C₇₉×#7439 #34119128

order	20224
nilpotency class	5
isoclinism	isoclinic class of #7396
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₇₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇₉×#7439

Subgroups

	subgroup	G/H
center	C₄×C₇₉	#353
commutator	C₂×M₁₆	C₂×C₄×C₇₉
Frattini, cosocle	#408	C₂²×C₇₉
socle	C₂×C₇₉	#1352
prime radical	C₂×#41×C₇₉	D₄
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇₉ ◁ C₂×C₄×C₇₉ ◁ C₂²×C₄×C₇₉ ◁ #408×C₇₉ ◁ G

170 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 42 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 — 2 generators on 143 points:

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

Conjugacy classes & character table

2449 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 427);