#34181129

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

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×#1275 ⋊ C₂
Sylow	C₇₉ ⋊ #8601

Subgroups

	subgroup	G/H
center	C₂×C₄	#14579118
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1275	C₂
socle	C₂²×C₇₉	#380
prime radical	C₂³×D₇₉	C₄⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1275	D₇₉
hypocenter	C₇₉	#8601
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #1275

462 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)(89,95)(90,99)(92,103)(93,104)(94,107)(96,109)(97,102)(98,110)(100,113)(101,114)(105,117)(106,118)(108,116)(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)(88,90,92,100,93,101,106,115)(89,94,96,98,97,108,105,111)(91,99,103,113,104,114,118,119)(95,107,109,110,102,116,117,112);(80,81,82,84,83,85,86,87)(88,89)(90,98)(91,102,104,95)(92,105)(93,97)(94,100)(96,106)(99,112,114,110)(101,111)(103,109,118,117)(107,113,116,119)(108,115)

Polycyclic presentation:

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

Conjugacy classes & character table

1780 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 62428);