#34178803

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #8157

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₇₉
commutator	C₂³×C₇₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1458	C₂
socle	C₂²×C₇₉	C₄×C₂²⋊C₄
prime radical	C₂⁴×C₇₉	C₄²
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1458	D₇₉
hypocenter	C₇₉	#8157
LCS	G ▷ C₂³×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1458

482 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)(81,84)(85,87)(88,91,92,102,93,103,104,115)(89,95,96,108,97,109,110,118)(90,99,101,114,100,113,105,116)(94,106,107,117,98,111,112,119);(80,81,82,85,83,84,86,87)(88,90)(89,94,97,98)(91,99)(92,101)(93,100)(95,106,109,111)(96,107,110,112)(102,114)(103,113)(104,105)(108,117,118,119)(115,116);(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)(88,89,92,96,93,97,104,110)(90,98,101,112,100,94,105,107)(91,95,102,108,103,109,115,118)(99,111,114,119,113,106,116,117)

Polycyclic presentation:

2^8,79|a^2=e,b^2=f,c^2=e,e^2=g,f^2=h,b^a=bd,f^a=fg,i^a=i^78,c^b=ch,d^b=dg,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, 60102);