#34191302

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #10352

Subgroups

	subgroup	G/H
center	C₂³	#14579131
commutator, cosocle	C₂×C₄²×C₇₉	C₂³
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1279	C₂
socle, prime radical	C₂³×C₇₉	#120
hypercenter	#1279	D₇₉
hypocenter	C₇₉	#10352
LCS	G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1279

416 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 20224

A minimal faithful permutation representation — 3 generators on 111 points:

(1,2)(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)(57,58)(59,60)(61,62)(63,64)(65,66)(67,68)(69,70)(71,72)(73,74)(75,76)(77,78)(89,92)(90,95)(93,94)(96,98,99,105)(97,102,101,110)(100,109,107,106)(103,111,108,104);(80,82,83,87)(81,84,85,86)(88,89)(90,92)(91,93)(94,95)(97,101)(102,110)(103,108)(104,111);(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)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81,83,85)(82,86,87,84)(90,95)(92,94)(96,97,99,101)(98,104,105,111)(100,108,107,103)(102,109,110,106)

Polycyclic presentation:

2^8,79|a^2=d,b^2=dg,c^2=g,e^2=g,f^2=h,b^a=bd,c^a=ce,e^a=eg,f^a=fh,i^a=i^78,c^b=cf,e^b=eh,f^b=fh,e^c=eg,f^c=fh,i^c=i^78

Conjugacy classes & character table

2230 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 72601);