Copied to
clipboard

## G = C4×S3×D9order 432 = 24·33

### Direct product of C4, S3 and D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — C4×S3×D9
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — S3×C18 — C2×S3×D9 — C4×S3×D9
 Lower central C3×C9 — C4×S3×D9
 Upper central C1 — C4

Generators and relations for C4×S3×D9
G = < a,b,c,d,e | a4=b3=c2=d9=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1068 in 178 conjugacy classes, 57 normal (41 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, S3, C6, C6, C2×C4, C23, C9, C9, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C22×C4, D9, D9, C18, C18, C3×S3, C3×S3, C3⋊S3, C3×C6, C4×S3, C4×S3, C2×Dic3, C2×C12, C22×S3, C3×C9, Dic9, Dic9, C36, C36, D18, D18, C2×C18, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, S3×C6, C2×C3⋊S3, S3×C2×C4, C3×D9, S3×C9, C9⋊S3, C3×C18, C4×D9, C4×D9, C2×Dic9, C2×C36, C22×D9, S3×Dic3, C6.D6, S3×C12, S3×C12, C4×C3⋊S3, C2×S32, C3×Dic9, C9×Dic3, C9⋊Dic3, C3×C36, S3×D9, C6×D9, S3×C18, C2×C9⋊S3, C2×C4×D9, C4×S32, Dic3×D9, C18.D6, S3×Dic9, C12×D9, S3×C36, C4×C9⋊S3, C2×S3×D9, C4×S3×D9
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, D9, C4×S3, C22×S3, D18, S32, S3×C2×C4, C4×D9, C22×D9, C2×S32, S3×D9, C2×C4×D9, C4×S32, C2×S3×D9, C4×S3×D9

Smallest permutation representation of C4×S3×D9
On 72 points
Generators in S72
(1 32 14 23)(2 33 15 24)(3 34 16 25)(4 35 17 26)(5 36 18 27)(6 28 10 19)(7 29 11 20)(8 30 12 21)(9 31 13 22)(37 64 46 55)(38 65 47 56)(39 66 48 57)(40 67 49 58)(41 68 50 59)(42 69 51 60)(43 70 52 61)(44 71 53 62)(45 72 54 63)
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 46)(8 47)(9 48)(10 45)(11 37)(12 38)(13 39)(14 40)(15 41)(16 42)(17 43)(18 44)(19 72)(20 64)(21 65)(22 66)(23 67)(24 68)(25 69)(26 70)(27 71)(28 63)(29 55)(30 56)(31 57)(32 58)(33 59)(34 60)(35 61)(36 62)
(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)(55 56 57 58 59 60 61 62 63)(64 65 66 67 68 69 70 71 72)
(1 48)(2 47)(3 46)(4 54)(5 53)(6 52)(7 51)(8 50)(9 49)(10 43)(11 42)(12 41)(13 40)(14 39)(15 38)(16 37)(17 45)(18 44)(19 70)(20 69)(21 68)(22 67)(23 66)(24 65)(25 64)(26 72)(27 71)(28 61)(29 60)(30 59)(31 58)(32 57)(33 56)(34 55)(35 63)(36 62)

G:=sub<Sym(72)| (1,32,14,23)(2,33,15,24)(3,34,16,25)(4,35,17,26)(5,36,18,27)(6,28,10,19)(7,29,11,20)(8,30,12,21)(9,31,13,22)(37,64,46,55)(38,65,47,56)(39,66,48,57)(40,67,49,58)(41,68,50,59)(42,69,51,60)(43,70,52,61)(44,71,53,62)(45,72,54,63), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,46)(8,47)(9,48)(10,45)(11,37)(12,38)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,72)(20,64)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,71)(28,63)(29,55)(30,56)(31,57)(32,58)(33,59)(34,60)(35,61)(36,62), (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)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72), (1,48)(2,47)(3,46)(4,54)(5,53)(6,52)(7,51)(8,50)(9,49)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,45)(18,44)(19,70)(20,69)(21,68)(22,67)(23,66)(24,65)(25,64)(26,72)(27,71)(28,61)(29,60)(30,59)(31,58)(32,57)(33,56)(34,55)(35,63)(36,62)>;

G:=Group( (1,32,14,23)(2,33,15,24)(3,34,16,25)(4,35,17,26)(5,36,18,27)(6,28,10,19)(7,29,11,20)(8,30,12,21)(9,31,13,22)(37,64,46,55)(38,65,47,56)(39,66,48,57)(40,67,49,58)(41,68,50,59)(42,69,51,60)(43,70,52,61)(44,71,53,62)(45,72,54,63), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,46)(8,47)(9,48)(10,45)(11,37)(12,38)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,72)(20,64)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,71)(28,63)(29,55)(30,56)(31,57)(32,58)(33,59)(34,60)(35,61)(36,62), (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)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72), (1,48)(2,47)(3,46)(4,54)(5,53)(6,52)(7,51)(8,50)(9,49)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,45)(18,44)(19,70)(20,69)(21,68)(22,67)(23,66)(24,65)(25,64)(26,72)(27,71)(28,61)(29,60)(30,59)(31,58)(32,57)(33,56)(34,55)(35,63)(36,62) );

