#34181918

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #8781

Subgroups

	subgroup	G/H
center	C₂×C₄	#14579119
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1564	C₂
socle	C₂²×C₇₉	#382
prime radical	C₂³×C₇₉	#117
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1564	D₇₉
hypocenter	C₇₉	#8781
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

310 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)(80,82,84,83)(81,85,87,86)(88,91)(89,95,96,108)(90,99)(92,103)(93,104)(94,106,98,110)(97,102,109,117)(100,113)(101,114)(105,118)(107,116,111,112)(115,119);(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)(88,90,92,100,93,101,105,115)(89,94,97,107,96,98,109,111)(91,99,103,113,104,114,118,119)(95,106,102,116,108,110,117,112);(80,81,83,86,84,87,82,85)(88,89)(90,98,101,94)(91,102,104,117)(92,97)(93,96)(95,103,108,118)(99,112)(100,111,115,107)(105,109)(106,113)(110,119)(114,116)

Polycyclic presentation:

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

Conjugacy classes & character table

2248 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 63217);