#34180519

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

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×#1488 ⋊ C₂
Sylow	C₇₉ ⋊ #8475

Subgroups

	subgroup	G/H
center	C₂²	#19098418
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1488	C₂
socle	C₂²×C₇₉	#379
prime radical	#23777193	C₂
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1488	D₇₉
hypocenter	C₇₉	#8475
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1488

932 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)(81,84)(82,86)(88,91)(89,95)(90,99)(92,107)(93,104)(94,109)(96,113)(97,102)(98,114)(100,117)(101,116)(103,108)(105,111)(106,115)(110,118)(112,119);(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)(82,86)(88,90)(89,94)(91,99)(92,106)(93,101)(95,109)(96,112)(97,110)(98,105)(100,108)(102,118)(103,117)(104,116)(107,115)(111,114)(113,119);(80,81,82,85,83,84,86,87)(88,89)(90,98)(91,102,104,95)(92,105)(93,97)(94,100)(96,108)(99,114,116,119)(101,112)(103,111,107,113)(106,110)(109,115,118,117)

Polycyclic presentation:

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

Conjugacy classes & character table

1522 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 61818);