metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C6xD4):9C4, (C6xQ8):9C4, C4oD4:6Dic3, (C2xD4):9Dic3, C4oD4.53D6, (C2xQ8):9Dic3, C12.453(C2xD4), (C2xC12).198D4, D4.7(C2xDic3), C12.86(C22xC4), Q8.12(C2xDic3), (C4xDic3):7C22, (C22xC6).114D4, (C22xC4).178D6, Q8:3Dic3:10C2, C12.38(C22:C4), (C2xC12).482C23, C3:4(C42:C22), C23.38(C3:D4), C4.Dic3:24C22, C4.16(C22xDic3), C4.23(C6.D4), C23.26D6:20C2, (C22xC12).208C22, C22.6(C6.D4), (C3xC4oD4):4C4, (C6xC4oD4).5C2, (C2xC6).40(C2xD4), (C2xC4oD4).11S3, (C3xD4).24(C2xC4), C4.144(C2xC3:D4), C6.85(C2xC22:C4), (C3xQ8).25(C2xC4), (C2xC12).126(C2xC4), (C2xC4).90(C3:D4), (C2xC4.Dic3):22C2, (C2xC4).29(C2xDic3), C22.12(C2xC3:D4), (C2xC6).27(C22:C4), (C2xC4).567(C22xS3), C2.21(C2xC6.D4), (C3xC4oD4).42C22, SmallGroup(192,795)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C6xD4):9C4
G = < a,b,c,d | a6=b4=c2=d4=1, ab=ba, ac=ca, dad-1=a-1b2, cbc=b-1, bd=db, dcd-1=bc >
Subgroups: 328 in 154 conjugacy classes, 67 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C23, Dic3, C12, C12, C2xC6, C2xC6, C42, C22:C4, C4:C4, C2xC8, M4(2), C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C4oD4, C3:C8, C2xDic3, C2xC12, C2xC12, C3xD4, C3xD4, C3xQ8, C3xQ8, C22xC6, C22xC6, C4wrC2, C42:C2, C2xM4(2), C2xC4oD4, C2xC3:C8, C4.Dic3, C4.Dic3, C4xDic3, C4:Dic3, C6.D4, C22xC12, C22xC12, C6xD4, C6xD4, C6xQ8, C3xC4oD4, C3xC4oD4, C42:C22, Q8:3Dic3, C2xC4.Dic3, C23.26D6, C6xC4oD4, (C6xD4):9C4
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, C23, Dic3, D6, C22:C4, C22xC4, C2xD4, C2xDic3, C3:D4, C22xS3, C2xC22:C4, C6.D4, C22xDic3, C2xC3:D4, C42:C22, C2xC6.D4, (C6xD4):9C4
►On 48 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)
(1 19 41 9)(2 20 42 10)(3 21 37 11)(4 22 38 12)(5 23 39 7)(6 24 40 8)(13 44 29 35)(14 45 30 36)(15 46 25 31)(16 47 26 32)(17 48 27 33)(18 43 28 34)
(1 29)(2 30)(3 25)(4 26)(5 27)(6 28)(7 33)(8 34)(9 35)(10 36)(11 31)(12 32)(13 41)(14 42)(15 37)(16 38)(17 39)(18 40)(19 44)(20 45)(21 46)(22 47)(23 48)(24 43)
(2 40)(3 5)(4 38)(6 42)(7 11)(8 20)(10 24)(12 22)(13 44 29 35)(14 34 30 43)(15 48 25 33)(16 32 26 47)(17 46 27 31)(18 36 28 45)(21 23)(37 39)
G:=sub<Sym(48)| (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), (1,19,41,9)(2,20,42,10)(3,21,37,11)(4,22,38,12)(5,23,39,7)(6,24,40,8)(13,44,29,35)(14,45,30,36)(15,46,25,31)(16,47,26,32)(17,48,27,33)(18,43,28,34), (1,29)(2,30)(3,25)(4,26)(5,27)(6,28)(7,33)(8,34)(9,35)(10,36)(11,31)(12,32)(13,41)(14,42)(15,37)(16,38)(17,39)(18,40)(19,44)(20,45)(21,46)(22,47)(23,48)(24,43), (2,40)(3,5)(4,38)(6,42)(7,11)(8,20)(10,24)(12,22)(13,44,29,35)(14,34,30,43)(15,48,25,33)(16,32,26,47)(17,46,27,31)(18,36,28,45)(21,23)(37,39)>;
