C₇₉×#8305 #34119994

order	20224
nilpotency class	3
isoclinism	isoclinic class of #8304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₇₉×#8305

Subgroups

	subgroup	G/H
center, socle	C₂³×C₇₉	C₂×C₂²⋊C₄
commutator	C₂⁴	C₂²×C₄×C₇₉
Frattini, cosocle	C₂³×C₄	C₂³×C₇₉
prime radical	C₂³×D₄×C₇₉	C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇₉ ◁ C₂²×C₄²×C₇₉ ◁ G

1154 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 1154 conjugacy classes of subgroups — too many to draw. Only its 182 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) = 107
transitive degree	μ_t(G) = 5056

A minimal faithful permutation representation — 3 generators on 107 points:

(1,2)(3,4)(5,6,7,10,8,9,11,12)(13,14)(15,21,23,18)(16,19)(17,20)(22,27,28,25)(24,26)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107);(6,9)(7,11)(13,15)(14,18)(16,22)(17,23)(19,25)(20,21)(24,28)(26,27);(2,4)(5,7,8,11)(6,10,9,12)(13,16,17,24)(14,19,20,26)(15,22,23,28)(18,25,21,27)

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, 1293);