C₇₉×#7438 #34119127

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×#7438

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

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

normal subgroups only

This group has 174 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)(3,13,15,42)(4,19)(5,24,27,10)(6,11)(7,12)(8,14,30,47)(9,21,36,26)(16,43,17,34)(18,44,51,61)(20,38,48,32)(22,41)(23,39)(25,58,46,62)(28,57,54,40)(29,35)(31,52,60,53)(33,49,59,50)(37,55,63,45)(56,64)(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);(1,3,5,25)(2,8,24,31)(4,20,21,47)(6,17,28,15)(7,18,27,16)(9,34,39,61)(10,38,12,33)(11,32,57,30)(13,41,43,63)(14,45,52,56)(19,42,36,62)(22,49,55,48)(23,53,26,50)(29,46,54,51)(35,60,40,59)(37,44,64,58)

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, 426);