C₇₉×#11497 #34123186

order	20224
nilpotency class	3
isoclinism	isoclinic class of #1510
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₇₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇₉×#11497

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇₉	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₇₉
Frattini, cosocle	C₂²×C₈	C₂³×C₇₉
socle	C₂²×C₇₉	C₂×C₂²⋊C₈
prime radical	C₂³×C₇₉	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₉ ◁ C₂×C₄×C₈×C₇₉ ◁ G

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

normal subgroups only

This group has 288 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) = 127
transitive degree	μ_t(G) = 10112

A minimal faithful permutation representation — 3 generators on 127 points:

(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,20,27,22,28,36,42,23,29,37,25,21,26,35,43)(19,30,33,46,32,41,48,45,34,44,31,47,38,24,39,40)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19)(18,24)(20,31)(21,38)(22,32)(23,34)(25,40)(26,44)(27,45)(28,30)(29,41)(33,37)(35,48)(36,39)(42,47)(43,46);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20,22,36,23,37,21,35)(18,25,28,43,29,27,26,42)(19,31,32,39,34,33,38,48)(24,40,30,46,41,45,44,47)

Conjugacy classes & character table

4108 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 4485);