C₇₉×#12412 #34124101

order	20224
nilpotency class	4
isoclinism	isoclinic class of #12324
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₉×#12412

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₉	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇₉
prime radical	C₂×D₈×C₇₉	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₄²×C₇₉ ◁ #446×C₇₉ ◁ G

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

normal subgroups only

This group has 280 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)(3,11,4,13)(5,15,16,9)(7,14,8,12)(17,18,22,27)(19,28,31,47)(20,24,34,26)(21,35,38,42)(23,39,41,30)(25,37,44,33)(29,43,32,46)(36,45,48,40)(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);(1,3,6,4)(2,7,10,8)(5,14,16,12)(9,13,15,11)(17,19,22,31)(18,23,27,41)(20,29,34,32)(21,36,38,48)(24,40,26,45)(25,43,44,46)(28,42,47,35)(30,37,39,33);(1,4,6,3)(2,8,10,7)(5,12,16,14)(9,11,15,13)(17,20,22,34)(18,24,27,26)(19,29,31,32)(21,37,38,33)(23,40,41,45)(25,35,44,42)(28,46,47,43)(30,36,39,48)

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, 5400);