p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4⋊C4.106D4, (C2×C8).162D4, (C2×D4).12Q8, (C2×C4).37SD16, C2.20(C8⋊8D4), C2.15(C8⋊2D4), C2.8(D4⋊2Q8), (C22×C4).154D4, C23.922(C2×D4), C2.10(D4.Q8), C4.35(C22⋊Q8), C4.153(C4⋊D4), C22.4Q16⋊25C2, C2.15(C4⋊SD16), (C22×C8).80C22, C4.33(C42⋊2C2), C2.20(D4.2D4), C22.116(C4○D8), (C2×C42).376C22, C22.101(C2×SD16), (C22×D4).89C22, C2.6(C23.Q8), C22.243(C4⋊D4), C22.145(C8⋊C22), (C22×C4).1456C23, C23.65C23⋊7C2, C22.109(C22⋊Q8), C24.3C22.17C2, (C2×C4⋊C8)⋊34C2, (C2×C4.Q8)⋊22C2, (C2×C4).283(C2×Q8), (C2×C4).1055(C2×D4), (C2×D4⋊C4).15C2, (C2×C4).620(C4○D4), (C2×C4⋊C4).141C22, SmallGroup(128,797)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4⋊C4.106D4
G = < a,b,c,d | a4=b4=c4=1, d2=a, bab-1=cac-1=a-1, ad=da, cbc-1=a2b-1, dbd-1=a-1b, dcd-1=ac-1 >
Subgroups: 344 in 142 conjugacy classes, 52 normal (44 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2.C42, D4⋊C4, C4⋊C8, C4.Q8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C22×D4, C22.4Q16, C23.65C23, C24.3C22, C2×D4⋊C4, C2×C4⋊C8, C2×C4.Q8, C4⋊C4.106D4
Quotients: C1, C2, C22, D4, Q8, C23, SD16, C2×D4, C2×Q8, C4○D4, C4⋊D4, C22⋊Q8, C42⋊2C2, C2×SD16, C4○D8, C8⋊C22, C23.Q8, C4⋊SD16, D4.2D4, C8⋊8D4, C8⋊2D4, D4⋊2Q8, D4.Q8, C4⋊C4.106D4
►On 64 points
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)
(1 61 46 16)(2 64 47 11)(3 59 48 14)(4 62 41 9)(5 57 42 12)(6 60 43 15)(7 63 44 10)(8 58 45 13)(17 55 26 40)(18 50 27 35)(19 53 28 38)(20 56 29 33)(21 51 30 36)(22 54 31 39)(23 49 32 34)(24 52 25 37)
(1 53 54 8)(2 7 55 52)(3 51 56 6)(4 5 49 50)(9 61 23 31)(10 30 24 60)(11 59 17 29)(12 28 18 58)(13 57 19 27)(14 26 20 64)(15 63 21 25)(16 32 22 62)(33 43 48 36)(34 35 41 42)(37 47 44 40)(38 39 45 46)
(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)
G:=sub<Sym(64)| (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (1,61,46,16)(2,64,47,11)(3,59,48,14)(4,62,41,9)(5,57,42,12)(6,60,43,15)(7,63,44,10)(8,58,45,13)(17,55,26,40)(18,50,27,35)(19,53,28,38)(20,56,29,33)(21,51,30,36)(22,54,31,39)(23,49,32,34)(24,52,25,37), (1,53,54,8)(2,7,55,52)(3,51,56,6)(4,5,49,50)(9,61,23,31)(10,30,24,60)(11,59,17,29)(12,28,18,58)(13,57,19,27)(14,26,20,64)(15,63,21,25)(16,32,22,62)(33,43,48,36)(34,35,41,42)(37,47,44,40)(38,39,45,46), (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)>;
G:=Group( (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (1,61,46,16)(2,64,47,11)(3,59,48,14)(4,62,41,9)(5,57,42,12)(6,60,43,15)(7,63,44,10)(8,58,45,13)(17,55,26,40)(18,50,27,35)(19,53,28,38)(20,56,29,33)(21,51,30,36)(22,54,31,39)(23,49,32,34)(24,52,25,37), (1,53,54,8)(2,7,55,52)(3,51,56,6)(4,5,49,50)(9,61,23,31)(10,30,24,60)(11,59,17,29)(12,28,18,58)(13,57,19,27)(14,26,20,64)(15,63,21,25)(16,32,22,62)(33,43,48,36)(34,35,41,42)(37,47,44,40)(38,39,45,46), (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) );
G=PermutationGroup([[(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64)], [(1,61,46,16),(2,64,47,11),(3,59,48,14),(4,62,41,9),(5,57,42,12),(6,60,43,15),(7,63,44,10),(8,58,45,13),(17,55,26,40),(18,50,27,35),(19,53,28,38),(20,56,29,33),(21,51,30,36),(22,54,31,39),(23,49,32,34),(24,52,25,37)], [(1,53,54,8),(2,7,55,52),(3,51,56,6),(4,5,49,50),(9,61,23,31),(10,30,24,60),(11,59,17,29),(12,28,18,58),(13,57,19,27),(14,26,20,64),(15,63,21,25),(16,32,22,62),(33,43,48,36),(34,35,41,42),(37,47,44,40),(38,39,45,46)], [(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)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | Q8 | SD16 | C4○D4 | C4○D8 | C8⋊C22 |
| kernel | C4⋊C4.106D4 | C22.4Q16 | C23.65C23 | C24.3C22 | C2×D4⋊C4 | C2×C4⋊C8 | C2×C4.Q8 | C4⋊C4 | C2×C8 | C22×C4 | C2×D4 | C2×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 6 | 4 | 2 |
Matrix representation of C4⋊C4.106D4 ►in GL6(𝔽17)
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 3 | 14 | 0 | 0 | 0 | 0 |
| 14 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 15 | 0 | 0 |
| 0 | 0 | 7 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 12 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 9 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 5 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,14,0,0,0,0,14,14,0,0,0,0,0,0,9,7,0,0,0,0,15,8,0,0,0,0,0,0,0,13,0,0,0,0,13,0],[12,12,0,0,0,0,12,5,0,0,0,0,0,0,1,9,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[12,5,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C4⋊C4.106D4 in GAP, Magma, Sage, TeX
C_4\rtimes C_4._{106}D_4 % in TeX
G:=Group("C4:C4.106D4"); // GroupNames label
G:=SmallGroup(128,797);
// by ID
G=gap.SmallGroup(128,797);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,168,141,64,422,387,2019,521,248,2804,718,172,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^2*b^-1,d*b*d^-1=a^-1*b,d*c*d^-1=a*c^-1>;
// generators/relations