C₂×C₇₉×#1413 #34125286

order	20224
nilpotency class	5
isoclinism	isoclinic class of #1412
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×C₇₉×#1413

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₉	#354
commutator	C₄²	C₂²×C₄×C₇₉
Frattini, cosocle	C₄⋊Q₈	C₂³×C₇₉
prime radical	C₂³×C₇₉	#99
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂³×C₇₉ ◁ C₂²×C₄×C₇₉ ◁ C₂×C₄⋊Q₈×C₇₉ ◁ G

314 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 113
transitive degree	μ_t(G) = 5056

A minimal faithful permutation representation — 3 generators on 113 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79)(80,81);(82,83,85,90,87,92,99,109)(84,93,95,111,94,110,112,107)(86,91,101,108,102,89,98,106)(88,103,104,96,105,97,113,100);(82,84,87,94)(83,88)(85,97,99,103)(86,100,102,96)(89,93)(90,107)(91,110)(92,105)(95,98,112,101)(104,106)(108,113)(109,111)

Conjugacy classes & character table

2212 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 6585);