#34182147

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

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #8838

Subgroups

	subgroup	G/H
center	C₂×C₄	#14579131
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1278	C₂
socle	C₂²×C₇₉	#385
prime radical	C₂³×C₇₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1278	D₇₉
hypocenter	C₇₉	#8838
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1278

386 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 103
transitive degree	μ_t(G) = 10112

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)(81,85)(82,86)(88,90,92,97)(89,94,95,103)(91,100,98,96)(93,101,99,102);(80,82,84,85,83,86,87,81)(88,89)(90,94)(91,99)(92,95)(93,98)(96,101)(97,103)(100,102);(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)(80,81,84,82,83,85,87,86)(89,93)(91,98)(94,102)(95,99)(96,100)(101,103)

Polycyclic presentation:

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

Conjugacy classes & character table

2236 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 63446);