p-group, metabelian, nilpotent (class 4), monomial
Aliases: C16⋊8D4, C22⋊1SD32, C23.28D8, C16⋊4C4⋊6C2, (C2×C4).61D8, C2.D16⋊2C2, (C2×C8).236D4, C8.101(C2×D4), C8⋊7D4.4C2, C2.9(C2×SD32), (C22×C16)⋊11C2, C2.Q32⋊2C2, (C2×SD32)⋊15C2, C4.19(C4○D8), C8.46(C4○D4), C8.18D4⋊2C2, (C2×D8).6C22, C2.12(C4○D16), C4.92(C4⋊D4), C2.20(C8⋊7D4), (C2×C8).522C23, C2.D8.7C22, (C2×C16).98C22, (C22×C4).592D4, C22.108(C2×D8), (C2×Q16).7C22, (C22×C8).532C22, (C2×C4).790(C2×D4), SmallGroup(128,949)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C16⋊8D4
G = < a,b,c | a16=b4=c2=1, bab-1=cac=a7, cbc=b-1 >
Subgroups: 216 in 82 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C16, C16, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, Q16, C22×C4, C2×D4, C2×Q8, D4⋊C4, Q8⋊C4, C2.D8, C2×C16, C2×C16, SD32, C4⋊D4, C22⋊Q8, C22×C8, C2×D8, C2×Q16, C2.D16, C2.Q32, C16⋊4C4, C8⋊7D4, C8.18D4, C22×C16, C2×SD32, C16⋊8D4
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C4○D4, SD32, C4⋊D4, C2×D8, C4○D8, C8⋊7D4, C2×SD32, C4○D16, C16⋊8D4
►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 63 27 38)(2 54 28 45)(3 61 29 36)(4 52 30 43)(5 59 31 34)(6 50 32 41)(7 57 17 48)(8 64 18 39)(9 55 19 46)(10 62 20 37)(11 53 21 44)(12 60 22 35)(13 51 23 42)(14 58 24 33)(15 49 25 40)(16 56 26 47)
(2 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(17 21)(18 28)(20 26)(22 24)(23 31)(25 29)(30 32)(33 60)(34 51)(35 58)(36 49)(37 56)(38 63)(39 54)(40 61)(41 52)(42 59)(43 50)(44 57)(45 64)(46 55)(47 62)(48 53)
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), (1,63,27,38)(2,54,28,45)(3,61,29,36)(4,52,30,43)(5,59,31,34)(6,50,32,41)(7,57,17,48)(8,64,18,39)(9,55,19,46)(10,62,20,37)(11,53,21,44)(12,60,22,35)(13,51,23,42)(14,58,24,33)(15,49,25,40)(16,56,26,47), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,21)(18,28)(20,26)(22,24)(23,31)(25,29)(30,32)(33,60)(34,51)(35,58)(36,49)(37,56)(38,63)(39,54)(40,61)(41,52)(42,59)(43,50)(44,57)(45,64)(46,55)(47,62)(48,53)>;
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), (1,63,27,38)(2,54,28,45)(3,61,29,36)(4,52,30,43)(5,59,31,34)(6,50,32,41)(7,57,17,48)(8,64,18,39)(9,55,19,46)(10,62,20,37)(11,53,21,44)(12,60,22,35)(13,51,23,42)(14,58,24,33)(15,49,25,40)(16,56,26,47), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,21)(18,28)(20,26)(22,24)(23,31)(25,29)(30,32)(33,60)(34,51)(35,58)(36,49)(37,56)(38,63)(39,54)(40,61)(41,52)(42,59)(43,50)(44,57)(45,64)(46,55)(47,62)(48,53) );
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)], [(1,63,27,38),(2,54,28,45),(3,61,29,36),(4,52,30,43),(5,59,31,34),(6,50,32,41),(7,57,17,48),(8,64,18,39),(9,55,19,46),(10,62,20,37),(11,53,21,44),(12,60,22,35),(13,51,23,42),(14,58,24,33),(15,49,25,40),(16,56,26,47)], [(2,8),(3,15),(4,6),(5,13),(7,11),(10,16),(12,14),(17,21),(18,28),(20,26),(22,24),(23,31),(25,29),(30,32),(33,60),(34,51),(35,58),(36,49),(37,56),(38,63),(39,54),(40,61),(41,52),(42,59),(43,50),(44,57),(45,64),(46,55),(47,62),(48,53)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 8A | ··· | 8H | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 16 | 2 | 2 | 2 | 2 | 16 | 16 | 16 | 2 | ··· | 2 | 2 | ··· | 2 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C4○D4 | D8 | D8 | C4○D8 | SD32 | C4○D16 |
| kernel | C16⋊8D4 | C2.D16 | C2.Q32 | C16⋊4C4 | C8⋊7D4 | C8.18D4 | C22×C16 | C2×SD32 | C16 | C2×C8 | C22×C4 | C8 | C2×C4 | C23 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 8 | 8 |
Matrix representation of C16⋊8D4 ►in GL4(𝔽17) generated by
| 14 | 3 | 0 | 0 |
| 14 | 14 | 0 | 0 |
| 0 | 0 | 6 | 15 |
| 0 | 0 | 1 | 8 |
| 0 | 4 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 8 |
| 0 | 0 | 0 | 13 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 16 | 16 |
G:=sub<GL(4,GF(17))| [14,14,0,0,3,14,0,0,0,0,6,1,0,0,15,8],[0,4,0,0,4,0,0,0,0,0,4,0,0,0,8,13],[1,0,0,0,0,16,0,0,0,0,1,16,0,0,0,16] >;
C16⋊8D4 in GAP, Magma, Sage, TeX
C_{16}\rtimes_8D_4 % in TeX
G:=Group("C16:8D4"); // GroupNames label
G:=SmallGroup(128,949);
// by ID
G=gap.SmallGroup(128,949);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,736,422,1123,360,4037,124]);
// Polycyclic
G:=Group<a,b,c|a^16=b^4=c^2=1,b*a*b^-1=c*a*c=a^7,c*b*c=b^-1>;
// generators/relations