p-group, metabelian, nilpotent (class 4), monomial
Aliases: D16⋊4C4, C42.134D4, C16⋊6(C2×C4), D8⋊3(C2×C4), (C4×D8)⋊37C2, C16⋊4C4⋊3C2, C16⋊5C4⋊2C2, C2.18(C4×D8), C4.30(C4×D4), (C2×D16).6C2, (C2×C4).111D8, (C2×C8).212D4, C2.D16⋊16C2, C4.16(C4○D8), C8.43(C4○D4), C8.40(C22×C4), C22.66(C2×D8), C2.5(C16⋊C22), (C4×C8).220C22, (C2×C16).21C22, (C2×C8).507C23, (C2×D8).106C22, C2.D8.155C22, (C2×C4).773(C2×D4), SmallGroup(128,909)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D16⋊4C4
G = < a,b,c | a16=b2=c4=1, bab=a-1, cac-1=a9, cbc-1=a8b >
Subgroups: 244 in 86 conjugacy classes, 40 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, C23, C16, C16, C42, C22⋊C4, C4⋊C4, C2×C8, D8, D8, C22×C4, C2×D4, C4×C8, D4⋊C4, C2.D8, C2×C16, D16, C4×D4, C2×D8, C16⋊5C4, C2.D16, C16⋊4C4, C4×D8, C2×D16, D16⋊4C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D8, C22×C4, C2×D4, C4○D4, C4×D4, C2×D8, C4○D8, C4×D8, C16⋊C22, D16⋊4C4
►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 17)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)(33 59)(34 58)(35 57)(36 56)(37 55)(38 54)(39 53)(40 52)(41 51)(42 50)(43 49)(44 64)(45 63)(46 62)(47 61)(48 60)
(1 46 25 62)(2 39 26 55)(3 48 27 64)(4 41 28 57)(5 34 29 50)(6 43 30 59)(7 36 31 52)(8 45 32 61)(9 38 17 54)(10 47 18 63)(11 40 19 56)(12 33 20 49)(13 42 21 58)(14 35 22 51)(15 44 23 60)(16 37 24 53)
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), (1,17)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(33,59)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,64)(45,63)(46,62)(47,61)(48,60), (1,46,25,62)(2,39,26,55)(3,48,27,64)(4,41,28,57)(5,34,29,50)(6,43,30,59)(7,36,31,52)(8,45,32,61)(9,38,17,54)(10,47,18,63)(11,40,19,56)(12,33,20,49)(13,42,21,58)(14,35,22,51)(15,44,23,60)(16,37,24,53)>;
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), (1,17)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(33,59)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,64)(45,63)(46,62)(47,61)(48,60), (1,46,25,62)(2,39,26,55)(3,48,27,64)(4,41,28,57)(5,34,29,50)(6,43,30,59)(7,36,31,52)(8,45,32,61)(9,38,17,54)(10,47,18,63)(11,40,19,56)(12,33,20,49)(13,42,21,58)(14,35,22,51)(15,44,23,60)(16,37,24,53) );
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)], [(1,17),(2,32),(3,31),(4,30),(5,29),(6,28),(7,27),(8,26),(9,25),(10,24),(11,23),(12,22),(13,21),(14,20),(15,19),(16,18),(33,59),(34,58),(35,57),(36,56),(37,55),(38,54),(39,53),(40,52),(41,51),(42,50),(43,49),(44,64),(45,63),(46,62),(47,61),(48,60)], [(1,46,25,62),(2,39,26,55),(3,48,27,64),(4,41,28,57),(5,34,29,50),(6,43,30,59),(7,36,31,52),(8,45,32,61),(9,38,17,54),(10,47,18,63),(11,40,19,56),(12,33,20,49),(13,42,21,58),(14,35,22,51),(15,44,23,60),(16,37,24,53)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 8A | 8B | 8C | 8D | 8E | 8F | 16A | ··· | 16H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | C4○D4 | D8 | C4○D8 | C16⋊C22 |
| kernel | D16⋊4C4 | C16⋊5C4 | C2.D16 | C16⋊4C4 | C4×D8 | C2×D16 | D16 | C42 | C2×C8 | C8 | C2×C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 1 | 1 | 2 | 4 | 4 | 4 |
Matrix representation of D16⋊4C4 ►in GL6(𝔽17)
| 16 | 2 | 0 | 0 | 0 | 0 |
| 16 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 11 | 15 | 12 |
| 0 | 0 | 16 | 10 | 14 | 16 |
| 0 | 0 | 10 | 10 | 8 | 14 |
| 0 | 0 | 13 | 11 | 16 | 4 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 16 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 12 | 1 | 1 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 12 | 5 | 16 | 15 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 5 | 13 | 12 | 12 |
G:=sub<GL(6,GF(17))| [16,16,0,0,0,0,2,1,0,0,0,0,0,0,12,16,10,13,0,0,11,10,10,11,0,0,15,14,8,16,0,0,12,16,14,4],[16,16,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,12,0,0,0,0,16,1,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,12,16,5,0,0,0,5,0,13,0,0,1,16,0,12,0,0,0,15,0,12] >;
D16⋊4C4 in GAP, Magma, Sage, TeX
D_{16}\rtimes_4C_4 % in TeX
G:=Group("D16:4C4"); // GroupNames label
G:=SmallGroup(128,909);
// by ID
G=gap.SmallGroup(128,909);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,1430,436,1123,570,360,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c|a^16=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^8*b>;
// generators/relations