p-group, metabelian, nilpotent (class 4), monomial
Aliases: C16.19D4, C22⋊1Q32, C23.27D8, (C2×Q32)⋊5C2, C16⋊3C4⋊3C2, (C2×C4).60D8, C2.6(C2×Q32), (C2×C8).235D4, C8.100(C2×D4), C2.Q32⋊1C2, C8.45(C4○D4), C4.18(C4○D8), C2.11(C4○D16), C2.19(C8⋊7D4), C4.91(C4⋊D4), (C2×C8).521C23, C2.D8.6C22, (C2×C16).82C22, (C22×C16).11C2, C8.18D4.4C2, C22.107(C2×D8), (C22×C4).591D4, (C2×Q16).6C22, (C22×C8).531C22, (C2×C4).789(C2×D4), SmallGroup(128,948)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C16.19D4
G = < a,b,c | a16=b4=1, c2=a8, bab-1=cac-1=a-1, cbc-1=a8b-1 >
Subgroups: 168 in 77 conjugacy classes, 34 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, Q8, C23, C16, C16, C22⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C22×C4, C2×Q8, Q8⋊C4, C2.D8, C2×C16, C2×C16, Q32, C22⋊Q8, C22×C8, C2×Q16, C2.Q32, C16⋊3C4, C8.18D4, C22×C16, C2×Q32, C16.19D4
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C4○D4, Q32, C4⋊D4, C2×D8, C4○D8, C8⋊7D4, C2×Q32, C4○D16, C16.19D4
►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 29 52 42)(2 28 53 41)(3 27 54 40)(4 26 55 39)(5 25 56 38)(6 24 57 37)(7 23 58 36)(8 22 59 35)(9 21 60 34)(10 20 61 33)(11 19 62 48)(12 18 63 47)(13 17 64 46)(14 32 49 45)(15 31 50 44)(16 30 51 43)
(1 34 9 42)(2 33 10 41)(3 48 11 40)(4 47 12 39)(5 46 13 38)(6 45 14 37)(7 44 15 36)(8 43 16 35)(17 64 25 56)(18 63 26 55)(19 62 27 54)(20 61 28 53)(21 60 29 52)(22 59 30 51)(23 58 31 50)(24 57 32 49)
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), (1,29,52,42)(2,28,53,41)(3,27,54,40)(4,26,55,39)(5,25,56,38)(6,24,57,37)(7,23,58,36)(8,22,59,35)(9,21,60,34)(10,20,61,33)(11,19,62,48)(12,18,63,47)(13,17,64,46)(14,32,49,45)(15,31,50,44)(16,30,51,43), (1,34,9,42)(2,33,10,41)(3,48,11,40)(4,47,12,39)(5,46,13,38)(6,45,14,37)(7,44,15,36)(8,43,16,35)(17,64,25,56)(18,63,26,55)(19,62,27,54)(20,61,28,53)(21,60,29,52)(22,59,30,51)(23,58,31,50)(24,57,32,49)>;
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), (1,29,52,42)(2,28,53,41)(3,27,54,40)(4,26,55,39)(5,25,56,38)(6,24,57,37)(7,23,58,36)(8,22,59,35)(9,21,60,34)(10,20,61,33)(11,19,62,48)(12,18,63,47)(13,17,64,46)(14,32,49,45)(15,31,50,44)(16,30,51,43), (1,34,9,42)(2,33,10,41)(3,48,11,40)(4,47,12,39)(5,46,13,38)(6,45,14,37)(7,44,15,36)(8,43,16,35)(17,64,25,56)(18,63,26,55)(19,62,27,54)(20,61,28,53)(21,60,29,52)(22,59,30,51)(23,58,31,50)(24,57,32,49) );
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)], [(1,29,52,42),(2,28,53,41),(3,27,54,40),(4,26,55,39),(5,25,56,38),(6,24,57,37),(7,23,58,36),(8,22,59,35),(9,21,60,34),(10,20,61,33),(11,19,62,48),(12,18,63,47),(13,17,64,46),(14,32,49,45),(15,31,50,44),(16,30,51,43)], [(1,34,9,42),(2,33,10,41),(3,48,11,40),(4,47,12,39),(5,46,13,38),(6,45,14,37),(7,44,15,36),(8,43,16,35),(17,64,25,56),(18,63,26,55),(19,62,27,54),(20,61,28,53),(21,60,29,52),(22,59,30,51),(23,58,31,50),(24,57,32,49)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | ··· | 8H | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 16 | 16 | 16 | 16 | 2 | ··· | 2 | 2 | ··· | 2 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C4○D4 | D8 | D8 | C4○D8 | Q32 | C4○D16 |
| kernel | C16.19D4 | C2.Q32 | C16⋊3C4 | C8.18D4 | C22×C16 | C2×Q32 | C16 | C2×C8 | C22×C4 | C8 | C2×C4 | C23 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 8 | 8 |
Matrix representation of C16.19D4 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 6 |
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,3,0,0,0,0,6],[0,16,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,0,16,0,0,1,0] >;
C16.19D4 in GAP, Magma, Sage, TeX
C_{16}._{19}D_4 % in TeX
G:=Group("C16.19D4"); // GroupNames label
G:=SmallGroup(128,948);
// by ID
G=gap.SmallGroup(128,948);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,288,422,1123,360,4037,124]);
// Polycyclic
G:=Group<a,b,c|a^16=b^4=1,c^2=a^8,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^8*b^-1>;
// generators/relations