p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C2×C8)⋊13D4, C8⋊2D4⋊4C2, C8⋊8D4⋊44C2, C8.120(C2×D4), C8.D4⋊4C2, (C2×D4).219D4, C4⋊C4.29C23, (C2×Q8).174D4, C23.78(C2×D4), (C2×C8).256C23, (C2×C4).264C24, (C2×D4).66C23, C4.158(C22×D4), (C2×Q8).54C23, C4.174(C4⋊D4), C2.17(D4○SD16), (C2×D8).161C22, C4⋊D4.20C22, C4.Q8.144C22, C22⋊Q8.20C22, C23.36D4⋊42C2, C22.14(C4⋊D4), (C22×C4).986C23, (C22×C8).261C22, (C2×Q16).156C22, C22.524(C22×D4), D4⋊C4.130C22, C22.31C24⋊5C2, Q8⋊C4.123C22, (C2×SD16).135C22, (C2×M4(2)).266C22, (C2×C8○D4)⋊3C2, (C2×C4○D8)⋊17C2, (C2×C4.Q8)⋊11C2, C4.31(C2×C4○D4), (C2×C4).132(C2×D4), C2.82(C2×C4⋊D4), (C2×C4).285(C4○D4), (C2×C4⋊C4).593C22, (C2×C4○D4).127C22, SmallGroup(128,1792)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×C8)⋊13D4
G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, ac=ca, dad=ab4, cbc-1=dbd=b3, dcd=c-1 >
Subgroups: 476 in 242 conjugacy classes, 100 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, D4⋊C4, Q8⋊C4, C4.Q8, C2×C4⋊C4, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22×C8, C22×C8, C2×M4(2), C2×M4(2), C8○D4, C2×D8, C2×SD16, C2×Q16, C4○D8, C2×C4○D4, C2×C4○D4, C23.36D4, C2×C4.Q8, C8⋊8D4, C8⋊2D4, C8.D4, C22.31C24, C2×C8○D4, C2×C4○D8, (C2×C8)⋊13D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C22×D4, C2×C4○D4, C2×C4⋊D4, D4○SD16, (C2×C8)⋊13D4
►On 64 points
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 9)(17 44)(18 45)(19 46)(20 47)(21 48)(22 41)(23 42)(24 43)(25 55)(26 56)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 61)(34 62)(35 63)(36 64)(37 57)(38 58)(39 59)(40 60)
(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 55 63)(2 41 56 58)(3 44 49 61)(4 47 50 64)(5 42 51 59)(6 45 52 62)(7 48 53 57)(8 43 54 60)(9 24 32 40)(10 19 25 35)(11 22 26 38)(12 17 27 33)(13 20 28 36)(14 23 29 39)(15 18 30 34)(16 21 31 37)
(2 4)(3 7)(6 8)(9 11)(10 14)(13 15)(17 33)(18 36)(19 39)(20 34)(21 37)(22 40)(23 35)(24 38)(25 29)(26 32)(28 30)(41 64)(42 59)(43 62)(44 57)(45 60)(46 63)(47 58)(48 61)(49 53)(50 56)(52 54)
G:=sub<Sym(64)| (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9)(17,44)(18,45)(19,46)(20,47)(21,48)(22,41)(23,42)(24,43)(25,55)(26,56)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,61)(34,62)(35,63)(36,64)(37,57)(38,58)(39,59)(40,60), (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), (1,46,55,63)(2,41,56,58)(3,44,49,61)(4,47,50,64)(5,42,51,59)(6,45,52,62)(7,48,53,57)(8,43,54,60)(9,24,32,40)(10,19,25,35)(11,22,26,38)(12,17,27,33)(13,20,28,36)(14,23,29,39)(15,18,30,34)(16,21,31,37), (2,4)(3,7)(6,8)(9,11)(10,14)(13,15)(17,33)(18,36)(19,39)(20,34)(21,37)(22,40)(23,35)(24,38)(25,29)(26,32)(28,30)(41,64)(42,59)(43,62)(44,57)(45,60)(46,63)(47,58)(48,61)(49,53)(50,56)(52,54)>;
