p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4.81(C4xD4), C4:C4.310D4, (C2xC4):12SD16, (C2xSD16):14C4, (C2xD4).204D4, C4.1(C4:D4), C2.10(C4xSD16), D4.2(C22:C4), C22.151(C4xD4), (C22xC4).687D4, C23.768(C2xD4), C22.4Q16:40C2, C2.2(D4.2D4), C2.5(C22:SD16), C22.93C22wrC2, C2.6(D4.7D4), C2.2(D4.D4), C22.55(C4oD8), (C22xC8).35C22, (C22xSD16).6C2, C22.58(C2xSD16), C22.76(C8:C22), (C2xC42).277C22, C2.10(SD16:C4), (C22xQ8).13C22, C22.114(C4:D4), (C22xC4).1364C23, C23.67C23:2C2, C22.7C42:27C2, C4.63(C22.D4), (C22xD4).460C22, C22.65(C8.C22), C2.16(C23.23D4), (C2xC8):20(C2xC4), (C2xQ8):6(C2xC4), (C2xC4xD4).21C2, (C2xQ8:C4):5C2, (C2xC4).996(C2xD4), C4.11(C2xC22:C4), (C2xD4:C4).3C2, (C2xD4).164(C2xC4), (C2xC4).561(C4oD4), (C2xC4:C4).765C22, (C2xC4).382(C22xC4), SmallGroup(128,609)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2xSD16):14C4
G = < a,b,c,d | a2=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd-1=ab-1, dcd-1=b4c >
Subgroups: 436 in 201 conjugacy classes, 66 normal (44 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2xC4, C2xC4, C2xC4, D4, D4, Q8, C23, C23, C42, C22:C4, C4:C4, C4:C4, C2xC8, C2xC8, SD16, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C2xQ8, C24, C2.C42, D4:C4, Q8:C4, C2xC42, C2xC22:C4, C2xC4:C4, C4xD4, C22xC8, C2xSD16, C2xSD16, C23xC4, C22xD4, C22xQ8, C22.7C42, C22.4Q16, C23.67C23, C2xD4:C4, C2xQ8:C4, C2xC4xD4, C22xSD16, (C2xSD16):14C4
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22:C4, SD16, C22xC4, C2xD4, C4oD4, C2xC22:C4, C4xD4, C22wrC2, C4:D4, C22.D4, C2xSD16, C4oD8, C8:C22, C8.C22, C23.23D4, C4xSD16, SD16:C4, C22:SD16, D4.7D4, D4.D4, D4.2D4, (C2xSD16):14C4
►On 64 points
(1 53)(2 54)(3 55)(4 56)(5 49)(6 50)(7 51)(8 52)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 38)(18 39)(19 40)(20 33)(21 34)(22 35)(23 36)(24 37)(25 58)(26 59)(27 60)(28 61)(29 62)(30 63)(31 64)(32 57)
(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 49)(2 52)(3 55)(4 50)(5 53)(6 56)(7 51)(8 54)(9 45)(10 48)(11 43)(12 46)(13 41)(14 44)(15 47)(16 42)(17 38)(18 33)(19 36)(20 39)(21 34)(22 37)(23 40)(24 35)(25 60)(26 63)(27 58)(28 61)(29 64)(30 59)(31 62)(32 57)
(1 43 59 38)(2 10 60 24)(3 41 61 36)(4 16 62 22)(5 47 63 34)(6 14 64 20)(7 45 57 40)(8 12 58 18)(9 28 23 55)(11 26 17 53)(13 32 19 51)(15 30 21 49)(25 39 52 44)(27 37 54 42)(29 35 56 48)(31 33 50 46)
G:=sub<Sym(64)| (1,53)(2,54)(3,55)(4,56)(5,49)(6,50)(7,51)(8,52)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,38)(18,39)(19,40)(20,33)(21,34)(22,35)(23,36)(24,37)(25,58)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,57), (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), (1,49)(2,52)(3,55)(4,50)(5,53)(6,56)(7,51)(8,54)(9,45)(10,48)(11,43)(12,46)(13,41)(14,44)(15,47)(16,42)(17,38)(18,33)(19,36)(20,39)(21,34)(22,37)(23,40)(24,35)(25,60)(26,63)(27,58)(28,61)(29,64)(30,59)(31,62)(32,57), (1,43,59,38)(2,10,60,24)(3,41,61,36)(4,16,62,22)(5,47,63,34)(6,14,64,20)(7,45,57,40)(8,12,58,18)(9,28,23,55)(11,26,17,53)(13,32,19,51)(15,30,21,49)(25,39,52,44)(27,37,54,42)(29,35,56,48)(31,33,50,46)>;
