C₇₉×#13572 #34125261

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

Aliases

direct product

C₇₉×#13572

Subgroups

	subgroup	G/H
center	C₄×C₇₉	AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₂²×C₄×C₇₉
Frattini, cosocle	C₄⋊Q₈	C₂³×C₇₉
socle	C₂×C₇₉	C₂×AΣL(1,𝔽₁₆)
prime radical	C₇₉×#3018	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇₉ ◁ C₂×C₄×C₇₉ ◁ C₂²×C₄×C₇₉ ◁ #532×C₇₉ ◁ G

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

normal subgroups only

This group has 482 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:

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

Conjugacy classes & character table

2686 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 6560);