C₇₉×#13724 #34125413

order	20224
nilpotency class	5
isoclinism	isoclinic class of #13697
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₉ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₇₉×#13724

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₇₉	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₇₉
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₄²×C₇₉ ◁ C₄×C₈×C₇₉ ◁ C₄×C₁₆×C₇₉ ◁ G

174 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 151
transitive degree	μ_t(G) = 20224

A minimal faithful permutation representation — 3 generators on 151 points:

(1,2)(3,5)(4,6)(7,8)(9,10,17,24)(11,25,31,19)(12,32,27,18)(13,21,38,45)(14,20,26,53)(15,41,43,72)(16,44,42,23)(22,56,57,36)(28,58,34,49)(29,64,66,71)(30,51,65,59)(33,50,61,46)(35,48,62,68)(37,70,39,55)(40,54,60,52)(47,69,67,63)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151);(1,3,4,7)(2,5,6,8)(9,11,26,34,15,29,37,62,42,65,40,63,56,33,13,27,17,31,14,28,43,66,39,35,16,30,60,69,36,61,38,12)(10,18,45,50,22,47,52,59,44,68,55,64,72,49,20,25,24,32,21,46,57,67,54,51,23,48,70,71,41,58,53,19);(2,6)(5,8)(9,12,13,33,36,69,40,65,16,35,37,29,43,28,26,11,17,27,38,61,56,63,60,30,42,62,39,66,15,34,14,31)(10,19,20,49,41,71,55,68,23,51,52,47,57,46,45,18,24,25,53,58,72,64,70,48,44,59,54,67,22,50,21,32)

Conjugacy classes & character table

3634 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 6712);