C₇₉×#7520 #34119209

order	20224
nilpotency class	6
isoclinism	isoclinic class of #7519
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₇₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇₉×#7520

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₉	#1402
commutator	C₄⋊Q₈	C₂×C₄×C₇₉
Frattini, cosocle	#492	C₂²×C₇₉
prime radical	#127×C₇₉	D₄
LCS	G ▷ C₄⋊Q₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₉ ◁ C₂²×C₇₉ ◁ C₂×C₄×C₇₉ ◁ C₄²×C₇₉ ◁ #492×C₇₉ ◁ G

156 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 30 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) = 95
transitive degree	μ_t(G) = 1264

A minimal faithful permutation representation — 2 generators on 95 points:

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

Conjugacy classes & character table

1738 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 508);