C₇₉×#12701 #34124390

order	20224
nilpotency class	4
isoclinism	isoclinic class of #11505
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₇₉×#12701

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₉	C₂×#99
commutator	C₂²×C₄	C₂²×C₄×C₇₉
Frattini, cosocle	C₂²×Q₈	C₂³×C₇₉
prime radical	C₂×#532×C₇₉	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂²×C₄×C₇₉ ◁ C₂²×#41×C₇₉ ◁ G

1070 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1070 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 99
transitive degree	μ_t(G) = 2528

A minimal faithful permutation representation — 3 generators on 99 points:

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

Conjugacy classes & character table

3160 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 5689);