#34205196

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

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×#1378 ⋊ C₂
Sylow	C₇₉ ⋊ #12989

Subgroups

	subgroup	G/H
center	C₄	#19098441
commutator	C₂³×C₇₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1378	C₂
socle	C₂×C₇₉	#1869
prime radical	#19099075	C₄
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#1378	D₇₉
hypocenter	C₇₉	#12989
LCS	G ▷ C₂³×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #1378

470 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)(80,82)(81,86)(83,95)(84,93)(85,91)(87,90)(88,102)(89,100)(92,97)(94,99)(96,108)(98,105)(101,104)(103,111)(106,109)(107,110);(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)(80,81,83,94,84,89,96,107,85,87,97,88,98,101,109,103)(82,86,95,99,93,100,108,110,91,90,92,102,105,104,106,111);(80,81,83,88,84,89,96,103,85,87,97,94,98,101,109,107)(82,90,92,102,93,104,106,111,91,86,95,99,105,100,108,110)

Polycyclic presentation:

2^8,79|a^2=f,b^2=df,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,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

1774 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 86495);