direct product, metabelian, supersoluble, monomial
Aliases: C6xC32:7D4, C63:3C2, C62:33D6, C33:36(C2xD4), C62:19(C2xC6), (C2xC62):12C6, (C2xC62):11S3, (C32xC6):14D4, C32:17(C6xD4), (C3xC62):15C22, (C32xC6).94C23, C6:3(C3xC3:D4), C3:4(C6xC3:D4), C6.61(S3xC2xC6), (C2xC6):13(S3xC6), (C3xC6):10(C3xD4), C23:3(C3xC3:S3), C22:4(C6xC3:S3), (C22xC6):6(C3xS3), (C3xC6):12(C3:D4), (C6xC3:S3):23C22, (C22xC3:S3):12C6, (C22xC6):3(C3:S3), (C2xC3:Dic3):14C6, C3:Dic3:11(C2xC6), (C6xC3:Dic3):18C2, C6.61(C22xC3:S3), C32:21(C2xC3:D4), (C3xC6).68(C22xC6), (C3xC6).183(C22xS3), (C3xC3:Dic3):25C22, (C2xC6xC3:S3):10C2, C2.10(C2xC6xC3:S3), (C2xC6):10(C2xC3:S3), (C2xC3:S3):11(C2xC6), SmallGroup(432,719)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6xC32:7D4
G = < a,b,c,d,e | a6=b3=c3=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ebe=b-1, dcd-1=ece=c-1, ede=d-1 >
Subgroups: 1284 in 452 conjugacy classes, 118 normal (22 characteristic)
C1, C2, C2, C2, C3, C3, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2xC4, D4, C23, C23, C32, C32, C32, Dic3, C12, D6, C2xC6, C2xC6, C2xC6, C2xD4, C3xS3, C3:S3, C3xC6, C3xC6, C3xC6, C2xDic3, C3:D4, C2xC12, C3xD4, C22xS3, C22xC6, C22xC6, C22xC6, C33, C3xDic3, C3:Dic3, S3xC6, C2xC3:S3, C2xC3:S3, C62, C62, C62, C2xC3:D4, C6xD4, C3xC3:S3, C32xC6, C32xC6, C32xC6, C6xDic3, C3xC3:D4, C2xC3:Dic3, C32:7D4, S3xC2xC6, C22xC3:S3, C2xC62, C2xC62, C2xC62, C3xC3:Dic3, C6xC3:S3, C6xC3:S3, C3xC62, C3xC62, C3xC62, C6xC3:D4, C2xC32:7D4, C6xC3:Dic3, C3xC32:7D4, C2xC6xC3:S3, C63, C6xC32:7D4
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2xC6, C2xD4, C3xS3, C3:S3, C3:D4, C3xD4, C22xS3, C22xC6, S3xC6, C2xC3:S3, C2xC3:D4, C6xD4, C3xC3:S3, C3xC3:D4, C32:7D4, S3xC2xC6, C22xC3:S3, C6xC3:S3, C6xC3:D4, C2xC32:7D4, C3xC32:7D4, C2xC6xC3:S3, C6xC32:7D4
►On 72 points
(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 27 16)(2 28 17)(3 29 18)(4 30 13)(5 25 14)(6 26 15)(7 34 39)(8 35 40)(9 36 41)(10 31 42)(11 32 37)(12 33 38)(19 70 59)(20 71 60)(21 72 55)(22 67 56)(23 68 57)(24 69 58)(43 51 62)(44 52 63)(45 53 64)(46 54 65)(47 49 66)(48 50 61)
(1 5 3)(2 6 4)(7 11 9)(8 12 10)(13 17 15)(14 18 16)(19 21 23)(20 22 24)(25 29 27)(26 30 28)(31 35 33)(32 36 34)(37 41 39)(38 42 40)(43 45 47)(44 46 48)(49 51 53)(50 52 54)(55 57 59)(56 58 60)(61 63 65)(62 64 66)(67 69 71)(68 70 72)
(1 71 35 44)(2 72 36 45)(3 67 31 46)(4 68 32 47)(5 69 33 48)(6 70 34 43)(7 51 15 59)(8 52 16 60)(9 53 17 55)(10 54 18 56)(11 49 13 57)(12 50 14 58)(19 39 62 26)(20 40 63 27)(21 41 64 28)(22 42 65 29)(23 37 66 30)(24 38 61 25)
