Copied to
clipboard

## G = C2×C92⋊2C3order 486 = 2·35

### Direct product of C2 and C92⋊2C3

direct product, metabelian, nilpotent (class 4), monomial, 3-elementary

Aliases: C2×C922C3, C9215C6, (C9×C18)⋊2C3, (C3×C6).2He3, He3⋊C35C6, C32.2(C2×He3), (C3×C18).17C32, C6.7(He3⋊C3), (C3×C9).18(C3×C6), (C2×He3⋊C3)⋊1C3, C3.7(C2×He3⋊C3), SmallGroup(486,86)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — C2×C92⋊2C3
 Chief series C1 — C3 — C32 — C3×C9 — C92 — C92⋊2C3 — C2×C92⋊2C3
 Lower central C1 — C3 — C32 — C3×C9 — C2×C92⋊2C3
 Upper central C1 — C6 — C3×C6 — C3×C18 — C2×C92⋊2C3

Generators and relations for C2×C922C3
G = < a,b,c,d | a2=b9=c9=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b7c-1, dcd-1=b3c >

Smallest permutation representation of C2×C922C3
On 54 points
Generators in S54
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 16)(14 17)(15 18)(19 54)(20 46)(21 47)(22 48)(23 49)(24 50)(25 51)(26 52)(27 53)(28 38)(29 39)(30 40)(31 41)(32 42)(33 43)(34 44)(35 45)(36 37)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21 22 23 24 25 26 27)(28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45)(46 47 48 49 50 51 52 53 54)
(1 18 4 3 17 6 2 16 5)(7 15 10 9 14 12 8 13 11)(19 20 21 22 23 24 25 26 27)(28 30 32 34 36 29 31 33 35)(37 39 41 43 45 38 40 42 44)(46 47 48 49 50 51 52 53 54)
(1 36 47)(2 30 53)(3 33 50)(4 31 46)(5 34 52)(6 28 49)(7 37 21)(8 40 27)(9 43 24)(10 41 20)(11 44 26)(12 38 23)(13 42 22)(14 45 19)(15 39 25)(16 32 48)(17 35 54)(18 29 51)

G:=sub<Sym(54)| (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,16)(14,17)(15,18)(19,54)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52)(27,53)(28,38)(29,39)(30,40)(31,41)(32,42)(33,43)(34,44)(35,45)(36,37), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21,22,23,24,25,26,27)(28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54), (1,18,4,3,17,6,2,16,5)(7,15,10,9,14,12,8,13,11)(19,20,21,22,23,24,25,26,27)(28,30,32,34,36,29,31,33,35)(37,39,41,43,45,38,40,42,44)(46,47,48,49,50,51,52,53,54), (1,36,47)(2,30,53)(3,33,50)(4,31,46)(5,34,52)(6,28,49)(7,37,21)(8,40,27)(9,43,24)(10,41,20)(11,44,26)(12,38,23)(13,42,22)(14,45,19)(15,39,25)(16,32,48)(17,35,54)(18,29,51)>;

G:=Group( (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,16)(14,17)(15,18)(19,54)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52)(27,53)(28,38)(29,39)(30,40)(31,41)(32,42)(33,43)(34,44)(35,45)(36,37), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21,22,23,24,25,26,27)(28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54), (1,18,4,3,17,6,2,16,5)(7,15,10,9,14,12,8,13,11)(19,20,21,22,23,24,25,26,27)(28,30,32,34,36,29,31,33,35)(37,39,41,43,45,38,40,42,44)(46,47,48,49,50,51,52,53,54), (1,36,47)(2,30,53)(3,33,50)(4,31,46)(5,34,52)(6,28,49)(7,37,21)(8,40,27)(9,43,24)(10,41,20)(11,44,26)(12,38,23)(13,42,22)(14,45,19)(15,39,25)(16,32,48)(17,35,54)(18,29,51) );

G=PermutationGroup([[(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,16),(14,17),(15,18),(19,54),(20,46),(21,47),(22,48),(23,49),(24,50),(25,51),(26,52),(27,53),(28,38),(29,39),(30,40),(31,41),(32,42),(33,43),(34,44),(35,45),(36,37)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21,22,23,24,25,26,27),(28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45),(46,47,48,49,50,51,52,53,54)], [(1,18,4,3,17,6,2,16,5),(7,15,10,9,14,12,8,13,11),(19,20,21,22,23,24,25,26,27),(28,30,32,34,36,29,31,33,35),(37,39,41,43,45,38,40,42,44),(46,47,48,49,50,51,52,53,54)], [(1,36,47),(2,30,53),(3,33,50),(4,31,46),(5,34,52),(6,28,49),(7,37,21),(8,40,27),(9,43,24),(10,41,20),(11,44,26),(12,38,23),(13,42,22),(14,45,19),(15,39,25),(16,32,48),(17,35,54),(18,29,51)]])

70 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E ··· 3J 6A 6B 6C 6D 6E ··· 6J 9A ··· 9X 18A ··· 18X order 1 2 3 3 3 3 3 ··· 3 6 6 6 6 6 ··· 6 9 ··· 9 18 ··· 18 size 1 1 1 1 3 3 27 ··· 27 1 1 3 3 27 ··· 27 3 ··· 3 3 ··· 3

70 irreducible representations

 dim 1 1 1 1 1 1 3 3 3 3 3 3 type + + image C1 C2 C3 C3 C6 C6 He3 C2×He3 He3⋊C3 C2×He3⋊C3 C92⋊2C3 C2×C92⋊2C3 kernel C2×C92⋊2C3 C92⋊2C3 C9×C18 C2×He3⋊C3 C92 He3⋊C3 C3×C6 C32 C6 C3 C2 C1 # reps 1 1 2 6 2 6 2 2 6 6 18 18

Matrix representation of C2×C922C3 in GL3(𝔽19) generated by

 18 0 0 0 18 0 0 0 18
,
 11 0 0 0 4 0 0 0 16
,
 9 0 0 0 4 0 0 0 9
,
 0 1 0 0 0 1 1 0 0
G:=sub<GL(3,GF(19))| [18,0,0,0,18,0,0,0,18],[11,0,0,0,4,0,0,0,16],[9,0,0,0,4,0,0,0,9],[0,0,1,1,0,0,0,1,0] >;

C2×C922C3 in GAP, Magma, Sage, TeX

C_2\times C_9^2\rtimes_2C_3
% in TeX

G:=Group("C2xC9^2:2C3");
// GroupNames label

G:=SmallGroup(486,86);
// by ID

G=gap.SmallGroup(486,86);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,224,824,873,453,3250]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^9=c^9=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^7*c^-1,d*c*d^-1=b^3*c>;
// generators/relations

Export

׿
×
𝔽