p-group, metabelian, nilpotent (class 2), monomial
Aliases: C16⋊6D4, C4⋊1M5(2), C4⋊C16⋊18C2, (C4×C16)⋊16C2, C4⋊C4.10C8, C4⋊C8.23C4, C2.17(C8×D4), (C8×D4).18C2, (C4×D4).20C4, (C2×D4).10C8, C4.177(C4×D4), C8.140(C2×D4), C22⋊C4.6C8, C22⋊C16⋊16C2, C2.8(D4○C16), C4.59(C8○D4), C22⋊C8.22C4, C23.11(C2×C8), C8.103(C4○D4), (C2×M5(2))⋊20C2, C42.273(C2×C4), (C4×C8).376C22, (C2×C8).634C23, (C2×C16).55C22, C2.10(C2×M5(2)), C22.53(C22×C8), (C22×C8).420C22, (C2×C4).30(C2×C8), (C2×C8).163(C2×C4), (C2×C4).619(C22×C4), (C22×C4).291(C2×C4), SmallGroup(128,901)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C16⋊6D4
G = < a,b,c | a16=b4=c2=1, ab=ba, cac=a9, cbc=b-1 >
Subgroups: 116 in 81 conjugacy classes, 50 normal (26 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, C23, C16, C16, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, C22×C4, C2×D4, C4×C8, C22⋊C8, C4⋊C8, C2×C16, C2×C16, M5(2), C4×D4, C22×C8, C4×C16, C22⋊C16, C4⋊C16, C8×D4, C2×M5(2), C16⋊6D4
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C2×C8, C22×C4, C2×D4, C4○D4, M5(2), C4×D4, C22×C8, C8○D4, C8×D4, C2×M5(2), D4○C16, C16⋊6D4
►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 53 24 45)(2 54 25 46)(3 55 26 47)(4 56 27 48)(5 57 28 33)(6 58 29 34)(7 59 30 35)(8 60 31 36)(9 61 32 37)(10 62 17 38)(11 63 18 39)(12 64 19 40)(13 49 20 41)(14 50 21 42)(15 51 22 43)(16 52 23 44)
(1 45)(2 38)(3 47)(4 40)(5 33)(6 42)(7 35)(8 44)(9 37)(10 46)(11 39)(12 48)(13 41)(14 34)(15 43)(16 36)(17 54)(18 63)(19 56)(20 49)(21 58)(22 51)(23 60)(24 53)(25 62)(26 55)(27 64)(28 57)(29 50)(30 59)(31 52)(32 61)
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), (1,53,24,45)(2,54,25,46)(3,55,26,47)(4,56,27,48)(5,57,28,33)(6,58,29,34)(7,59,30,35)(8,60,31,36)(9,61,32,37)(10,62,17,38)(11,63,18,39)(12,64,19,40)(13,49,20,41)(14,50,21,42)(15,51,22,43)(16,52,23,44), (1,45)(2,38)(3,47)(4,40)(5,33)(6,42)(7,35)(8,44)(9,37)(10,46)(11,39)(12,48)(13,41)(14,34)(15,43)(16,36)(17,54)(18,63)(19,56)(20,49)(21,58)(22,51)(23,60)(24,53)(25,62)(26,55)(27,64)(28,57)(29,50)(30,59)(31,52)(32,61)>;
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), (1,53,24,45)(2,54,25,46)(3,55,26,47)(4,56,27,48)(5,57,28,33)(6,58,29,34)(7,59,30,35)(8,60,31,36)(9,61,32,37)(10,62,17,38)(11,63,18,39)(12,64,19,40)(13,49,20,41)(14,50,21,42)(15,51,22,43)(16,52,23,44), (1,45)(2,38)(3,47)(4,40)(5,33)(6,42)(7,35)(8,44)(9,37)(10,46)(11,39)(12,48)(13,41)(14,34)(15,43)(16,36)(17,54)(18,63)(19,56)(20,49)(21,58)(22,51)(23,60)(24,53)(25,62)(26,55)(27,64)(28,57)(29,50)(30,59)(31,52)(32,61) );
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)], [(1,53,24,45),(2,54,25,46),(3,55,26,47),(4,56,27,48),(5,57,28,33),(6,58,29,34),(7,59,30,35),(8,60,31,36),(9,61,32,37),(10,62,17,38),(11,63,18,39),(12,64,19,40),(13,49,20,41),(14,50,21,42),(15,51,22,43),(16,52,23,44)], [(1,45),(2,38),(3,47),(4,40),(5,33),(6,42),(7,35),(8,44),(9,37),(10,46),(11,39),(12,48),(13,41),(14,34),(15,43),(16,36),(17,54),(18,63),(19,56),(20,49),(21,58),(22,51),(23,60),(24,53),(25,62),(26,55),(27,64),(28,57),(29,50),(30,59),(31,52),(32,61)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 8M | 8N | 8O | 8P | 16A | ··· | 16P | 16Q | ··· | 16X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | C8 | D4 | C4○D4 | M5(2) | C8○D4 | D4○C16 |
| kernel | C16⋊6D4 | C4×C16 | C22⋊C16 | C4⋊C16 | C8×D4 | C2×M5(2) | C22⋊C8 | C4⋊C8 | C4×D4 | C22⋊C4 | C4⋊C4 | C2×D4 | C16 | C8 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 2 | 2 | 8 | 4 | 4 | 2 | 2 | 8 | 4 | 8 |
Matrix representation of C16⋊6D4 ►in GL4(𝔽17) generated by
| 0 | 1 | 0 | 0 |
| 8 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 11 | 0 | 0 |
| 3 | 0 | 0 | 0 |
| 0 | 0 | 10 | 15 |
| 0 | 0 | 8 | 7 |
| 0 | 11 | 0 | 0 |
| 14 | 0 | 0 | 0 |
| 0 | 0 | 10 | 15 |
| 0 | 0 | 7 | 7 |
G:=sub<GL(4,GF(17))| [0,8,0,0,1,0,0,0,0,0,16,0,0,0,0,16],[0,3,0,0,11,0,0,0,0,0,10,8,0,0,15,7],[0,14,0,0,11,0,0,0,0,0,10,7,0,0,15,7] >;
C16⋊6D4 in GAP, Magma, Sage, TeX
C_{16}\rtimes_6D_4 % in TeX
G:=Group("C16:6D4"); // GroupNames label
G:=SmallGroup(128,901);
// by ID
G=gap.SmallGroup(128,901);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,1430,723,100,102,124]);
// Polycyclic
G:=Group<a,b,c|a^16=b^4=c^2=1,a*b=b*a,c*a*c=a^9,c*b*c=b^-1>;
// generators/relations