C₇₉×#9411 #34121100

order	20224
nilpotency class	3
isoclinism	isoclinic class of #9385
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇₉×#9411

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₉	#386
commutator	C₂²×C₄	C₂²×C₄×C₇₉
Frattini, cosocle	C₂³×C₄	C₂³×C₇₉
prime radical	C₂⁵×C₇₉	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂³×C₄×C₇₉ ◁ G

606 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 122 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 — 3 generators on 111 points:

(1,2)(3,6,10,15)(4,11,13,7)(5,8)(9,14)(12,16)(17,18,21,24,22,25,20,30)(19,26,29,31,27,32,28,23)(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)(2,6,8,15)(4,12)(5,10)(7,16,11,14)(9,13)(17,19)(18,23,25,31)(20,28)(21,29)(22,27)(24,26,30,32);(1,4)(2,7)(3,9)(5,13)(6,14)(8,11)(10,12)(15,16)(19,27)(23,31)(26,32)(28,29)

Conjugacy classes & character table

2923 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 2399);