#34183150

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×#1301 ⋊ C₂
Sylow	C₇₉ ⋊ #8994

Subgroups

	subgroup	G/H
center	C₂×C₄	#14579131
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1301	C₂
socle	C₂²×C₇₉	#386
prime radical	C₂×#11929624	D₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1301	D₇₉
hypocenter	C₇₉	#8994
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1301

430 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 10112

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

Polycyclic presentation:

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

Conjugacy classes & character table

1534 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 64449);