C₇₉×#12804 #34124493

order	20224
nilpotency class	4
isoclinism	isoclinic class of #11508
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇₉×#12804

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₉	C₂×#99
commutator	C₂²×C₄	C₂²×C₄×C₇₉
Frattini, cosocle	C₂²×Q₈	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

432 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 115
transitive degree	μ_t(G) = 5056

A minimal faithful permutation representation — 3 generators on 115 points:

(2,4)(5,6,8,14,10,15,21,31)(7,11,18,26,19,27,16,32)(9,13,22,12,23,29,20,30)(17,25,33,28,35,36,34,24)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115);(1,2,3,4)(5,7)(6,11)(8,18)(9,17)(10,19)(12,28)(13,25)(14,26)(15,27)(16,21)(20,34)(22,33)(23,35)(24,30)(29,36)(31,32);(5,8,10,21)(6,12,15,30)(7,16,19,18)(9,22,23,20)(11,24,27,28)(13,31,29,14)(17,34,35,33)(25,26,36,32)

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, 5792);