C₇₃×#11462 #32920866

order	18688
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₇₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇₃×#11462

Subgroups

	subgroup	G/H
center	C₄²×C₇₃	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₇₃
Frattini, cosocle	C₂²×C₈	C₂³×C₇₃
socle	C₂²×C₇₃	C₂×C₂²⋊C₈
prime radical	C₂×#41×C₇₃	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄²×C₇₃ ◁ C₂×C₄×C₈×C₇₃ ◁ G

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

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 121
transitive degree	μ_t(G) = 9344

A minimal faithful permutation representation — 3 generators on 121 points:

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

Conjugacy classes & character table

5548 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 4450);