C₇₉×#13618 #34125307

order	20224
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇₉×#13618

Subgroups

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

290 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 126 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)(7,18)(8,14)(9,26)(10,32)(11,35)(12,17)(13,20)(15,29)(16,28)(19,46)(21,53)(22,55)(23,47)(24,56)(25,44)(27,43)(30,63)(31,45)(33,41)(34,64)(36,48)(37,57)(38,52)(39,51)(40,58)(42,50)(49,68)(54,67)(59,62)(60,65)(61,66)(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);(5,7,12,23)(6,13,17,38)(8,22,11,19)(9,27,31,61)(10,33,34,49)(14,39,35,36)(15,41,42,68)(16,43,44,66)(18,45,47,26)(20,50,52,29)(21,25,54,28)(24,57,30,60)(32,37,64,65)(40,67,59,53)(46,56,55,63)(48,58,51,62);(5,8,12,11)(6,14,17,35)(7,19,23,22)(9,24,31,30)(10,25,34,28)(13,36,38,39)(15,40,42,59)(16,32,44,64)(18,46,47,55)(20,48,52,51)(21,49,54,33)(26,56,45,63)(27,60,61,57)(29,58,50,62)(37,43,65,66)(41,53,68,67)

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, 6606);