C₇₉×#13684 #34125373

order	20224
nilpotency class	5
isoclinism	isoclinic class of #13681
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇₉×#13684

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₉	#2196
commutator, Frattini, cosocle	C₄⋊Q₈	C₂³×C₇₉
prime radical	C₇₉×#3293	C₂
LCS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₉ ◁ C₂²×C₇₉ ◁ C₂×C₄×C₇₉ ◁ C₄⋊Q₈×C₇₉ ◁ G

378 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 58 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:

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

Conjugacy classes & character table

1975 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 6672);