#34203549

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

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #12648

Subgroups

	subgroup	G/H
center	C₂²	#19098675
commutator	C₈×C₇₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#2141	C₂
socle	C₂²×C₇₉	#442
prime radical	C₂³×C₇₉	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#2141	D₇₉
hypocenter	C₇₉	#12648
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2141

422 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 115
transitive degree	μ_t(G) = 5056

A minimal faithful permutation representation — 3 generators on 115 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)(86,97,101,100)(87,89)(91,108,111,110)(92,94)(96,112,109,115)(98,114,99,113)(102,105)(104,106);(80,81,82,83)(84,86,89,99,90,100,106,113,88,101,87,98,103,97,104,114)(85,91,94,109,95,110,105,115,93,111,92,96,107,108,102,112);(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)(84,85,88,93)(86,96)(87,102,89,105)(90,107,103,95)(91,99)(92,106,94,104)(97,112)(98,111)(100,115)(101,109)(108,113)(110,114)

Polycyclic presentation:

2^8,79|a^2=e,b^2=f,c^2=eh,d^2=g,f^2=g,g^2=h,b^a=bd,d^a=dg,f^a=fg,g^a=gh,i^a=i^78,c^b=cg,e^b=eh,d^c=dh,f^c=fh,i^c=i^78

Conjugacy classes & character table

1294 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 84848);