p-group, metabelian, nilpotent (class 4), monomial
Aliases: D4.11D8, D8.12D4, Q8.11D8, Q16.3D4, M4(2).19D4, M5(2).2C22, Q8○D8.C2, D4.C8⋊4C2, (C2×Q32)⋊3C2, C4.40(C2×D8), C8.69(C2×D4), C4○D4.12D4, Q32⋊C2⋊3C2, C4.27C22≀C2, D4.5D4⋊2C2, D8.C4⋊2C2, C8.17D4⋊2C2, (C2×C16).3C22, C4○D8.8C22, C8○D4.4C22, (C2×C8).234C23, C2.35(C22⋊D8), C22.7(C8⋊C22), C8.C4.4C22, (C2×Q16).45C22, (C2×C4).42(C2×D4), SmallGroup(128,927)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D4.11D8
G = < a,b,c,d | a4=b2=1, c8=d2=a2, bab=cac-1=dad-1=a-1, cbc-1=ab, dbd-1=a-1b, dcd-1=c7 >
Subgroups: 240 in 103 conjugacy classes, 32 normal (30 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×5], C22, C22 [×2], C8 [×2], C8 [×2], C2×C4, C2×C4 [×8], D4, D4 [×5], Q8, Q8 [×8], C16 [×2], C2×C8, C2×C8, M4(2), M4(2) [×2], D8, SD16 [×4], Q16, Q16 [×2], Q16 [×6], C2×Q8 [×5], C4○D4, C4○D4 [×6], C4.10D4, C8.C4, C2×C16, M5(2), SD32, Q32 [×3], C8○D4, C2×Q16 [×2], C2×Q16, C4○D8, C4○D8, C8.C22 [×4], 2- 1+4, D4.C8, D8.C4, C8.17D4, D4.5D4, C2×Q32, Q32⋊C2, Q8○D8, D4.11D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×2], C2×D4 [×3], C22≀C2, C2×D8, C8⋊C22, C22⋊D8, D4.11D8
Character table of D4.11D8
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 8A | 8B | 8C | 8D | 8E | 16A | 16B | 16C | 16D | 16E | 16F | |
| size | 1 | 1 | 2 | 4 | 8 | 2 | 2 | 4 | 8 | 8 | 8 | 16 | 2 | 2 | 4 | 8 | 16 | 4 | 4 | 4 | 4 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ9 | 2 | 2 | -2 | 0 | 2 | 2 | -2 | 0 | 0 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 2 | 0 | -2 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | -2 | 0 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | -2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | -2 | 0 | 2 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | -2 | -2 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | √2 | -√2 | orthogonal lifted from D8 |
| ρ16 | 2 | 2 | -2 | 2 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | -√2 | √2 | orthogonal lifted from D8 |
| ρ17 | 2 | 2 | -2 | -2 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | -√2 | √2 | orthogonal lifted from D8 |
| ρ18 | 2 | 2 | -2 | 2 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | √2 | -√2 | orthogonal lifted from D8 |
| ρ19 | 4 | 4 | 4 | 0 | 0 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ20 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | -ζ167+ζ165-ζ163+ζ16 | ζ1615-ζ169-ζ165+ζ163 | ζ167-ζ165+ζ163-ζ16 | -ζ1615+ζ169+ζ165-ζ163 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ21 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | ζ167-ζ165+ζ163-ζ16 | -ζ1615+ζ169+ζ165-ζ163 | -ζ167+ζ165-ζ163+ζ16 | ζ1615-ζ169-ζ165+ζ163 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ22 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | -ζ1615+ζ169+ζ165-ζ163 | -ζ167+ζ165-ζ163+ζ16 | ζ1615-ζ169-ζ165+ζ163 | ζ167-ζ165+ζ163-ζ16 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ23 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | ζ1615-ζ169-ζ165+ζ163 | ζ167-ζ165+ζ163-ζ16 | -ζ1615+ζ169+ζ165-ζ163 | -ζ167+ζ165-ζ163+ζ16 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 64 points
Generators in S64
(1 46 9 38)(2 39 10 47)(3 48 11 40)(4 41 12 33)(5 34 13 42)(6 43 14 35)(7 36 15 44)(8 45 16 37)(17 55 25 63)(18 64 26 56)(19 57 27 49)(20 50 28 58)(21 59 29 51)(22 52 30 60)(23 61 31 53)(24 54 32 62)
(1 58)(2 29)(3 60)(4 31)(5 62)(6 17)(7 64)(8 19)(9 50)(10 21)(11 52)(12 23)(13 54)(14 25)(15 56)(16 27)(18 36)(20 38)(22 40)(24 42)(26 44)(28 46)(30 48)(32 34)(33 53)(35 55)(37 57)(39 59)(41 61)(43 63)(45 49)(47 51)
