C₈₃×#7381 #35235205

order	21248
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆×C₈₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₈₃×#7381

Subgroups

	subgroup	G/H
center	C₁₆×C₈₃	D₈
commutator	C₈	C₂×C₁₆×C₈₃
Frattini, cosocle	C₄×C₁₆	C₂²×C₈₃
socle	C₂×C₈₃	D₄⋊C₁₆
prime radical	#135×C₈₃	C₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆×C₈₃ ◁ C₂×C₁₆×C₈₃ ◁ C₄×C₁₆×C₈₃ ◁ G

120 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 147
transitive degree	μ_t(G) = 5312

A minimal faithful permutation representation — 2 generators on 147 points:

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

Conjugacy classes & character table

7304 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21248, 369);