#34205221

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

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×#1380 ⋊ C₂
Sylow	C₇₉ ⋊ #12994

Subgroups

	subgroup	G/H
center	C₂³	D₈×D₇₉
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1380	C₂
socle	C₂³×C₇₉	C₂×D₈
prime radical	C₂³×D₇₉	D₈
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#1380	D₇₉
hypocenter	C₇₉	#12994
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

530 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 119
transitive degree	μ_t(G) = 10112

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

Polycyclic presentation:

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

Conjugacy classes & character table

1774 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 86520);