,106)(80,105)(85,109)(86,96)(87,92)(90,103)(91,113)(95,114)(99,104)(100,117)(101,116)(110,112)(111,115);(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)(54,56,60,71)(55,62,65,67)(57,73,75,102)(58,69,80,101)(59,81,61,70)(63,87,89,96)(64,85,77,95)(66,92,83,86)(68,82,98,76)(72,84,93,94)(74,104,105,117)(78,100,79,99)(88,112,108,115)(90,111,91,110)(97,106,116,107)(103,109,113,114);(54,55,61,83,60,65,59,66)(56,67,81,86,71,62,70,92)(57,72,82,63,75,93,76,89)(58,77,79,90,80,64,78,91)(68,96,102,94,98,87,73,84)(69,85,100,111,101,95,99,110)(74,103,107,108,105,113,106,88)(97,115,104,109,116,112,117,114)

Polycyclic presentation:

2^8,53|a^2=fh,b^2=h,f^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,f^c=fh,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, 207116);