C₂×C₈₃×SD₁₂₈ #35241563

order	21248
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂×C₈₃×SD₁₂₈

Subgroups

	subgroup	G/H
center, socle	C₂²×C₈₃	D₃₂
commutator, Frattini, cosocle	C₃₂	C₂³×C₈₃
prime radical	C₂×D₃₂×C₈₃	C₂
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₃ ◁ C₂×C₄×C₈₃ ◁ C₂×C₈×C₈₃ ◁ C₂×C₁₆×C₈₃ ◁ C₂×C₃₂×C₈₃ ◁ G

134 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 149
transitive degree	μ_t(G) = 10624

A minimal faithful permutation representation — 3 generators on 149 points:

(1,2)(67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149);(3,4,9,15)(5,10,20,32)(6,21,24,55)(7,25,27,58)(8,28,26,14)(11,33,36,16)(12,37,39,17)(13,40,42,22)(18,41,51,66)(19,43,50,31)(23,56,47,54)(29,60,62,45)(30,63,64,44)(34,65,61,49)(35,46,59,52)(38,48,53,57);(4,10)(5,16)(6,17)(7,22)(8,26)(11,12)(13,41)(14,43)(15,32)(18,49)(19,52)(20,33)(21,29)(23,57)(24,37)(25,30)(27,40)(28,31)(34,54)(35,53)(36,39)(38,59)(42,66)(44,45)(46,50)(47,48)(51,65)(55,62)(56,61)(58,64)(60,63)

Conjugacy classes & character table

5810 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21248, 6727);