C₇₉×#7524 #34119213

order	20224
nilpotency class	6
isoclinism	isoclinic class of #7523
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₇₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇₉×#7524

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₉	#1404
commutator	#125	C₂×C₄×C₇₉
Frattini, cosocle	#490	C₂²×C₇₉
prime radical	#127×C₇₉	D₄
LCS	G ▷ #125 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₉ ◁ C₂²×C₇₉ ◁ C₂×C₄×C₇₉ ◁ C₄²×C₇₉ ◁ #490×C₇₉ ◁ G

134 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 30 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) = 111
transitive degree	μ_t(G) = 2528

A minimal faithful permutation representation — 2 generators on 111 points:

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

Conjugacy classes & character table

1501 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 512);