C₇₉×#13648 #34125337

order	20224
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆×C₇₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₉×#13648

Subgroups

	subgroup	G/H
center	C₁₆×C₇₉	C₂×D₄
commutator	C₄	C₂²×C₁₆×C₇₉
Frattini, cosocle	C₂×C₁₆	C₂³×C₇₉
socle	C₂×C₇₉	D₄×C₁₆
prime radical	#135×C₇₉	C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆×C₇₉ ◁ C₂×C₃₂×C₇₉ ◁ G

184 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 118 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:

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

Conjugacy classes & character table

6952 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 6636);