#27540897

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2178 ⋊ C₂
Sylow	C₅₃ ⋊ #33859

Subgroups

	subgroup	G/H
center	C₂	D₈×D₂₁₂
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2178	C₂
socle	C₂×C₅₃	D₄×D₈
hypercenter	#2178	D₅₃
hypocenter	C₅₃	#33859
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #2178

646 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)(54,57)(55,63)(56,70)(58,81)(59,77)(60,85)(61,78)(62,89)(64,96)(65,74)(66,99)(67,94)(68,102)(69,104)(71,106)(72,109)(73,110)(75,80)(76,83)(79,93)(82,108)(84,115)(86,105)(87,107)(88,111)(90,116)(91,113)(92,95)(97,114)(98,117)(100,101)(103,112);(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)(54,57)(55,63)(56,69)(58,76)(59,85)(60,77)(61,78)(62,88)(64,93)(65,99)(66,74)(67,94)(68,101)(70,104)(71,105)(72,107)(73,75)(79,96)(80,110)(81,83)(82,115)(84,108)(86,106)(87,109)(89,111)(90,112)(91,92)(95,113)(97,117)(98,114)(100,102)(103,116);(54,56,61,73)(55,62,67,91)(57,75,78,69)(58,80,84,104)(59,72,60,86)(63,92,94,88)(64,95,98,111)(65,90,66,100)(68,97,103,79)(70,108,110,76)(71,82,87,83)(74,112,99,102)(77,107,85,106)(81,109,115,105)(89,114,113,93)(96,116,117,101);(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)(54,55,61,67)(56,68,73,103)(57,74,78,99)(58,79,84,97)(59,66,60,65)(62,87,91,71)(63,85,94,77)(64,82,98,83)(69,89,75,113)(70,88,110,92)(72,111,86,95)(76,114,108,93)(80,100,104,90)(81,96,115,117)(101,107,116,106)(102,109,112,105)

Polycyclic presentation:

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

Conjugacy classes & character table

1187 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 326373);