direct product, p-group, metabelian, nilpotent (class 4), monomial
Aliases: C2×C2.D16, C23.57D8, C22.12D16, C22.10SD32, D8⋊7(C2×C4), (C2×D8)⋊9C4, C8.90(C2×D4), C2.1(C2×D16), (C2×C4).77D8, (C22×C16)⋊4C2, (C2×C8).246D4, C2.1(C2×SD32), C4.8(C2×SD16), (C2×C16)⋊16C22, C8.27(C22×C4), (C2×C4).75SD16, (C22×D8).6C2, C22.51(C2×D8), C2.D8⋊42C22, C8.24(C22⋊C4), (C2×C8).492C23, (C2×D8).99C22, (C22×C4).582D4, C4.23(D4⋊C4), (C22×C8).526C22, C22.52(D4⋊C4), (C2×C2.D8)⋊15C2, (C2×C8).174(C2×C4), (C2×C4).754(C2×D4), C4.48(C2×C22⋊C4), C2.26(C2×D4⋊C4), (C2×C4).269(C22⋊C4), SmallGroup(128,868)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C2.D16
G = < a,b,c,d | a2=b2=c16=1, d2=b, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=bc-1 >
Subgroups: 404 in 136 conjugacy classes, 60 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C16, C4⋊C4, C2×C8, C2×C8, D8, D8, C22×C4, C22×C4, C2×D4, C24, C2.D8, C2.D8, C2×C16, C2×C16, C2×C4⋊C4, C22×C8, C2×D8, C2×D8, C22×D4, C2.D16, C2×C2.D8, C22×C16, C22×D8, C2×C2.D16
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, D8, SD16, C22×C4, C2×D4, D4⋊C4, D16, SD32, C2×C22⋊C4, C2×D8, C2×SD16, C2.D16, C2×D4⋊C4, C2×D16, C2×SD32, C2×C2.D16
►On 64 points
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 33)(18 34)(19 35)(20 36)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 17)(8 18)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(33 55)(34 56)(35 57)(36 58)(37 59)(38 60)(39 61)(40 62)(41 63)(42 64)(43 49)(44 50)(45 51)(46 52)(47 53)(48 54)
(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 26 27 16)(2 15 28 25)(3 24 29 14)(4 13 30 23)(5 22 31 12)(6 11 32 21)(7 20 17 10)(8 9 18 19)(33 58 55 36)(34 35 56 57)(37 54 59 48)(38 47 60 53)(39 52 61 46)(40 45 62 51)(41 50 63 44)(42 43 64 49)
G:=sub<Sym(64)| (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,17)(8,18)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(33,55)(34,56)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62)(41,63)(42,64)(43,49)(44,50)(45,51)(46,52)(47,53)(48,54), (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), (1,26,27,16)(2,15,28,25)(3,24,29,14)(4,13,30,23)(5,22,31,12)(6,11,32,21)(7,20,17,10)(8,9,18,19)(33,58,55,36)(34,35,56,57)(37,54,59,48)(38,47,60,53)(39,52,61,46)(40,45,62,51)(41,50,63,44)(42,43,64,49)>;
G:=Group( (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,17)(8,18)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(33,55)(34,56)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62)(41,63)(42,64)(43,49)(44,50)(45,51)(46,52)(47,53)(48,54), (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), (1,26,27,16)(2,15,28,25)(3,24,29,14)(4,13,30,23)(5,22,31,12)(6,11,32,21)(7,20,17,10)(8,9,18,19)(33,58,55,36)(34,35,56,57)(37,54,59,48)(38,47,60,53)(39,52,61,46)(40,45,62,51)(41,50,63,44)(42,43,64,49) );
G=PermutationGroup([[(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,55),(8,56),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,33),(18,34),(19,35),(20,36),(21,37),(22,38),(23,39),(24,40),(25,41),(26,42),(27,43),(28,44),(29,45),(30,46),(31,47),(32,48)], [(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,17),(8,18),(9,19),(10,20),(11,21),(12,22),(13,23),(14,24),(15,25),(16,26),(33,55),(34,56),(35,57),(36,58),(37,59),(38,60),(39,61),(40,62),(41,63),(42,64),(43,49),(44,50),(45,51),(46,52),(47,53),(48,54)], [(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)], [(1,26,27,16),(2,15,28,25),(3,24,29,14),(4,13,30,23),(5,22,31,12),(6,11,32,21),(7,20,17,10),(8,9,18,19),(33,58,55,36),(34,35,56,57),(37,54,59,48),(38,47,60,53),(39,52,61,46),(40,45,62,51),(41,50,63,44),(42,43,64,49)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | ··· | 8H | 16A | ··· | 16P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D8 | SD16 | D8 | D16 | SD32 |
| kernel | C2×C2.D16 | C2.D16 | C2×C2.D8 | C22×C16 | C22×D8 | C2×D8 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 3 | 1 | 2 | 4 | 2 | 8 | 8 |
Matrix representation of C2×C2.D16 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 16 | 7 |
| 0 | 0 | 10 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 16 | 7 |
| 0 | 0 | 7 | 1 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,4,0,0,0,0,16,10,0,0,7,16],[1,0,0,0,0,13,0,0,0,0,16,7,0,0,7,1] >;
C2×C2.D16 in GAP, Magma, Sage, TeX
C_2\times C_2.D_{16} % in TeX
G:=Group("C2xC2.D16"); // GroupNames label
G:=SmallGroup(128,868);
// by ID
G=gap.SmallGroup(128,868);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,1123,570,360,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^16=1,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=b*c^-1>;
// generators/relations