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), (1,19,41,9)(2,20,42,10)(3,21,37,11)(4,22,38,12)(5,23,39,7)(6,24,40,8)(13,44,29,35)(14,45,30,36)(15,46,25,31)(16,47,26,32)(17,48,27,33)(18,43,28,34), (1,29)(2,30)(3,25)(4,26)(5,27)(6,28)(7,33)(8,34)(9,35)(10,36)(11,31)(12,32)(13,41)(14,42)(15,37)(16,38)(17,39)(18,40)(19,44)(20,45)(21,46)(22,47)(23,48)(24,43), (2,40)(3,5)(4,38)(6,42)(7,11)(8,20)(10,24)(12,22)(13,44,29,35)(14,34,30,43)(15,48,25,33)(16,32,26,47)(17,46,27,31)(18,36,28,45)(21,23)(37,39) );
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)], [(1,19,41,9),(2,20,42,10),(3,21,37,11),(4,22,38,12),(5,23,39,7),(6,24,40,8),(13,44,29,35),(14,45,30,36),(15,46,25,31),(16,47,26,32),(17,48,27,33),(18,43,28,34)], [(1,29),(2,30),(3,25),(4,26),(5,27),(6,28),(7,33),(8,34),(9,35),(10,36),(11,31),(12,32),(13,41),(14,42),(15,37),(16,38),(17,39),(18,40),(19,44),(20,45),(21,46),(22,47),(23,48),(24,43)], [(2,40),(3,5),(4,38),(6,42),(7,11),(8,20),(10,24),(12,22),(13,44,29,35),(14,34,30,43),(15,48,25,33),(16,32,26,47),(17,46,27,31),(18,36,28,45),(21,23),(37,39)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 6A | 6B | 6C | 6D | ··· | 6I | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | - | - | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D4 | D4 | D6 | Dic3 | Dic3 | Dic3 | D6 | C3:D4 | C3:D4 | C42:C22 | (C6xD4):9C4 |
| kernel | (C6xD4):9C4 | Q8:3Dic3 | C2xC4.Dic3 | C23.26D6 | C6xC4oD4 | C6xD4 | C6xQ8 | C3xC4oD4 | C2xC4oD4 | C2xC12 | C22xC6 | C22xC4 | C2xD4 | C2xQ8 | C4oD4 | C4oD4 | C2xC4 | C23 | C3 | C1 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 3 | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 2 | 2 | 4 |
Matrix representation of (C6xD4):9C4 ►in GL4(F73) generated by
| 60 | 30 | 0 | 0 |
| 43 | 30 | 0 | 0 |
| 0 | 0 | 60 | 30 |
| 0 | 0 | 43 | 30 |
| 27 | 0 | 46 | 0 |
| 0 | 27 | 0 | 46 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 |
| 7 | 14 | 66 | 59 |
| 59 | 66 | 14 | 7 |
| 14 | 28 | 66 | 59 |
| 45 | 59 | 14 | 7 |
| 0 | 72 | 0 | 60 |
| 72 | 0 | 60 | 0 |
| 0 | 0 | 0 | 46 |
| 0 | 0 | 46 | 0 |
G:=sub<GL(4,GF(73))| [60,43,0,0,30,30,0,0,0,0,60,43,0,0,30,30],[27,0,0,0,0,27,0,0,46,0,46,0,0,46,0,46],[7,59,14,45,14,66,28,59,66,14,66,14,59,7,59,7],[0,72,0,0,72,0,0,0,0,60,0,46,60,0,46,0] >;
(C6xD4):9C4 in GAP, Magma, Sage, TeX
(C_6\times D_4)\rtimes_9C_4
% in TeX
G:=Group("(C6xD4):9C4"); // GroupNames label
G:=SmallGroup(192,795);
// by ID
G=gap.SmallGroup(192,795);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,422,387,136,1684,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^4=c^2=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b*c>;
// generators/relations