G:=Group( (1,53)(2,54)(3,55)(4,56)(5,49)(6,50)(7,51)(8,52)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,38)(18,39)(19,40)(20,33)(21,34)(22,35)(23,36)(24,37)(25,58)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,57), (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), (1,49)(2,52)(3,55)(4,50)(5,53)(6,56)(7,51)(8,54)(9,45)(10,48)(11,43)(12,46)(13,41)(14,44)(15,47)(16,42)(17,38)(18,33)(19,36)(20,39)(21,34)(22,37)(23,40)(24,35)(25,60)(26,63)(27,58)(28,61)(29,64)(30,59)(31,62)(32,57), (1,43,59,38)(2,10,60,24)(3,41,61,36)(4,16,62,22)(5,47,63,34)(6,14,64,20)(7,45,57,40)(8,12,58,18)(9,28,23,55)(11,26,17,53)(13,32,19,51)(15,30,21,49)(25,39,52,44)(27,37,54,42)(29,35,56,48)(31,33,50,46) );
G=PermutationGroup([[(1,53),(2,54),(3,55),(4,56),(5,49),(6,50),(7,51),(8,52),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(17,38),(18,39),(19,40),(20,33),(21,34),(22,35),(23,36),(24,37),(25,58),(26,59),(27,60),(28,61),(29,62),(30,63),(31,64),(32,57)], [(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)], [(1,49),(2,52),(3,55),(4,50),(5,53),(6,56),(7,51),(8,54),(9,45),(10,48),(11,43),(12,46),(13,41),(14,44),(15,47),(16,42),(17,38),(18,33),(19,36),(20,39),(21,34),(22,37),(23,40),(24,35),(25,60),(26,63),(27,58),(28,61),(29,64),(30,59),(31,62),(32,57)], [(1,43,59,38),(2,10,60,24),(3,41,61,36),(4,16,62,22),(5,47,63,34),(6,14,64,20),(7,45,57,40),(8,12,58,18),(9,28,23,55),(11,26,17,53),(13,32,19,51),(15,30,21,49),(25,39,52,44),(27,37,54,42),(29,35,56,48),(31,33,50,46)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4N | 4O | 4P | 4Q | 4R | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | SD16 | C4oD4 | C4oD8 | C8:C22 | C8.C22 |
| kernel | (C2xSD16):14C4 | C22.7C42 | C22.4Q16 | C23.67C23 | C2xD4:C4 | C2xQ8:C4 | C2xC4xD4 | C22xSD16 | C2xSD16 | C4:C4 | C22xC4 | C2xD4 | C2xC4 | C2xC4 | C22 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 |
Matrix representation of (C2xSD16):14C4 ►in GL5(F17)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 12 | 0 | 0 |
| 0 | 5 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 1 | 0 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,12,5,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,16,0],[16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1],[4,0,0,0,0,0,0,13,0,0,0,13,0,0,0,0,0,0,16,0,0,0,0,0,16] >;
(C2xSD16):14C4 in GAP, Magma, Sage, TeX
(C_2\times {\rm SD}_{16})\rtimes_{14}C_4 % in TeX
G:=Group("(C2xSD16):14C4"); // GroupNames label
G:=SmallGroup(128,609);
// by ID
G=gap.SmallGroup(128,609);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,1018,521,248,1411,718,172,2028,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d^-1=a*b^-1,d*c*d^-1=b^4*c>;
// generators/relations