G:=Group( (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9)(17,44)(18,45)(19,46)(20,47)(21,48)(22,41)(23,42)(24,43)(25,55)(26,56)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,61)(34,62)(35,63)(36,64)(37,57)(38,58)(39,59)(40,60), (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), (1,46,55,63)(2,41,56,58)(3,44,49,61)(4,47,50,64)(5,42,51,59)(6,45,52,62)(7,48,53,57)(8,43,54,60)(9,24,32,40)(10,19,25,35)(11,22,26,38)(12,17,27,33)(13,20,28,36)(14,23,29,39)(15,18,30,34)(16,21,31,37), (2,4)(3,7)(6,8)(9,11)(10,14)(13,15)(17,33)(18,36)(19,39)(20,34)(21,37)(22,40)(23,35)(24,38)(25,29)(26,32)(28,30)(41,64)(42,59)(43,62)(44,57)(45,60)(46,63)(47,58)(48,61)(49,53)(50,56)(52,54) );
G=PermutationGroup([[(1,10),(2,11),(3,12),(4,13),(5,14),(6,15),(7,16),(8,9),(17,44),(18,45),(19,46),(20,47),(21,48),(22,41),(23,42),(24,43),(25,55),(26,56),(27,49),(28,50),(29,51),(30,52),(31,53),(32,54),(33,61),(34,62),(35,63),(36,64),(37,57),(38,58),(39,59),(40,60)], [(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)], [(1,46,55,63),(2,41,56,58),(3,44,49,61),(4,47,50,64),(5,42,51,59),(6,45,52,62),(7,48,53,57),(8,43,54,60),(9,24,32,40),(10,19,25,35),(11,22,26,38),(12,17,27,33),(13,20,28,36),(14,23,29,39),(15,18,30,34),(16,21,31,37)], [(2,4),(3,7),(6,8),(9,11),(10,14),(13,15),(17,33),(18,36),(19,39),(20,34),(21,37),(22,40),(23,35),(24,38),(25,29),(26,32),(28,30),(41,64),(42,59),(43,62),(44,57),(45,60),(46,63),(47,58),(48,61),(49,53),(50,56),(52,54)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C4○D4 | D4○SD16 |
| kernel | (C2×C8)⋊13D4 | C23.36D4 | C2×C4.Q8 | C8⋊8D4 | C8⋊2D4 | C8.D4 | C22.31C24 | C2×C8○D4 | C2×C4○D8 | C2×C8 | C2×D4 | C2×Q8 | C2×C4 | C2 |
| # reps | 1 | 2 | 1 | 4 | 2 | 2 | 2 | 1 | 1 | 4 | 3 | 1 | 4 | 4 |
Matrix representation of (C2×C8)⋊13D4 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 2 |
| 0 | 0 | 1 | 0 | 1 | 1 |
| 0 | 0 | 16 | 1 | 0 | 16 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 5 | 7 | 0 | 0 |
| 0 | 0 | 0 | 5 | 12 | 5 |
| 0 | 0 | 12 | 5 | 12 | 12 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 3 | 0 | 2 |
| 0 | 0 | 0 | 10 | 16 | 1 |
| 0 | 0 | 10 | 6 | 0 | 9 |
| 0 | 0 | 10 | 7 | 10 | 16 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 16 | 0 | 0 | 16 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,16,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,2,1,16,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,5,0,12,0,0,7,7,5,5,0,0,0,0,12,12,0,0,0,0,5,12],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,8,0,10,10,0,0,3,10,6,7,0,0,0,16,0,10,0,0,2,1,9,16],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,16,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;
(C2×C8)⋊13D4 in GAP, Magma, Sage, TeX
(C_2\times C_8)\rtimes_{13}D_4 % in TeX
G:=Group("(C2xC8):13D4"); // GroupNames label
G:=SmallGroup(128,1792);
// by ID
G=gap.SmallGroup(128,1792);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,521,248,2804,172]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^4=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a*b^4,c*b*c^-1=d*b*d=b^3,d*c*d=c^-1>;
// generators/relations