#34181589

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄×#107×C₇₉ ⋊ C₂
Sylow	C₇₉ ⋊ #8697

Subgroups

	subgroup	G/H
center	C₂²	#19098420
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₄×#107×C₇₉	C₂
socle	C₂²×C₇₉	#380
prime radical	#14579135	D₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₄×#107	D₇₉
hypocenter	C₇₉	#8697
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#107

384 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 103
transitive degree	μ_t(G) = 5056

A minimal faithful permutation representation — 3 generators on 103 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)(80,83)(81,85)(82,87)(84,86)(88,91)(89,94)(90,98)(92,102)(93,96)(95,101)(97,100)(99,103);(80,82,85,84)(81,86,83,87)(88,90,93,100,94,101,92,99)(89,95,102,103,91,98,96,97);(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)(80,81,84,87,85,83,82,86)(88,89,93,96,94,91,92,102)(90,97,100,98,101,103,99,95)

Polycyclic presentation:

2^8,79|a^2=f,b^2=dgh,d^2=g,e^2=gh,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,i^a=i^78,e^c=egh,f^c=fg,i^c=i^78

Conjugacy classes & character table

2230 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 62888);