#34205157

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×#1376 ⋊ C₂
Sylow	C₇₉ ⋊ #12982

Subgroups

	subgroup	G/H
center	C₈	#14579131
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1376	C₂
socle	C₂×C₇₉	#1869
prime radical	#19099113	C₄
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#1376	D₇₉
hypocenter	C₇₉	#12982
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1376

474 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 2528

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

Polycyclic presentation:

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

Conjugacy classes & character table

2248 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 86456);