#34175449

order	20224
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #7271

Subgroups

	subgroup	G/H
center	C₂²	#19098311
commutator	C₄²×C₇₉	C₄²
Frattini	C₂×C₄×C₈	C₂×D₇₉
Fitting	C₂×#364×C₇₉	C₂
socle	C₂²×C₇₉	#333
prime radical	C₂³×C₇₉	#100
cosocle	C₂×C₄×C₈×C₇₉	C₂²
hypercenter	C₂×#364	D₇₉
hypocenter	C₇₉	#7271
LCS	G ▷ C₄²×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

222 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 115
transitive degree	μ_t(G) = 5056

A minimal faithful permutation representation — 2 generators on 115 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)(84,86,87,97,100,98,113,112,88,96,101,111,89,95,102,110)(85,90,92,105,103,94,108,109,91,104,106,115,93,107,99,114);(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)(80,81,82,83)(84,85)(86,94,96,107)(87,99)(88,91)(89,103)(90,95,104,98)(92,102)(93,100)(97,105,111,115)(101,108)(106,113)(109,112,114,110)

Polycyclic presentation:

2^8,79|a^2=d,b^2=e,c^2=fh,e^2=fg,g^2=h,b^a=bc,c^a=cf,e^a=efgh,g^a=gh,i^a=i^78,c^b=cg,d^b=dh,f^b=fh,g^b=gh,i^b=i^78

Conjugacy classes & character table

2230 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 56748);