(1 47)(2 48)(3 43)(4 44)(5 45)(6 46)(7 56)(8 57)(9 58)(10 59)(11 60)(12 55)(13 52)(14 53)(15 54)(16 49)(17 50)(18 51)(19 42)(20 37)(21 38)(22 39)(23 40)(24 41)(25 64)(26 65)(27 66)(28 61)(29 62)(30 63)(31 70)(32 71)(33 72)(34 67)(35 68)(36 69)
G:=sub<Sym(72)| (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), (1,27,16)(2,28,17)(3,29,18)(4,30,13)(5,25,14)(6,26,15)(7,34,39)(8,35,40)(9,36,41)(10,31,42)(11,32,37)(12,33,38)(19,70,59)(20,71,60)(21,72,55)(22,67,56)(23,68,57)(24,69,58)(43,51,62)(44,52,63)(45,53,64)(46,54,65)(47,49,66)(48,50,61), (1,5,3)(2,6,4)(7,11,9)(8,12,10)(13,17,15)(14,18,16)(19,21,23)(20,22,24)(25,29,27)(26,30,28)(31,35,33)(32,36,34)(37,41,39)(38,42,40)(43,45,47)(44,46,48)(49,51,53)(50,52,54)(55,57,59)(56,58,60)(61,63,65)(62,64,66)(67,69,71)(68,70,72), (1,71,35,44)(2,72,36,45)(3,67,31,46)(4,68,32,47)(5,69,33,48)(6,70,34,43)(7,51,15,59)(8,52,16,60)(9,53,17,55)(10,54,18,56)(11,49,13,57)(12,50,14,58)(19,39,62,26)(20,40,63,27)(21,41,64,28)(22,42,65,29)(23,37,66,30)(24,38,61,25), (1,47)(2,48)(3,43)(4,44)(5,45)(6,46)(7,56)(8,57)(9,58)(10,59)(11,60)(12,55)(13,52)(14,53)(15,54)(16,49)(17,50)(18,51)(19,42)(20,37)(21,38)(22,39)(23,40)(24,41)(25,64)(26,65)(27,66)(28,61)(29,62)(30,63)(31,70)(32,71)(33,72)(34,67)(35,68)(36,69)>;
G:=Group( (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), (1,27,16)(2,28,17)(3,29,18)(4,30,13)(5,25,14)(6,26,15)(7,34,39)(8,35,40)(9,36,41)(10,31,42)(11,32,37)(12,33,38)(19,70,59)(20,71,60)(21,72,55)(22,67,56)(23,68,57)(24,69,58)(43,51,62)(44,52,63)(45,53,64)(46,54,65)(47,49,66)(48,50,61), (1,5,3)(2,6,4)(7,11,9)(8,12,10)(13,17,15)(14,18,16)(19,21,23)(20,22,24)(25,29,27)(26,30,28)(31,35,33)(32,36,34)(37,41,39)(38,42,40)(43,45,47)(44,46,48)(49,51,53)(50,52,54)(55,57,59)(56,58,60)(61,63,65)(62,64,66)(67,69,71)(68,70,72), (1,71,35,44)(2,72,36,45)(3,67,31,46)(4,68,32,47)(5,69,33,48)(6,70,34,43)(7,51,15,59)(8,52,16,60)(9,53,17,55)(10,54,18,56)(11,49,13,57)(12,50,14,58)(19,39,62,26)(20,40,63,27)(21,41,64,28)(22,42,65,29)(23,37,66,30)(24,38,61,25), (1,47)(2,48)(3,43)(4,44)(5,45)(6,46)(7,56)(8,57)(9,58)(10,59)(11,60)(12,55)(13,52)(14,53)(15,54)(16,49)(17,50)(18,51)(19,42)(20,37)(21,38)(22,39)(23,40)(24,41)(25,64)(26,65)(27,66)(28,61)(29,62)(30,63)(31,70)(32,71)(33,72)(34,67)(35,68)(36,69) );
