C₇₉×#12020 #34123709

order	20224
nilpotency class	4
isoclinism	isoclinic class of #12019
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₉ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₇₉×#12020

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₉	C₂×#106
commutator	C₂×C₈	C₂²×C₄×C₇₉
Frattini, cosocle	C₄×C₈	C₂³×C₇₉
prime radical	C₂³×C₇₉	#106
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂×C₄²×C₇₉ ◁ C₂×C₄×C₈×C₇₉ ◁ G

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

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 10112

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

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

Conjugacy classes & character table

4108 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 5008);