C₇₉×#13656 #34125345

order	20224
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₉×#13656

Subgroups

	subgroup	G/H
center	C₄×C₇₉	C₂×D₁₆
commutator	C₁₆	C₂²×C₄×C₇₉
Frattini, cosocle	C₂×C₁₆	C₂³×C₇₉
socle	C₂×C₇₉	C₄×D₁₆
prime radical	C₇₉×#2262	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇₉ ◁ C₂×C₈×C₇₉ ◁ C₄×C₈×C₇₉ ◁ C₄×C₁₆×C₇₉ ◁ G

184 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 143
transitive degree	μ_t(G) = 5056

A minimal faithful permutation representation — 3 generators on 143 points:

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

Conjugacy classes & character table

4108 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 6644);