direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C22×Q16, C4.3C24, C8.12C23, C23.62D4, Q8.1C23, (C2×C4).89D4, C4.18(C2×D4), (C2×C8).85C22, (C22×C8).10C2, C2.25(C22×D4), C22.66(C2×D4), (C22×Q8).9C2, (C2×C4).137C23, (C2×Q8).68C22, (C22×C4).131C22, SmallGroup(64,252)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22×Q16
G = < a,b,c,d | a2=b2=c8=1, d2=c4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 169 in 129 conjugacy classes, 89 normal (7 characteristic)
C1, C2, C2, C4, C4, C4, C22, C8, C2×C4, C2×C4, Q8, Q8, C23, C2×C8, Q16, C22×C4, C22×C4, C2×Q8, C2×Q8, C22×C8, C2×Q16, C22×Q8, C22×Q16
Quotients: C1, C2, C22, D4, C23, Q16, C2×D4, C24, C2×Q16, C22×D4, C22×Q16
Character table of C22×Q16
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ5 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | linear of order 2 |
| ρ6 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | linear of order 2 |
| ρ7 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ9 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ10 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | linear of order 2 |
| ρ11 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ12 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ13 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | linear of order 2 |
| ρ14 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ15 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ16 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ17 | 2 | 2 | -2 | 2 | 2 | -2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | -2 | -2 | -2 | 2 | 2 | -2 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | √2 | √2 | -√2 | -√2 | -√2 | √2 | symplectic lifted from Q16, Schur index 2 |
| ρ22 | 2 | -2 | -2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | √2 | √2 | -√2 | √2 | -√2 | -√2 | -√2 | symplectic lifted from Q16, Schur index 2 |
| ρ23 | 2 | -2 | -2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | -√2 | -√2 | √2 | -√2 | √2 | √2 | √2 | symplectic lifted from Q16, Schur index 2 |
| ρ24 | 2 | -2 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | -√2 | -√2 | √2 | √2 | √2 | -√2 | symplectic lifted from Q16, Schur index 2 |
| ρ25 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | √2 | -√2 | √2 | -√2 | symplectic lifted from Q16, Schur index 2 |
| ρ26 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | √2 | -√2 | -√2 | -√2 | -√2 | √2 | √2 | symplectic lifted from Q16, Schur index 2 |
| ρ27 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | -√2 | √2 | √2 | √2 | √2 | -√2 | -√2 | symplectic lifted from Q16, Schur index 2 |
| ρ28 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | -√2 | √2 | -√2 | √2 | symplectic lifted from Q16, Schur index 2 |
►Regular action on 64 points
Generators in S64
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 49)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 25)(17 60)(18 61)(19 62)(20 63)(21 64)(22 57)(23 58)(24 59)(33 44)(34 45)(35 46)(36 47)(37 48)(38 41)(39 42)(40 43)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 57)(8 58)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 33)(17 51)(18 52)(19 53)(20 54)(21 55)(22 56)(23 49)(24 50)(25 44)(26 45)(27 46)(28 47)(29 48)(30 41)(31 42)(32 43)
(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 40 5 36)(2 39 6 35)(3 38 7 34)(4 37 8 33)(9 61 13 57)(10 60 14 64)(11 59 15 63)(12 58 16 62)(17 31 21 27)(18 30 22 26)(19 29 23 25)(20 28 24 32)(41 56 45 52)(42 55 46 51)(43 54 47 50)(44 53 48 49)
G:=sub<Sym(64)| (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,49)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,25)(17,60)(18,61)(19,62)(20,63)(21,64)(22,57)(23,58)(24,59)(33,44)(34,45)(35,46)(36,47)(37,48)(38,41)(39,42)(40,43), (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,57)(8,58)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,33)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,49)(24,50)(25,44)(26,45)(27,46)(28,47)(29,48)(30,41)(31,42)(32,43), (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), (1,40,5,36)(2,39,6,35)(3,38,7,34)(4,37,8,33)(9,61,13,57)(10,60,14,64)(11,59,15,63)(12,58,16,62)(17,31,21,27)(18,30,22,26)(19,29,23,25)(20,28,24,32)(41,56,45,52)(42,55,46,51)(43,54,47,50)(44,53,48,49)>;
G:=Group( (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,49)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,25)(17,60)(18,61)(19,62)(20,63)(21,64)(22,57)(23,58)(24,59)(33,44)(34,45)(35,46)(36,47)(37,48)(38,41)(39,42)(40,43), (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,57)(8,58)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,33)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,49)(24,50)(25,44)(26,45)(27,46)(28,47)(29,48)(30,41)(31,42)(32,43), (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), (1,40,5,36)(2,39,6,35)(3,38,7,34)(4,37,8,33)(9,61,13,57)(10,60,14,64)(11,59,15,63)(12,58,16,62)(17,31,21,27)(18,30,22,26)(19,29,23,25)(20,28,24,32)(41,56,45,52)(42,55,46,51)(43,54,47,50)(44,53,48,49) );
G=PermutationGroup([[(1,50),(2,51),(3,52),(4,53),(5,54),(6,55),(7,56),(8,49),(9,26),(10,27),(11,28),(12,29),(13,30),(14,31),(15,32),(16,25),(17,60),(18,61),(19,62),(20,63),(21,64),(22,57),(23,58),(24,59),(33,44),(34,45),(35,46),(36,47),(37,48),(38,41),(39,42),(40,43)], [(1,59),(2,60),(3,61),(4,62),(5,63),(6,64),(7,57),(8,58),(9,34),(10,35),(11,36),(12,37),(13,38),(14,39),(15,40),(16,33),(17,51),(18,52),(19,53),(20,54),(21,55),(22,56),(23,49),(24,50),(25,44),(26,45),(27,46),(28,47),(29,48),(30,41),(31,42),(32,43)], [(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)], [(1,40,5,36),(2,39,6,35),(3,38,7,34),(4,37,8,33),(9,61,13,57),(10,60,14,64),(11,59,15,63),(12,58,16,62),(17,31,21,27),(18,30,22,26),(19,29,23,25),(20,28,24,32),(41,56,45,52),(42,55,46,51),(43,54,47,50),(44,53,48,49)]])
C22×Q16 is a maximal subgroup of
(C2×C4)⋊9Q16 (C2×C4)⋊6Q16 (C2×Q16)⋊10C4 M4(2).33D4 C23⋊2Q16 (C2×C8).41D4 (C2×C4)⋊2Q16 M4(2).6D4 C4⋊C4.98D4 (C2×C4)⋊3Q16 C23.41D8 Q16⋊7D4 Q16.8D4 C42.279C23 Q8.(C2×D4) C42.17C23 C8.D4⋊C2 M4(2).20D4 Q16⋊9D4 Q16⋊12D4
C22×Q16 is a maximal quotient of
C42.224D4 C42.367D4 C23⋊3Q16 C42.267D4 C42.282D4 C42.297D4 D4⋊5Q16 D4⋊6Q16 Q8⋊5Q16 Q8⋊6Q16
Matrix representation of C22×Q16 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 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 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 3 |
| 0 | 0 | 0 | 0 | 14 | 14 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 0 | 0 | 0 | 0 | 13 | 0 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,14,14,0,0,0,0,3,14],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,13,0,0,0,0,13,0] >;
C22×Q16 in GAP, Magma, Sage, TeX
C_2^2\times Q_{16} % in TeX
G:=Group("C2^2xQ16"); // GroupNames label
G:=SmallGroup(64,252);
// by ID
G=gap.SmallGroup(64,252);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,-2,192,217,199,1444,730,88]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^8=1,d^2=c^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
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