C₇₉×#11728 #34123417

order	20224
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇₉×#11728

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇₉	#99
commutator	C₂²×C₄	C₂²×C₄×C₇₉
Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₇₉
socle	C₂²×C₇₉	C₂×#99
prime radical	C₂²×D₄×C₇₉	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₉ ◁ C₂²×C₄×C₇₉ ◁ C₂×C₄×D₄×C₇₉ ◁ G

470 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 5056

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

(2,6,7,8)(9,10)(11,19)(12,16)(13,22,23,17)(14,18)(15,20,24,21)(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);(1,2)(3,6)(4,8)(5,7)(9,11,14,12)(10,15,18,24)(13,20,23,21)(16,17,19,22);(1,3,5,4)(2,6,7,8)(9,12,14,11)(10,16,18,19)(13,21,23,20)(15,17,24,22)

Conjugacy classes & character table

3634 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 4716);