C₇₉×#13666 #34125355

order	20224
nilpotency class	5
isoclinism	isoclinic class of #13661
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇₉×#13666

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₉	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₇₉
prime radical	C₂×#508×C₇₉	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂²×C₄×C₇₉ ◁ C₂²×C₈×C₇₉ ◁ C₂²×C₁₆×C₇₉ ◁ G

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

normal subgroups only

This group has 288 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) = 147
transitive degree	μ_t(G) = 10112

A minimal faithful permutation representation — 3 generators on 147 points:

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

Conjugacy classes & character table

3871 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 6654);