p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.90D4, C42.17Q8, C42.627C23, C4⋊C8.17C4, C8⋊2C8⋊26C2, C4.5(C4.Q8), C4.43(C8○D4), C22⋊C8.15C4, C4.99(C2×SD16), (C22×C4).29Q8, C4⋊C8.267C22, C23.52(C4⋊C4), C42.123(C2×C4), (C4×C8).236C22, (C2×C4).118SD16, (C22×C4).747D4, C22.5(C4.Q8), C4⋊M4(2).23C2, (C2×C42).226C22, C2.5(M4(2).C4), C42.12C4.39C2, C2.5(C42.6C22), (C2×C4⋊C8).19C2, C2.5(C2×C4.Q8), (C2×C4).34(C4⋊C4), (C2×C8).100(C2×C4), C22.84(C2×C4⋊C4), (C2×C4).154(C2×Q8), (C2×C4).1463(C2×D4), (C2×C4).509(C22×C4), (C22×C4).248(C2×C4), SmallGroup(128,302)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.90D4
G = < a,b,c,d | a4=b4=1, c4=a2, d2=a2b, ab=ba, cac-1=ab2, dad-1=a-1b2, bc=cb, bd=db, dcd-1=c3 >
Subgroups: 132 in 84 conjugacy classes, 54 normal (26 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C2×C8, C2×C8, M4(2), C22×C4, C4×C8, C22⋊C8, C4⋊C8, C4⋊C8, C2×C42, C22×C8, C2×M4(2), C8⋊2C8, C2×C4⋊C8, C4⋊M4(2), C42.12C4, C42.90D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, SD16, C22×C4, C2×D4, C2×Q8, C4.Q8, C2×C4⋊C4, C8○D4, C2×SD16, C42.6C22, C2×C4.Q8, M4(2).C4, C42.90D4
►On 64 points
(1 3 5 7)(2 50 6 54)(4 52 8 56)(9 59 13 63)(10 12 14 16)(11 61 15 57)(17 29 21 25)(18 20 22 24)(19 31 23 27)(26 28 30 32)(33 35 37 39)(34 42 38 46)(36 44 40 48)(41 43 45 47)(49 51 53 55)(58 60 62 64)
(1 39 55 45)(2 40 56 46)(3 33 49 47)(4 34 50 48)(5 35 51 41)(6 36 52 42)(7 37 53 43)(8 38 54 44)(9 17 57 27)(10 18 58 28)(11 19 59 29)(12 20 60 30)(13 21 61 31)(14 22 62 32)(15 23 63 25)(16 24 64 26)
(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 27 35 13 55 17 41 61)(2 30 36 16 56 20 42 64)(3 25 37 11 49 23 43 59)(4 28 38 14 50 18 44 62)(5 31 39 9 51 21 45 57)(6 26 40 12 52 24 46 60)(7 29 33 15 53 19 47 63)(8 32 34 10 54 22 48 58)
G:=sub<Sym(64)| (1,3,5,7)(2,50,6,54)(4,52,8,56)(9,59,13,63)(10,12,14,16)(11,61,15,57)(17,29,21,25)(18,20,22,24)(19,31,23,27)(26,28,30,32)(33,35,37,39)(34,42,38,46)(36,44,40,48)(41,43,45,47)(49,51,53,55)(58,60,62,64), (1,39,55,45)(2,40,56,46)(3,33,49,47)(4,34,50,48)(5,35,51,41)(6,36,52,42)(7,37,53,43)(8,38,54,44)(9,17,57,27)(10,18,58,28)(11,19,59,29)(12,20,60,30)(13,21,61,31)(14,22,62,32)(15,23,63,25)(16,24,64,26), (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), (1,27,35,13,55,17,41,61)(2,30,36,16,56,20,42,64)(3,25,37,11,49,23,43,59)(4,28,38,14,50,18,44,62)(5,31,39,9,51,21,45,57)(6,26,40,12,52,24,46,60)(7,29,33,15,53,19,47,63)(8,32,34,10,54,22,48,58)>;
G:=Group( (1,3,5,7)(2,50,6,54)(4,52,8,56)(9,59,13,63)(10,12,14,16)(11,61,15,57)(17,29,21,25)(18,20,22,24)(19,31,23,27)(26,28,30,32)(33,35,37,39)(34,42,38,46)(36,44,40,48)(41,43,45,47)(49,51,53,55)(58,60,62,64), (1,39,55,45)(2,40,56,46)(3,33,49,47)(4,34,50,48)(5,35,51,41)(6,36,52,42)(7,37,53,43)(8,38,54,44)(9,17,57,27)(10,18,58,28)(11,19,59,29)(12,20,60,30)(13,21,61,31)(14,22,62,32)(15,23,63,25)(16,24,64,26), (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), (1,27,35,13,55,17,41,61)(2,30,36,16,56,20,42,64)(3,25,37,11,49,23,43,59)(4,28,38,14,50,18,44,62)(5,31,39,9,51,21,45,57)(6,26,40,12,52,24,46,60)(7,29,33,15,53,19,47,63)(8,32,34,10,54,22,48,58) );
G=PermutationGroup([[(1,3,5,7),(2,50,6,54),(4,52,8,56),(9,59,13,63),(10,12,14,16),(11,61,15,57),(17,29,21,25),(18,20,22,24),(19,31,23,27),(26,28,30,32),(33,35,37,39),(34,42,38,46),(36,44,40,48),(41,43,45,47),(49,51,53,55),(58,60,62,64)], [(1,39,55,45),(2,40,56,46),(3,33,49,47),(4,34,50,48),(5,35,51,41),(6,36,52,42),(7,37,53,43),(8,38,54,44),(9,17,57,27),(10,18,58,28),(11,19,59,29),(12,20,60,30),(13,21,61,31),(14,22,62,32),(15,23,63,25),(16,24,64,26)], [(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)], [(1,27,35,13,55,17,41,61),(2,30,36,16,56,20,42,64),(3,25,37,11,49,23,43,59),(4,28,38,14,50,18,44,62),(5,31,39,9,51,21,45,57),(6,26,40,12,52,24,46,60),(7,29,33,15,53,19,47,63),(8,32,34,10,54,22,48,58)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | 4L | 8A | ··· | 8P | 8Q | 8R | 8S | 8T |
| order | 1 | 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 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | - | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | Q8 | D4 | Q8 | SD16 | C8○D4 | M4(2).C4 |
| kernel | C42.90D4 | C8⋊2C8 | C2×C4⋊C8 | C4⋊M4(2) | C42.12C4 | C22⋊C8 | C4⋊C8 | C42 | C42 | C22×C4 | C22×C4 | C2×C4 | C4 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 8 | 8 | 2 |
Matrix representation of C42.90D4 ►in GL4(𝔽17) generated by
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 5 | 12 | 0 | 0 |
| 5 | 5 | 0 | 0 |
| 0 | 0 | 0 | 15 |
| 0 | 0 | 8 | 0 |
| 0 | 13 | 0 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[5,5,0,0,12,5,0,0,0,0,0,8,0,0,15,0],[0,13,0,0,13,0,0,0,0,0,0,1,0,0,4,0] >;
C42.90D4 in GAP, Magma, Sage, TeX
C_4^2._{90}D_4 % in TeX
G:=Group("C4^2.90D4"); // GroupNames label
G:=SmallGroup(128,302);
// by ID
G=gap.SmallGroup(128,302);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,64,723,1123,136,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2,d^2=a^2*b,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations