#27304303

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #13558

Subgroups

	subgroup	G/H
center	C₁₆	C₂×C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂²×C₁₆
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₂×C₅₃×#504	C₂
socle	C₂×C₅₃	C₂×C₂²⋊C₁₆
prime radical	C₂×#41×C₅₃	C₈
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	C₂×#504	D₅₃
hypocenter	C₅₃	#13558
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ C₂×#504

290 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 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)(55,63)(62,83)(64,67)(65,86)(66,87)(68,88)(84,91)(85,108)(89,93)(90,94)(92,110)(95,103)(96,111)(109,113)(112,114)(115,117);(54,56)(55,62)(57,73)(58,76)(59,71)(60,69)(61,72)(63,83)(64,88)(65,91)(66,85)(67,68)(70,79)(74,97)(75,101)(77,104)(78,82)(80,98)(81,100)(84,86)(87,108)(89,95)(90,111)(92,113)(93,103)(94,96)(99,106)(102,116)(105,107)(109,110)(112,115)(114,117);(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)(54,55,58,65,59,66,77,92,60,67,78,93,80,94,105,114,61,63,79,86,81,87,106,110,57,64,74,89,75,90,102,112)(56,68,70,95,71,96,99,115,69,83,97,91,98,108,116,113,72,88,76,103,100,111,104,117,73,62,82,84,101,85,107,109)

Polycyclic presentation:

2^8,53|a^2=e,d^2=h,e^2=f,f^2=g,g^2=h,b^a=bd,c^a=ch,i^a=i^52,d^b=dh,e^b=eh

Conjugacy classes & character table

2168 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 89779);