C₇₉×#13655 #34125344

order	20224
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₉×#13655

Subgroups

	subgroup	G/H
center	C₈×C₇₉	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 ◁ C₈×C₇₉ ◁ C₂×C₈×C₇₉ ◁ C₄×C₈×C₇₉ ◁ C₄×C₁₆×C₇₉ ◁ G

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

normal subgroups only

This group has 186 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:

(1,2,6,12,8,14,7,30)(3,15,18,45,20,47,31,46)(4,10,22,38,24,39,23,13)(5,25,27,37,29,56,28,59)(9,32,34,51,36,57,35,17)(11,40,42,54,43,53,55,21)(16,41,49,63,50,64,19,52)(26,44,48,33,58,61,60,62)(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)(5,26)(6,18)(7,31)(8,20)(10,33)(11,41)(12,34)(13,44)(14,36)(15,25)(17,21)(19,23)(22,49)(24,50)(27,48)(28,60)(29,58)(30,35)(32,40)(37,45)(38,61)(39,62)(42,63)(43,64)(46,59)(47,56)(51,54)(52,55)(53,57);(1,4,6,22,8,24,7,23)(2,10,12,38,14,39,30,13)(3,16,18,49,20,50,31,19)(5,21,27,40,29,54,28,53)(9,33,34,61,36,62,35,44)(11,37,42,56,43,59,55,25)(15,41,45,63,47,64,46,52)(17,48,32,58,51,60,57,26)

Conjugacy classes & character table

6004 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 6643);