C₇₉×#11574 #34123263

order	20224
nilpotency class	4
isoclinism	isoclinic class of #11549
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₉ · Schur multiplier (H₂) = C₄²

Aliases

direct product

C₇₉×#11574

Subgroups

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

374 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 106 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 — 3 generators on 111 points:

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

Conjugacy classes & character table

2923 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 4562);