C₇₉×#18731 #34130420

order	20224
nilpotency class	3
isoclinism	isoclinic class of #18712
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₇₉×#18731

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₉	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇₉
prime radical	C₂×#127×C₇₉	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂×C₄²×C₇₉ ◁ G

884 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 884 conjugacy classes of subgroups — too many to draw. Only its 230 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) = 5056

A minimal faithful permutation representation — 4 generators on 111 points:

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

Conjugacy classes & character table

3160 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 11719);