direct product, p-group, metabelian, nilpotent (class 4), monomial
Aliases: C2×Q32, C4.8D8, C8.11D4, C8.8C23, C16.5C22, C22.16D8, Q16.1C22, C4.9(C2×D4), (C2×C16).4C2, (C2×C4).83D4, C2.14(C2×D8), (C2×Q16).4C2, (C2×C8).84C22, SmallGroup(64,188)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×Q32
G = < a,b,c | a2=b16=1, c2=b8, ab=ba, ac=ca, cbc-1=b-1 >
Character table of C2×Q32
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 8A | 8B | 8C | 8D | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 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 | trivial |
| ρ2 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | √2 | -√2 | √2 | -√2 | -√2 | orthogonal lifted from D8 |
| ρ12 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | √2 | -√2 | -√2 | √2 | orthogonal lifted from D8 |
| ρ13 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | -√2 | √2 | -√2 | √2 | √2 | orthogonal lifted from D8 |
| ρ14 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | -√2 | √2 | √2 | -√2 | orthogonal lifted from D8 |
| ρ15 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | -√2 | √2 | √2 | ζ165-ζ163 | -ζ1615+ζ169 | -ζ165+ζ163 | ζ165-ζ163 | ζ1615-ζ169 | -ζ165+ζ163 | ζ1615-ζ169 | -ζ1615+ζ169 | symplectic lifted from Q32, Schur index 2 |
| ρ16 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | -√2 | √2 | -ζ165+ζ163 | ζ1615-ζ169 | ζ165-ζ163 | ζ165-ζ163 | ζ1615-ζ169 | -ζ165+ζ163 | -ζ1615+ζ169 | -ζ1615+ζ169 | symplectic lifted from Q32, Schur index 2 |
| ρ17 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | √2 | -√2 | -ζ1615+ζ169 | -ζ165+ζ163 | ζ1615-ζ169 | ζ1615-ζ169 | -ζ165+ζ163 | -ζ1615+ζ169 | ζ165-ζ163 | ζ165-ζ163 | symplectic lifted from Q32, Schur index 2 |
| ρ18 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | √2 | -√2 | -√2 | ζ1615-ζ169 | ζ165-ζ163 | -ζ1615+ζ169 | ζ1615-ζ169 | -ζ165+ζ163 | -ζ1615+ζ169 | -ζ165+ζ163 | ζ165-ζ163 | symplectic lifted from Q32, Schur index 2 |
| ρ19 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | √2 | -√2 | ζ1615-ζ169 | ζ165-ζ163 | -ζ1615+ζ169 | -ζ1615+ζ169 | ζ165-ζ163 | ζ1615-ζ169 | -ζ165+ζ163 | -ζ165+ζ163 | symplectic lifted from Q32, Schur index 2 |
| ρ20 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | -√2 | √2 | √2 | -ζ165+ζ163 | ζ1615-ζ169 | ζ165-ζ163 | -ζ165+ζ163 | -ζ1615+ζ169 | ζ165-ζ163 | -ζ1615+ζ169 | ζ1615-ζ169 | symplectic lifted from Q32, Schur index 2 |
| ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | -√2 | √2 | ζ165-ζ163 | -ζ1615+ζ169 | -ζ165+ζ163 | -ζ165+ζ163 | -ζ1615+ζ169 | ζ165-ζ163 | ζ1615-ζ169 | ζ1615-ζ169 | symplectic lifted from Q32, Schur index 2 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | √2 | -√2 | -√2 | -ζ1615+ζ169 | -ζ165+ζ163 | ζ1615-ζ169 | -ζ1615+ζ169 | ζ165-ζ163 | ζ1615-ζ169 | ζ165-ζ163 | -ζ165+ζ163 | symplectic lifted from Q32, Schur index 2 |
►Regular action on 64 points
Generators in S64
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 17)(33 49)(34 50)(35 51)(36 52)(37 53)(38 54)(39 55)(40 56)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 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 47 9 39)(2 46 10 38)(3 45 11 37)(4 44 12 36)(5 43 13 35)(6 42 14 34)(7 41 15 33)(8 40 16 48)(17 64 25 56)(18 63 26 55)(19 62 27 54)(20 61 28 53)(21 60 29 52)(22 59 30 51)(23 58 31 50)(24 57 32 49)
G:=sub<Sym(64)| (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,17)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64), (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), (1,47,9,39)(2,46,10,38)(3,45,11,37)(4,44,12,36)(5,43,13,35)(6,42,14,34)(7,41,15,33)(8,40,16,48)(17,64,25,56)(18,63,26,55)(19,62,27,54)(20,61,28,53)(21,60,29,52)(22,59,30,51)(23,58,31,50)(24,57,32,49)>;
G:=Group( (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,17)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64), (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), (1,47,9,39)(2,46,10,38)(3,45,11,37)(4,44,12,36)(5,43,13,35)(6,42,14,34)(7,41,15,33)(8,40,16,48)(17,64,25,56)(18,63,26,55)(19,62,27,54)(20,61,28,53)(21,60,29,52)(22,59,30,51)(23,58,31,50)(24,57,32,49) );
G=PermutationGroup([[(1,18),(2,19),(3,20),(4,21),(5,22),(6,23),(7,24),(8,25),(9,26),(10,27),(11,28),(12,29),(13,30),(14,31),(15,32),(16,17),(33,49),(34,50),(35,51),(36,52),(37,53),(38,54),(39,55),(40,56),(41,57),(42,58),(43,59),(44,60),(45,61),(46,62),(47,63),(48,64)], [(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)], [(1,47,9,39),(2,46,10,38),(3,45,11,37),(4,44,12,36),(5,43,13,35),(6,42,14,34),(7,41,15,33),(8,40,16,48),(17,64,25,56),(18,63,26,55),(19,62,27,54),(20,61,28,53),(21,60,29,52),(22,59,30,51),(23,58,31,50),(24,57,32,49)]])
C2×Q32 is a maximal subgroup of
Q32⋊2C4 C16.18D4 Q32⋊4C4 Q16.8D4 D8.10D4 D8.12D4 Q16.4D4 Q16.5D4 C16.19D4 C16.D4 D4.4D8 C4⋊Q32 C8.21D8 C8.7D8 Q64⋊C2 Q8○D16
C2×Q32 is a maximal quotient of
Q16.8D4 Q16.4D4 C16.19D4 C4.Q32 C23.51D8 C4.SD32 C4⋊Q32 C16⋊2Q8
Matrix representation of C2×Q32 ►in GL3(𝔽17) generated by
| 16 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 16 |
| 1 | 0 | 0 |
| 0 | 2 | 8 |
| 0 | 13 | 10 |
| 1 | 0 | 0 |
| 0 | 7 | 10 |
| 0 | 12 | 10 |
G:=sub<GL(3,GF(17))| [16,0,0,0,16,0,0,0,16],[1,0,0,0,2,13,0,8,10],[1,0,0,0,7,12,0,10,10] >;
C2×Q32 in GAP, Magma, Sage, TeX
C_2\times Q_{32} % in TeX
G:=Group("C2xQ32"); // GroupNames label
G:=SmallGroup(64,188);
// by ID
G=gap.SmallGroup(64,188);
# by ID
G:=PCGroup([6,-2,2,2,-2,-2,-2,192,121,199,579,297,165,1444,730,88]);
// Polycyclic
G:=Group<a,b,c|a^2=b^16=1,c^2=b^8,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export
Subgroup lattice of C2×Q32 in TeX
Character table of C2×Q32 in TeX