#34179684

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×#1476 ⋊ C₂
Sylow	C₇₉ ⋊ #8321

Subgroups

	subgroup	G/H
center	C₂×C₄	#14579131
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1476	C₂
socle	C₂²×C₇₉	#379
prime radical	#19099141	C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1476	D₇₉
hypocenter	C₇₉	#8321
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1476

710 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 10112

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)(82,88)(85,91)(89,94)(92,95)(96,98,100,105)(97,102,103,111)(99,108,106,104)(101,109,107,110);(80,82)(81,85)(83,89)(84,88)(86,92)(87,91)(90,94)(93,95)(96,97)(98,102)(99,107)(100,103)(101,106)(104,109)(105,111)(108,110);(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)(80,81,83,86,84,87,90,93)(82,85,89,92,88,91,94,95)(97,101)(99,106)(102,110)(103,107)(104,108)(109,111)

Polycyclic presentation:

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

Conjugacy classes & character table

2236 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 60983);