C₇₉×#11538 #34123227

order	20224
nilpotency class	4
isoclinism	isoclinic class of #11522
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₉×#11538

Subgroups

	subgroup	G/H
center	C₄×C₇₉	#410
commutator	C₂²×C₄	C₂²×C₄×C₇₉
Frattini, cosocle	C₂×M₁₆	C₂³×C₇₉
socle	C₂×C₇₉	#1466
prime radical	#525×C₇₉	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇₉ ◁ C₂×C₈×C₇₉ ◁ #405×C₇₉ ◁ G

334 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 102 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:

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

Conjugacy classes & character table

2686 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 4526);