C₇₉×#12034 #34123723

order	20224
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₉ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₇₉×#12034

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇₉	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₇₉
Frattini, cosocle	C₄×C₈	C₂³×C₇₉
socle	C₂²×C₇₉	C₂×#107
prime radical	C₂³×C₇₉	#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₉ ◁ C₂×C₄²×C₇₉ ◁ C₂×C₄×C₈×C₇₉ ◁ G

204 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 118 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:

(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,28)(20,24)(21,34)(22,27)(23,31)(25,29)(26,33)(30,41)(32,43)(35,39)(36,44)(37,48)(38,47)(40,45)(42,46)(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)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,29,44,20,30,35,47,22,32,45,48,33,46,21,31)(18,23,34,42,26,37,40,43,27,38,39,41,24,36,25,28);(3,9)(6,12)(10,15)(13,16)(17,20,22,33)(18,24,27,26)(19,30,32,46)(21,29,35,45)(23,36,38,37)(25,39,40,34)(28,41,43,42)(31,44,47,48)

Conjugacy classes & character table

4582 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 5022);