C₇₉×#7427 #34119116

order	20224
nilpotency class	5
isoclinism	isoclinic class of #7426
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₇₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇₉×#7427

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₉	#1351
commutator	C₂×C₂²⋊C₄	C₂×C₄×C₇₉
Frattini, cosocle	#381	C₂²×C₇₉
prime radical	C₂⁴×C₇₉	C₂²⋊C₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₉ ◁ C₂³×C₇₉ ◁ C₂⁴×C₇₉ ◁ #381×C₇₉ ◁ G

214 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 42 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 — 2 generators on 111 points:

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

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, 415);