#34207696

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

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #13421

Subgroups

	subgroup	G/H
center	C₂²	#19098751
commutator, prime radical, cosocle	C₂²×D₄×C₇₉	C₂³
Frattini	C₂²×D₄	C₂²×D₇₉
Fitting	C₂×#352×C₇₉	C₂
socle	C₂²×C₇₉	B(4,𝔽₂)
hypercenter	C₂×#352	D₇₉
hypocenter	C₇₉	#13421
LCS	G ▷ C₂²×D₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×D₄ ◁ C₂×#352

496 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 75 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) = 99
transitive degree	μ_t(G) = 5056

A minimal faithful permutation representation — 3 generators on 99 points:

(1,3,4,2)(5,8)(6,11)(7,13)(9,16)(10,17)(12,15)(14,20)(18,19)(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)(95,96)(97,98);(1,2)(3,4)(5,7,10,6)(8,13,17,11)(9,12,18,14)(15,16,20,19)(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)(96,97)(98,99);(2,3)(5,6,10,7)(8,15,17,20)(9,12,18,14)(11,19,13,16)

Polycyclic presentation:

2^8,79|a^2=h,b^2=h,c^2=d,b^a=bd,c^a=ce,f^a=fgh,c^b=cf,e^b=eg,i^b=i^78,g^c=gh,i^c=i^78,f^e=fh

Conjugacy classes & character table

1045 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 88995);