#34201646

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×#1339 ⋊ C₂
Sylow	C₇₉ ⋊ #12307

Subgroups

	subgroup	G/H
center	C₂	#23762438
commutator, cosocle	C₂×M₁₆×C₇₉	C₂³
Frattini	C₂×M₁₆	C₂²×D₇₉
Fitting	C₇₉×#1339	C₂
socle	C₂×C₇₉	#1616
prime radical	#23774777	C₂
hypercenter	#1339	D₇₉
hypocenter	C₇₉	#12307
LCS	G ▷ C₂×M₁₆×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #1339

318 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 95
transitive degree	μ_t(G) = 1264

A minimal faithful permutation representation — 3 generators on 95 points:

(2,8,11,10)(7,16,12,9)(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)(81,82)(83,84)(85,86)(87,88)(89,90)(91,92)(93,94);(2,7)(3,13)(4,5)(8,16)(9,10)(11,12)(14,15);(1,2,5,10,6,11,4,8)(3,12,14,9,15,7,13,16)(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)(82,83)(84,85)(86,87)(88,89)(90,91)(92,93)(94,95)

Polycyclic presentation:

2^8,79|a^2=fg,c^2=g,d^2=f,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fh,g^a=gh,i^a=i^78,d^b=dfh,e^b=egh,f^b=fh,d^c=dg,i^c=i^78,e^d=eh

Conjugacy classes & character table

805 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 82945);