C₇₉×#9413 #34121102

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

Aliases

direct product

C₇₉×#9413

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₉	#386
commutator	C₂²×C₄	C₂²×C₄×C₇₉
Frattini, cosocle	C₂³×C₄	C₂³×C₇₉
prime radical	C₇₉×#2539	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂³×C₄×C₇₉ ◁ G

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

normal subgroups only

This group has 720 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) = 103
transitive degree	μ_t(G) = 2528

A minimal faithful permutation representation — 3 generators on 103 points:

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

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, 2401);