#34194801

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₂×#344×C₇₉ ⋊ C₂
Sylow	C₇₉ ⋊ #10974

Subgroups

	subgroup	G/H
center	C₂²	#19098441
commutator	C₂⁴×C₇₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₇₉
Fitting	C₂×#344×C₇₉	C₂
socle	C₂²×C₇₉	#386
prime radical	C₂²×#11929624	C₂²
cosocle	C₂³×C₄×C₇₉	C₂³
hypercenter	C₂×#344	D₇₉
hypocenter	C₇₉	#10974
LCS	G ▷ C₂⁴×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#344

1258 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 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)(81,83)(84,87)(85,91)(86,95)(88,99)(89,98)(90,102)(92,105)(93,106)(94,104)(96,110)(97,109)(100,112)(101,111)(103,113)(107,115)(108,114);(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)(84,86,89,97)(85,90,92,94)(87,95,98,109)(88,100,101,96)(91,104,105,102)(93,108,107,103)(99,112,111,110)(106,113,115,114);(80,81)(82,83)(84,85,88,93,89,92,101,107)(86,94,96,108,97,90,100,103)(87,91,99,106,98,105,111,115)(95,104,110,114,109,102,112,113)

Polycyclic presentation:

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

Conjugacy classes & character table

1762 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 76100);