(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 8 9 16)(2 15 10 7)(3 6 11 14)(4 13 12 5)(17 30 25 22)(18 21 26 29)(19 28 27 20)(23 24 31 32)(33 42 41 34)(35 40 43 48)(36 47 44 39)(37 38 45 46)(49 58 57 50)(51 56 59 64)(52 63 60 55)(53 54 61 62)
G:=sub<Sym(64)| (1,46,9,38)(2,39,10,47)(3,48,11,40)(4,41,12,33)(5,34,13,42)(6,43,14,35)(7,36,15,44)(8,45,16,37)(17,55,25,63)(18,64,26,56)(19,57,27,49)(20,50,28,58)(21,59,29,51)(22,52,30,60)(23,61,31,53)(24,54,32,62), (1,58)(2,29)(3,60)(4,31)(5,62)(6,17)(7,64)(8,19)(9,50)(10,21)(11,52)(12,23)(13,54)(14,25)(15,56)(16,27)(18,36)(20,38)(22,40)(24,42)(26,44)(28,46)(30,48)(32,34)(33,53)(35,55)(37,57)(39,59)(41,61)(43,63)(45,49)(47,51), (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), (1,8,9,16)(2,15,10,7)(3,6,11,14)(4,13,12,5)(17,30,25,22)(18,21,26,29)(19,28,27,20)(23,24,31,32)(33,42,41,34)(35,40,43,48)(36,47,44,39)(37,38,45,46)(49,58,57,50)(51,56,59,64)(52,63,60,55)(53,54,61,62)>;
G:=Group( (1,46,9,38)(2,39,10,47)(3,48,11,40)(4,41,12,33)(5,34,13,42)(6,43,14,35)(7,36,15,44)(8,45,16,37)(17,55,25,63)(18,64,26,56)(19,57,27,49)(20,50,28,58)(21,59,29,51)(22,52,30,60)(23,61,31,53)(24,54,32,62), (1,58)(2,29)(3,60)(4,31)(5,62)(6,17)(7,64)(8,19)(9,50)(10,21)(11,52)(12,23)(13,54)(14,25)(15,56)(16,27)(18,36)(20,38)(22,40)(24,42)(26,44)(28,46)(30,48)(32,34)(33,53)(35,55)(37,57)(39,59)(41,61)(43,63)(45,49)(47,51), (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), (1,8,9,16)(2,15,10,7)(3,6,11,14)(4,13,12,5)(17,30,25,22)(18,21,26,29)(19,28,27,20)(23,24,31,32)(33,42,41,34)(35,40,43,48)(36,47,44,39)(37,38,45,46)(49,58,57,50)(51,56,59,64)(52,63,60,55)(53,54,61,62) );
G=PermutationGroup([(1,46,9,38),(2,39,10,47),(3,48,11,40),(4,41,12,33),(5,34,13,42),(6,43,14,35),(7,36,15,44),(8,45,16,37),(17,55,25,63),(18,64,26,56),(19,57,27,49),(20,50,28,58),(21,59,29,51),(22,52,30,60),(23,61,31,53),(24,54,32,62)], [(1,58),(2,29),(3,60),(4,31),(5,62),(6,17),(7,64),(8,19),(9,50),(10,21),(11,52),(12,23),(13,54),(14,25),(15,56),(16,27),(18,36),(20,38),(22,40),(24,42),(26,44),(28,46),(30,48),(32,34),(33,53),(35,55),(37,57),(39,59),(41,61),(43,63),(45,49),(47,51)], [(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)], [(1,8,9,16),(2,15,10,7),(3,6,11,14),(4,13,12,5),(17,30,25,22),(18,21,26,29),(19,28,27,20),(23,24,31,32),(33,42,41,34),(35,40,43,48),(36,47,44,39),(37,38,45,46),(49,58,57,50),(51,56,59,64),(52,63,60,55),(53,54,61,62)])
Matrix representation of D4.11D8 ►in GL4(𝔽17) generated by
| 0 | 0 | 1 | 0 |
| 16 | 16 | 16 | 15 |
| 16 | 0 | 0 | 0 |
| 1 | 1 | 0 | 1 |
| 1 | 11 | 1 | 2 |
| 7 | 0 | 1 | 0 |
| 10 | 9 | 10 | 3 |
| 16 | 7 | 6 | 6 |
| 13 | 15 | 3 | 1 |
| 0 | 10 | 15 | 12 |
| 0 | 14 | 2 | 12 |
| 4 | 7 | 2 | 9 |
| 0 | 10 | 15 | 12 |
| 13 | 15 | 3 | 1 |
| 13 | 15 | 10 | 8 |
| 4 | 7 | 2 | 9 |
G:=sub<GL(4,GF(17))| [0,16,16,1,0,16,0,1,1,16,0,0,0,15,0,1],[1,7,10,16,11,0,9,7,1,1,10,6,2,0,3,6],[13,0,0,4,15,10,14,7,3,15,2,2,1,12,12,9],[0,13,13,4,10,15,15,7,15,3,10,2,12,1,8,9] >;
D4.11D8 in GAP, Magma, Sage, TeX
D_4._{11}D_8 % in TeX
G:=Group("D4.11D8"); // GroupNames label
G:=SmallGroup(128,927);
// by ID
G=gap.SmallGroup(128,927);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,422,352,1123,570,360,2804,718,172,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=1,c^8=d^2=a^2,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^7>;
// generators/relations