C₇₉×#12397 #34124086

order	20224
nilpotency class	4
isoclinism	isoclinic class of #12316
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₉×#12397

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₉	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇₉
prime radical	C₂×D₄×C₇₉	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₄²×C₇₉ ◁ C₈×D₄×C₇₉ ◁ G

278 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 127
transitive degree	μ_t(G) = 10112

A minimal faithful permutation representation — 3 generators on 127 points:

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

Conjugacy classes & character table

3160 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 5385);