p-group, metabelian, nilpotent (class 4), monomial
Aliases: D16⋊4C22, C16.2C23, C8.15C24, Q32⋊4C22, D8.4C23, C23.21D8, SD32⋊3C22, Q16.4C23, M5(2)⋊7C22, C4○D16⋊3C2, C4.77(C2×D8), (C2×C4).56D8, C8.58(C2×D4), C4○(C16⋊C22), C16⋊C22⋊7C2, (C2×C16)⋊4C22, C4○(Q32⋊C2), Q32⋊C2⋊7C2, (C2×C8).149D4, C4○D8⋊6C22, (C2×D8)⋊53C22, (C2×M5(2))⋊5C2, C22.26(C2×D8), C4.21(C22×D4), C2.30(C22×D8), (C2×C8).293C23, (C2×Q16)⋊57C22, (C22×C4).534D4, (C22×C8).296C22, (C2×C4○D8)⋊28C2, (C2×C4).660(C2×D4), SmallGroup(128,2146)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 404 in 182 conjugacy classes, 90 normal (16 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×9], C8 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×14], Q8 [×6], C23, C23 [×2], C16 [×4], C2×C8 [×2], C2×C8 [×4], D8 [×4], D8 [×2], SD16 [×8], Q16 [×4], Q16 [×2], C22×C4, C22×C4 [×2], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×12], C2×C16 [×2], M5(2) [×4], D16 [×4], SD32 [×8], Q32 [×4], C22×C8, C2×D8 [×2], C2×SD16 [×2], C2×Q16 [×2], C4○D8 [×8], C4○D8 [×4], C2×C4○D4 [×2], C2×M5(2), C4○D16 [×4], C16⋊C22 [×4], Q32⋊C2 [×4], C2×C4○D8 [×2], D16⋊C22
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D8 [×4], C2×D4 [×6], C24, C2×D8 [×6], C22×D4, C22×D8, D16⋊C22
Generators and relations
G = < a,b,c,d | a16=b2=c2=d2=1, bab=a-1, cac=a9, ad=da, cbc=dbd=a8b, cd=dc >
►On 32 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)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 18)(19 32)(20 31)(21 30)(22 29)(23 28)(24 27)(25 26)
(2 10)(4 12)(6 14)(8 16)(17 25)(19 27)(21 29)(23 31)
(1 22)(2 23)(3 24)(4 25)(5 26)(6 27)(7 28)(8 29)(9 30)(10 31)(11 32)(12 17)(13 18)(14 19)(15 20)(16 21)
G:=sub<Sym(32)| (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), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,18)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26), (2,10)(4,12)(6,14)(8,16)(17,25)(19,27)(21,29)(23,31), (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,17)(13,18)(14,19)(15,20)(16,21)>;
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), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,18)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26), (2,10)(4,12)(6,14)(8,16)(17,25)(19,27)(21,29)(23,31), (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,17)(13,18)(14,19)(15,20)(16,21) );
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)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9),(17,18),(19,32),(20,31),(21,30),(22,29),(23,28),(24,27),(25,26)], [(2,10),(4,12),(6,14),(8,16),(17,25),(19,27),(21,29),(23,31)], [(1,22),(2,23),(3,24),(4,25),(5,26),(6,27),(7,28),(8,29),(9,30),(10,31),(11,32),(12,17),(13,18),(14,19),(15,20),(16,21)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 0 | 0 | 0 | 4 |
| 0 | 0 | 4 | 0 |
| 12 | 5 | 0 | 0 |
| 5 | 5 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 4 | 0 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 0 | 4 | 0 |
G:=sub<GL(4,GF(17))| [0,0,12,5,0,0,5,5,0,4,0,0,4,0,0,0],[0,0,13,0,0,0,0,13,4,0,0,0,0,4,0,0],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[0,13,0,0,4,0,0,0,0,0,0,4,0,0,13,0] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 8A | 8B | 8C | 8D | 8E | 8F | 16A | ··· | 16H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 1 | 1 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D8 | D8 | D16⋊C22 |
| kernel | D16⋊C22 | C2×M5(2) | C4○D16 | C16⋊C22 | Q32⋊C2 | C2×C4○D8 | C2×C8 | C22×C4 | C2×C4 | C23 | C1 |
| # reps | 1 | 1 | 4 | 4 | 4 | 2 | 3 | 1 | 6 | 2 | 4 |
In GAP, Magma, Sage, TeX
D_{16}\rtimes C_2^2 % in TeX
G:=Group("D16:C2^2"); // GroupNames label
G:=SmallGroup(128,2146);
// by ID
G=gap.SmallGroup(128,2146);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,-2,-2,253,1430,248,1684,851,242,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c,d|a^16=b^2=c^2=d^2=1,b*a*b=a^-1,c*a*c=a^9,a*d=d*a,c*b*c=d*b*d=a^8*b,c*d=d*c>;
// generators/relations