C₇₉×#12031 #34123720

order	20224
nilpotency class	4
isoclinism	isoclinic class of #12028
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₉ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₇₉×#12031

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₉	C₂×#106
commutator	C₂×C₈	C₂²×C₄×C₇₉
Frattini, cosocle	C₄×C₈	C₂³×C₇₉
prime radical	C₂³×C₇₉	#106
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂×C₄²×C₇₉ ◁ C₂×C₄×C₈×C₇₉ ◁ G

200 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 114 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)(19,28,31,43)(20,24)(21,35)(22,27)(23,30,38,46)(25,33)(26,34)(29,42,32,48)(36,40)(37,45,39,47)(41,44)(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)(17,19,21,30,20,32,33,47,22,31,36,46,34,29,44,45)(18,23,25,28,26,37,35,42,27,38,41,43,24,39,40,48);(3,9)(6,12)(10,15)(13,16)(17,20,22,34)(18,24,27,26)(19,29,31,32)(21,33,36,44)(23,37,38,39)(25,40,41,35)(28,42,43,48)(30,45,46,47)

Conjugacy classes & character table

3634 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 5019);