p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8:12SD16, D4.1M4(2), C42.639C23, Q8:C8:6C2, C8:C8:17C2, C8:2C8:28C2, C4.Q8.7C4, D4:C8.12C2, C8:4Q8:28C2, (C8xD4).15C2, (C2xC8).378D4, C2.5(C4xSD16), D4:C4.6C4, C4.31(C8oD4), C2.9(C8:9D4), Q8:C4.6C4, (C2xSD16).3C4, (C4xSD16).9C2, C4.129(C4oD8), (C4xQ8).9C22, C2.7(C8.26D4), C4:C8.221C22, (C4xC8).238C22, C4.101(C2xSD16), C22.130(C4xD4), C4.25(C2xM4(2)), (C4xD4).272C22, C4:C4.133(C2xC4), (C2xC8).102(C2xC4), (C2xQ8).51(C2xC4), (C2xD4).153(C2xC4), (C2xC4).1475(C2xD4), (C2xC4).500(C4oD4), (C2xC4).331(C22xC4), SmallGroup(128,314)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8:12SD16
G = < a,b,c | a8=b8=c2=1, bab-1=a5, ac=ca, cbc=b3 >
Subgroups: 152 in 84 conjugacy classes, 44 normal (40 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2xC4, C2xC4, D4, D4, Q8, C23, C42, C42, C22:C4, C4:C4, C4:C4, C2xC8, C2xC8, SD16, C22xC4, C2xD4, C2xQ8, C4xC8, C8:C4, C22:C8, D4:C4, Q8:C4, C4:C8, C4:C8, C4.Q8, C4xD4, C4xQ8, C22xC8, C2xSD16, C8:C8, D4:C8, Q8:C8, C8:2C8, C8xD4, C4xSD16, C8:4Q8, C8:12SD16
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, M4(2), SD16, C22xC4, C2xD4, C4oD4, C4xD4, C2xM4(2), C8oD4, C2xSD16, C4oD8, C8:9D4, C4xSD16, C8.26D4, C8:12SD16
►On 64 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)
(1 46 36 62 31 20 12 49)(2 43 37 59 32 17 13 54)(3 48 38 64 25 22 14 51)(4 45 39 61 26 19 15 56)(5 42 40 58 27 24 16 53)(6 47 33 63 28 21 9 50)(7 44 34 60 29 18 10 55)(8 41 35 57 30 23 11 52)
(9 33)(10 34)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(17 54)(18 55)(19 56)(20 49)(21 50)(22 51)(23 52)(24 53)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 64)
G:=sub<Sym(64)| (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), (1,46,36,62,31,20,12,49)(2,43,37,59,32,17,13,54)(3,48,38,64,25,22,14,51)(4,45,39,61,26,19,15,56)(5,42,40,58,27,24,16,53)(6,47,33,63,28,21,9,50)(7,44,34,60,29,18,10,55)(8,41,35,57,30,23,11,52), (9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,54)(18,55)(19,56)(20,49)(21,50)(22,51)(23,52)(24,53)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64)>;
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), (1,46,36,62,31,20,12,49)(2,43,37,59,32,17,13,54)(3,48,38,64,25,22,14,51)(4,45,39,61,26,19,15,56)(5,42,40,58,27,24,16,53)(6,47,33,63,28,21,9,50)(7,44,34,60,29,18,10,55)(8,41,35,57,30,23,11,52), (9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,54)(18,55)(19,56)(20,49)(21,50)(22,51)(23,52)(24,53)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64) );
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)], [(1,46,36,62,31,20,12,49),(2,43,37,59,32,17,13,54),(3,48,38,64,25,22,14,51),(4,45,39,61,26,19,15,56),(5,42,40,58,27,24,16,53),(6,47,33,63,28,21,9,50),(7,44,34,60,29,18,10,55),(8,41,35,57,30,23,11,52)], [(9,33),(10,34),(11,35),(12,36),(13,37),(14,38),(15,39),(16,40),(17,54),(18,55),(19,56),(20,49),(21,50),(22,51),(23,52),(24,53),(41,57),(42,58),(43,59),(44,60),(45,61),(46,62),(47,63),(48,64)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 8A | 8B | 8C | 8D | 8E | ··· | 8R | 8S | 8T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | SD16 | C4oD4 | M4(2) | C8oD4 | C4oD8 | C8.26D4 |
| kernel | C8:12SD16 | C8:C8 | D4:C8 | Q8:C8 | C8:2C8 | C8xD4 | C4xSD16 | C8:4Q8 | D4:C4 | Q8:C4 | C4.Q8 | C2xSD16 | C2xC8 | C8 | C2xC4 | D4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 4 | 2 |
Matrix representation of C8:12SD16 ►in GL4(F17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 9 |
| 12 | 5 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 8 |
| 0 | 0 | 2 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,8,0,0,0,0,9],[12,12,0,0,5,12,0,0,0,0,0,2,0,0,8,0],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16] >;
C8:12SD16 in GAP, Magma, Sage, TeX
C_8\rtimes_{12}{\rm SD}_{16} % in TeX
G:=Group("C8:12SD16"); // GroupNames label
G:=SmallGroup(128,314);
// by ID
G=gap.SmallGroup(128,314);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,268,1123,570,136,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^8=c^2=1,b*a*b^-1=a^5,a*c=c*a,c*b*c=b^3>;
// generators/relations