#34180663

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

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×#1512 ⋊ C₂
Sylow	C₇₉ ⋊ #8498

Subgroups

	subgroup	G/H
center	C₂²	#19098418
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1512	C₂
socle	C₂²×C₇₉	#379
prime radical	#23777740	C₂
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1512	D₇₉
hypocenter	C₇₉	#8498
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1512

692 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 119
transitive degree	μ_t(G) = 5056

A minimal faithful permutation representation — 3 generators on 119 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)(82,87)(84,86)(88,91,94,107)(89,96,97,111)(90,101,103,117)(92,106,93,105)(95,109,100,114)(98,112,99,104)(102,118,108,116)(110,115,113,119);(80,81)(82,86)(83,85)(84,87)(88,90)(89,95)(91,101)(92,102)(93,108)(94,103)(96,109)(97,100)(98,110)(99,113)(104,119)(105,116)(106,118)(107,117)(111,114)(112,115);(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)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(81,84,85,86)(88,89,93,99,94,97,92,98)(90,100,102,113,103,95,108,110)(91,104,106,96,107,112,105,111)(101,115,116,109,117,119,118,114)

Polycyclic presentation:

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

Conjugacy classes & character table

1048 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 61962);