C₇₉×#7568 #34119257

order	20224
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₇₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇₉×#7568

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₇₉	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₇₉
Frattini, cosocle	C₂×C₄²	C₂³×C₇₉
socle, prime radical	C₂³×C₇₉	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇₉ ◁ C₄³×C₇₉ ◁ G

444 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 170 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) = 20224

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

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

Conjugacy classes & character table

6004 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 556);