G=PermutationGroup([[(1,32,14,23),(2,33,15,24),(3,34,16,25),(4,35,17,26),(5,36,18,27),(6,28,10,19),(7,29,11,20),(8,30,12,21),(9,31,13,22),(37,64,46,55),(38,65,47,56),(39,66,48,57),(40,67,49,58),(41,68,50,59),(42,69,51,60),(43,70,52,61),(44,71,53,62),(45,72,54,63)], [(1,7,4),(2,8,5),(3,9,6),(10,16,13),(11,17,14),(12,18,15),(19,25,22),(20,26,23),(21,27,24),(28,34,31),(29,35,32),(30,36,33),(37,40,43),(38,41,44),(39,42,45),(46,49,52),(47,50,53),(48,51,54),(55,58,61),(56,59,62),(57,60,63),(64,67,70),(65,68,71),(66,69,72)], [(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,46),(8,47),(9,48),(10,45),(11,37),(12,38),(13,39),(14,40),(15,41),(16,42),(17,43),(18,44),(19,72),(20,64),(21,65),(22,66),(23,67),(24,68),(25,69),(26,70),(27,71),(28,63),(29,55),(30,56),(31,57),(32,58),(33,59),(34,60),(35,61),(36,62)], [(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),(55,56,57,58,59,60,61,62,63),(64,65,66,67,68,69,70,71,72)], [(1,48),(2,47),(3,46),(4,54),(5,53),(6,52),(7,51),(8,50),(9,49),(10,43),(11,42),(12,41),(13,40),(14,39),(15,38),(16,37),(17,45),(18,44),(19,70),(20,69),(21,68),(22,67),(23,66),(24,65),(25,64),(26,72),(27,71),(28,61),(29,60),(30,59),(31,58),(32,57),(33,56),(34,55),(35,63),(36,62)]])

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D 6E 6F 6G 9A 9B 9C 9D 9E 9F 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 18A 18B 18C 18D 18E 18F 18G ··· 18L 36A ··· 36F 36G ··· 36L 36M ··· 36R order 1 2 2 2 2 2 2 2 3 3 3 4 4 4 4 4 4 4 4 6 6 6 6 6 6 6 9 9 9 9 9 9 12 12 12 12 12 12 12 12 12 12 18 18 18 18 18 18 18 ··· 18 36 ··· 36 36 ··· 36 36 ··· 36 size 1 1 3 3 9 9 27 27 2 2 4 1 1 3 3 9 9 27 27 2 2 4 6 6 18 18 2 2 2 4 4 4 2 2 2 2 4 4 6 6 18 18 2 2 2 4 4 4 6 ··· 6 2 ··· 2 4 ··· 4 6 ··· 6

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 S3 S3 D6 D6 D6 D6 D6 D6 D9 C4×S3 C4×S3 D18 D18 D18 C4×D9 S32 C2×S32 S3×D9 C4×S32 C2×S3×D9 C4×S3×D9 kernel C4×S3×D9 Dic3×D9 C18.D6 S3×Dic9 C12×D9 S3×C36 C4×C9⋊S3 C2×S3×D9 S3×D9 C4×D9 S3×C12 Dic9 C36 D18 C3×Dic3 C3×C12 S3×C6 C4×S3 D9 C3×S3 Dic3 C12 D6 S3 C12 C6 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 8 1 1 1 1 1 1 1 1 3 4 4 3 3 3 12 1 1 3 2 3 6

Matrix representation of C4×S3×D9 in GL4(𝔽37) generated by

 31 0 0 0 0 31 0 0 0 0 36 0 0 0 0 36
,
 36 1 0 0 36 0 0 0 0 0 1 0 0 0 0 1
,
 0 36 0 0 36 0 0 0 0 0 36 0 0 0 0 36
,
 1 0 0 0 0 1 0 0 0 0 26 6 0 0 31 20
,
 36 0 0 0 0 36 0 0 0 0 26 6 0 0 17 11
G:=sub<GL(4,GF(37))| [31,0,0,0,0,31,0,0,0,0,36,0,0,0,0,36],[36,36,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,36,0,0,36,0,0,0,0,0,36,0,0,0,0,36],[1,0,0,0,0,1,0,0,0,0,26,31,0,0,6,20],[36,0,0,0,0,36,0,0,0,0,26,17,0,0,6,11] >;

C4×S3×D9 in GAP, Magma, Sage, TeX

C_4\times S_3\times D_9
% in TeX

G:=Group("C4xS3xD9");
// GroupNames label

G:=SmallGroup(432,290);
// by ID

G=gap.SmallGroup(432,290);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,58,3091,662,4037,7069]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^3=c^2=d^9=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