C₇₉×#13592 #34125281

order	20224
nilpotency class	5
isoclinism	isoclinic class of #13584
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₉ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇₉×#13592

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₉	C₂×#353
commutator	C₄²	C₂²×C₄×C₇₉
Frattini, cosocle	C₄⋊Q₈	C₂³×C₇₉
prime radical	#532×C₇₉	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₉ ◁ C₂×C₄×C₇₉ ◁ C₂²×C₄×C₇₉ ◁ #532×C₇₉ ◁ G

398 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 2528

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

(1,2,5,10,7,12,20,28)(3,13,15,26,17,23,30,31)(4,9,18,25,6,11,21,27)(8,19,22,14,24,29,32,16)(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);(1,3,7,17)(2,8,12,24)(4,16,6,14)(5,19,20,29)(9,23,11,13)(10,26,28,31)(15,18,30,21)(22,27,32,25);(1,4,7,6)(2,9,12,11)(3,14,17,16)(5,18,20,21)(8,13,24,23)(10,25,28,27)(15,29,30,19)(22,26,32,31)

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, 6580);