C₇₉×#7424 #34119113

order	20224
nilpotency class	5
isoclinism	isoclinic class of #7422
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₇₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇₉×#7424

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₉	#1347
commutator	C₂×C₁₆	C₂×C₄×C₇₉
Frattini, cosocle	C₂×M₃₂	C₂²×C₇₉
prime radical	C₇₉×#2218	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₉ ◁ C₂×C₄×C₇₉ ◁ C₂²×C₈×C₇₉ ◁ C₂×M₃₂×C₇₉ ◁ G

146 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 46 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:

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

Conjugacy classes & character table

2449 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 412);