G=PermutationGroup([[(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)], [(1,27,16),(2,28,17),(3,29,18),(4,30,13),(5,25,14),(6,26,15),(7,34,39),(8,35,40),(9,36,41),(10,31,42),(11,32,37),(12,33,38),(19,70,59),(20,71,60),(21,72,55),(22,67,56),(23,68,57),(24,69,58),(43,51,62),(44,52,63),(45,53,64),(46,54,65),(47,49,66),(48,50,61)], [(1,5,3),(2,6,4),(7,11,9),(8,12,10),(13,17,15),(14,18,16),(19,21,23),(20,22,24),(25,29,27),(26,30,28),(31,35,33),(32,36,34),(37,41,39),(38,42,40),(43,45,47),(44,46,48),(49,51,53),(50,52,54),(55,57,59),(56,58,60),(61,63,65),(62,64,66),(67,69,71),(68,70,72)], [(1,71,35,44),(2,72,36,45),(3,67,31,46),(4,68,32,47),(5,69,33,48),(6,70,34,43),(7,51,15,59),(8,52,16,60),(9,53,17,55),(10,54,18,56),(11,49,13,57),(12,50,14,58),(19,39,62,26),(20,40,63,27),(21,41,64,28),(22,42,65,29),(23,37,66,30),(24,38,61,25)], [(1,47),(2,48),(3,43),(4,44),(5,45),(6,46),(7,56),(8,57),(9,58),(10,59),(11,60),(12,55),(13,52),(14,53),(15,54),(16,49),(17,50),(18,51),(19,42),(20,37),(21,38),(22,39),(23,40),(24,41),(25,64),(26,65),(27,66),(28,61),(29,62),(30,63),(31,70),(32,71),(33,72),(34,67),(35,68),(36,69)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | ··· | 3N | 4A | 4B | 6A | ··· | 6F | 6G | ··· | 6CP | 6CQ | 6CR | 6CS | 6CT | 12A | 12B | 12C | 12D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 18 | 18 | 1 | 1 | 2 | ··· | 2 | 18 | 18 | 1 | ··· | 1 | 2 | ··· | 2 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | 18 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | S3 | D4 | D6 | C3xS3 | C3:D4 | C3xD4 | S3xC6 | C3xC3:D4 |
| kernel | C6xC32:7D4 | C6xC3:Dic3 | C3xC32:7D4 | C2xC6xC3:S3 | C63 | C2xC32:7D4 | C2xC3:Dic3 | C32:7D4 | C22xC3:S3 | C2xC62 | C2xC62 | C32xC6 | C62 | C22xC6 | C3xC6 | C3xC6 | C2xC6 | C6 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 4 | 2 | 12 | 8 | 16 | 4 | 24 | 32 |
Matrix representation of C6xC32:7D4 ►in GL4(F13) generated by
| 10 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 5 | 9 |
| 3 | 0 | 0 | 0 |
| 9 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 3 | 2 | 0 | 0 |
| 9 | 10 | 0 | 0 |
| 0 | 0 | 8 | 7 |
| 0 | 0 | 0 | 5 |
| 10 | 11 | 0 | 0 |
| 4 | 3 | 0 | 0 |
| 0 | 0 | 8 | 7 |
| 0 | 0 | 4 | 5 |
G:=sub<GL(4,GF(13))| [10,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,3,5,0,0,0,9],[3,9,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[3,9,0,0,2,10,0,0,0,0,8,0,0,0,7,5],[10,4,0,0,11,3,0,0,0,0,8,4,0,0,7,5] >;
C6xC32:7D4 in GAP, Magma, Sage, TeX
C_6\times C_3^2\rtimes_7D_4
% in TeX
G:=Group("C6xC3^2:7D4"); // GroupNames label
G:=SmallGroup(432,719);
// by ID
G=gap.SmallGroup(432,719);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,590,4037,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^3=c^3=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=e*b*e=b^-1,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations