C₇₉×#33564 #34145253

order	20224
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇₉ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₇₉×#33564

Subgroups

	subgroup	G/H
center	C₄×C₇₉	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×Q₈	C₂⁴×C₇₉
socle	C₂×C₇₉	C₂×B(4,𝔽₂)
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇₉ ◁ C₂×C₄×C₇₉ ◁ C₂×#41×C₇₉ ◁ G

1112 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 218 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 — 4 generators on 111 points:

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

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, 26552);