#34180129

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

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×#1506 ⋊ C₂
Sylow	C₇₉ ⋊ #8413

Subgroups

	subgroup	G/H
center	C₂²	#19098418
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1506	C₂
socle	C₂²×C₇₉	#379
prime radical	#23777409	C₂
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1506	D₇₉
hypocenter	C₇₉	#8413
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1506

702 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)(88,91,93,104)(89,95,96,109)(90,99,101,116)(92,105,106,103)(94,107,98,112)(97,111,110,102)(100,117,118,115)(108,114,113,119);(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)(80,81)(82,86)(83,85)(84,87)(88,90)(89,94)(91,99)(92,100)(93,101)(95,107)(96,98)(97,108)(102,119)(103,115)(104,116)(105,117)(106,118)(109,112)(110,113)(111,114);(81,84,85,86)(88,89)(90,98,101,94)(91,102,104,111)(92,97)(93,96)(95,103,109,105)(99,114)(100,113,118,108)(106,110)(107,115)(112,117)(116,119)

Polycyclic presentation:

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

Conjugacy classes & character table

1291 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 61428);