C₇₉×#8699 #34120388

order	20224
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₇₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇₉×#8699

Subgroups

	subgroup	G/H
center, socle	C₂³×C₇₉	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₈×C₇₉
Frattini, cosocle	C₂³×C₄	C₂³×C₇₉
prime radical	C₂⁴×C₇₉	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇₉ ◁ C₂²×C₄²×C₇₉ ◁ G

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

normal subgroups only

This group has 598 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) = 111
transitive degree	μ_t(G) = 10112

A minimal faithful permutation representation — 3 generators on 111 points:

(1,2,3,5,4,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,27,28,23)(21,24)(25,31,32,29)(26,30)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111);(10,13)(14,16)(17,19,21,26)(18,22,24,30)(20,25,28,32)(23,29,27,31);(9,11,12,15)(10,14,13,16)(17,20)(18,23)(19,25)(21,28)(22,29)(24,27)(26,32)(30,31)

Conjugacy classes & character table

5056 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 1687);