Copied to
clipboard

## G = C2×C32⋊D9order 324 = 22·34

### Direct product of C2 and C32⋊D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — C2×C32⋊D9
 Chief series C1 — C3 — C32 — C3×C9 — C32⋊C9 — C32⋊D9 — C2×C32⋊D9
 Lower central C3×C9 — C2×C32⋊D9
 Upper central C1 — C2

Generators and relations for C2×C32⋊D9
G = < a,b,c,d,e | a2=b3=c3=d9=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 511 in 83 conjugacy classes, 26 normal (18 characteristic)
C1, C2, C2 [×2], C3 [×2], C3 [×2], C3 [×2], C22, S3 [×8], C6 [×2], C6 [×2], C6 [×4], C9 [×2], C32 [×2], C32 [×4], D6 [×4], C2×C6, D9 [×2], C18 [×2], C3×S3 [×8], C3⋊S3 [×2], C3×C6 [×2], C3×C6 [×4], C3×C9, C3×C9, C33, D18, S3×C6 [×4], C2×C3⋊S3, C9⋊S3 [×2], C3×C18, C3×C18, C3×C3⋊S3 [×2], C32×C6, C32⋊C9, C2×C9⋊S3, C6×C3⋊S3, C32⋊D9 [×2], C2×C32⋊C9, C2×C32⋊D9
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], D6, C2×C6, D9, C3×S3, D18, S3×C6, C3×D9, C32⋊C6, C9⋊C6 [×2], C6×D9, C2×C32⋊C6, C2×C9⋊C6 [×2], C32⋊D9, C2×C32⋊D9

Smallest permutation representation of C2×C32⋊D9
On 54 points
Generators in S54
(1 25)(2 26)(3 27)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 32)(11 33)(12 34)(13 35)(14 36)(15 28)(16 29)(17 30)(18 31)(37 54)(38 46)(39 47)(40 48)(41 49)(42 50)(43 51)(44 52)(45 53)
(2 36 44)(3 45 28)(5 30 38)(6 39 31)(8 33 41)(9 42 34)(11 49 23)(12 24 50)(14 52 26)(15 27 53)(17 46 20)(18 21 47)
(1 35 43)(2 36 44)(3 28 45)(4 29 37)(5 30 38)(6 31 39)(7 32 40)(8 33 41)(9 34 42)(10 48 22)(11 49 23)(12 50 24)(13 51 25)(14 52 26)(15 53 27)(16 54 19)(17 46 20)(18 47 21)
(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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 27)(8 26)(9 25)(10 45)(11 44)(12 43)(13 42)(14 41)(15 40)(16 39)(17 38)(18 37)(28 48)(29 47)(30 46)(31 54)(32 53)(33 52)(34 51)(35 50)(36 49)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 3G 3H 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 9A ··· 9I 18A ··· 18I order 1 2 2 2 3 3 3 3 3 3 3 3 6 6 6 6 6 6 6 6 6 6 6 6 9 ··· 9 18 ··· 18 size 1 1 27 27 2 2 2 2 3 3 6 6 2 2 2 2 3 3 6 6 27 27 27 27 6 ··· 6 6 ··· 6

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 6 6 6 6 type + + + + + + + + + + + image C1 C2 C2 C3 C6 C6 S3 D6 D9 C3×S3 D18 S3×C6 C3×D9 C6×D9 C32⋊C6 C9⋊C6 C2×C32⋊C6 C2×C9⋊C6 kernel C2×C32⋊D9 C32⋊D9 C2×C32⋊C9 C2×C9⋊S3 C9⋊S3 C3×C18 C32×C6 C33 C3×C6 C3×C6 C32 C32 C6 C3 C6 C6 C3 C3 # reps 1 2 1 2 4 2 1 1 3 2 3 2 6 6 1 2 1 2

Matrix representation of C2×C32⋊D9 in GL8(𝔽19)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 18
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 18 18 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 18 18
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 18 18 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 18 18 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 18 18 0 0 0 0 0 0 1 0
,
 14 2 0 0 0 0 0 0 17 12 0 0 0 0 0 0 0 0 0 0 0 0 18 18 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 18 18 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 18 18 0 0
,
 14 2 0 0 0 0 0 0 7 5 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 18 0 0 0 0 1 1 0 0 0 0 0 0 0 18 0 0 0 0 1 1 0 0 0 0 0 0 0 18 0 0 0 0

G:=sub<GL(8,GF(19))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,18],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,1,18],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,18,0],[14,17,0,0,0,0,0,0,2,12,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,1,18,0,0,18,1,0,0,0,0,0,0,18,0,0,0,0,0],[14,7,0,0,0,0,0,0,2,5,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,18,0,0,0,0,1,0,0,0,0,0,0,0,1,18,0,0,0,0,1,0,0,0,0,0,0,0,1,18,0,0,0,0] >;

C2×C32⋊D9 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes D_9
% in TeX

G:=Group("C2xC3^2:D9");
// GroupNames label

G:=SmallGroup(324,63);
// by ID

G=gap.SmallGroup(324,63);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,3171,303,453,2164,7781]);
// Polycyclic

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

׿
×
𝔽