p-group, metabelian, nilpotent (class 4), monomial
Aliases: Q16⋊1C4, C8.15D4, C2.1Q32, C4.2SD16, C2.2SD32, C22.9D8, C8.8(C2×C4), (C2×C16).1C2, (C2×C4).61D4, C2.D8.1C2, (C2×Q16).1C2, C4.2(C22⋊C4), (C2×C8).70C22, C2.7(D4⋊C4), SmallGroup(64,39)
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=ab-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 | i | -1 | -i | 1 | 1 | -1 | 1 | -1 | i | -i | -i | i | i | i | -i | -i | linear of order 4 |
| ρ6 | 1 | -1 | -1 | 1 | -1 | 1 | -i | -1 | i | 1 | 1 | -1 | 1 | -1 | -i | i | i | -i | -i | -i | i | i | linear of order 4 |
| ρ7 | 1 | -1 | -1 | 1 | -1 | 1 | -i | 1 | i | -1 | 1 | -1 | 1 | -1 | i | -i | -i | i | i | i | -i | -i | linear of order 4 |
| ρ8 | 1 | -1 | -1 | 1 | -1 | 1 | i | 1 | -i | -1 | 1 | -1 | 1 | -1 | -i | i | i | -i | -i | -i | i | i | linear of order 4 |
| 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 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | -√2 | √2 | ζ165-ζ163 | -ζ1615+ζ169 | ζ165-ζ163 | -ζ1615+ζ169 | -ζ165+ζ163 | ζ1615-ζ169 | -ζ165+ζ163 | ζ1615-ζ169 | symplectic lifted from Q32, Schur index 2 |
| ρ14 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | √2 | -√2 | ζ1615-ζ169 | ζ165-ζ163 | ζ1615-ζ169 | ζ165-ζ163 | -ζ1615+ζ169 | -ζ165+ζ163 | -ζ1615+ζ169 | -ζ165+ζ163 | symplectic lifted from Q32, Schur index 2 |
| ρ15 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | -√2 | √2 | -ζ165+ζ163 | ζ1615-ζ169 | -ζ165+ζ163 | ζ1615-ζ169 | ζ165-ζ163 | -ζ1615+ζ169 | ζ165-ζ163 | -ζ1615+ζ169 | symplectic lifted from Q32, Schur index 2 |
| ρ16 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | √2 | -√2 | -ζ1615+ζ169 | -ζ165+ζ163 | -ζ1615+ζ169 | -ζ165+ζ163 | ζ1615-ζ169 | ζ165-ζ163 | ζ1615-ζ169 | ζ165-ζ163 | symplectic lifted from Q32, Schur index 2 |
| ρ17 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | -√-2 | √-2 | √-2 | -√-2 | √-2 | √-2 | -√-2 | complex lifted from SD16 |
| ρ18 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | √-2 | -√-2 | -√-2 | √-2 | -√-2 | -√-2 | √-2 | complex lifted from SD16 |
| ρ19 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | -√2 | √2 | √2 | ζ1613+ζ1611 | ζ1615+ζ169 | ζ165+ζ163 | ζ167+ζ16 | ζ165+ζ163 | ζ1615+ζ169 | ζ1613+ζ1611 | ζ167+ζ16 | complex lifted from SD32 |
| ρ20 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | √2 | -√2 | -√2 | ζ167+ζ16 | ζ1613+ζ1611 | ζ1615+ζ169 | ζ165+ζ163 | ζ1615+ζ169 | ζ1613+ζ1611 | ζ167+ζ16 | ζ165+ζ163 | complex lifted from SD32 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | √2 | -√2 | -√2 | ζ1615+ζ169 | ζ165+ζ163 | ζ167+ζ16 | ζ1613+ζ1611 | ζ167+ζ16 | ζ165+ζ163 | ζ1615+ζ169 | ζ1613+ζ1611 | complex lifted from SD32 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | -√2 | √2 | √2 | ζ165+ζ163 | ζ167+ζ16 | ζ1613+ζ1611 | ζ1615+ζ169 | ζ1613+ζ1611 | ζ167+ζ16 | ζ165+ζ163 | ζ1615+ζ169 | complex lifted from SD32 |
►Regular action on 64 points
Generators in S64
(1 63)(2 64)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 55)(10 56)(11 57)(12 58)(13 59)(14 60)(15 61)(16 62)(17 34)(18 35)(19 36)(20 37)(21 38)(22 39)(23 40)(24 41)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 33)
(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 18 9 26)(2 34 10 42)(3 32 11 24)(4 48 12 40)(5 30 13 22)(6 46 14 38)(7 28 15 20)(8 44 16 36)(17 56 25 64)(19 54 27 62)(21 52 29 60)(23 50 31 58)(33 57 41 49)(35 55 43 63)(37 53 45 61)(39 51 47 59)
G:=sub<Sym(64)| (1,63)(2,64)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,41)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,33), (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), (1,18,9,26)(2,34,10,42)(3,32,11,24)(4,48,12,40)(5,30,13,22)(6,46,14,38)(7,28,15,20)(8,44,16,36)(17,56,25,64)(19,54,27,62)(21,52,29,60)(23,50,31,58)(33,57,41,49)(35,55,43,63)(37,53,45,61)(39,51,47,59)>;
G:=Group( (1,63)(2,64)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,41)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,33), (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), (1,18,9,26)(2,34,10,42)(3,32,11,24)(4,48,12,40)(5,30,13,22)(6,46,14,38)(7,28,15,20)(8,44,16,36)(17,56,25,64)(19,54,27,62)(21,52,29,60)(23,50,31,58)(33,57,41,49)(35,55,43,63)(37,53,45,61)(39,51,47,59) );
G=PermutationGroup([[(1,63),(2,64),(3,49),(4,50),(5,51),(6,52),(7,53),(8,54),(9,55),(10,56),(11,57),(12,58),(13,59),(14,60),(15,61),(16,62),(17,34),(18,35),(19,36),(20,37),(21,38),(22,39),(23,40),(24,41),(25,42),(26,43),(27,44),(28,45),(29,46),(30,47),(31,48),(32,33)], [(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)], [(1,18,9,26),(2,34,10,42),(3,32,11,24),(4,48,12,40),(5,30,13,22),(6,46,14,38),(7,28,15,20),(8,44,16,36),(17,56,25,64),(19,54,27,62),(21,52,29,60),(23,50,31,58),(33,57,41,49),(35,55,43,63),(37,53,45,61),(39,51,47,59)]])
C2.Q32 is a maximal subgroup of
C23.24D8 C23.39D8 C23.41D8 C4×SD32 C4×Q32 SD32⋊3C4 Q32⋊4C4 Q16⋊7D4 D8⋊8D4 Q16.8D4 D8.10D4 D8.4D4 Q16.4D4 D8.5D4 Q16.5D4 C16.19D4 C16⋊8D4 C16.D4 C16⋊2D4 Q16⋊Q8 Q16.Q8 C23.50D8 C23.51D8 C23.20D8 C8.22SD16 C8.12SD16 C8.14SD16 D5.Q32
C2p.Q32: C4.Q32 C4.SD32 C6.Q32 C2.Dic24 C6.5Q32 C10.Q32 C40.78D4 C40.15D4 ...
C2.Q32 is a maximal quotient of
C23.32D8 C8.7C42 D5.Q32
C2p.Q32: Q16⋊1C8 C8.27D8 C4.6Q32 C6.Q32 C2.Dic24 C6.5Q32 C10.Q32 C40.78D4 ...
Matrix representation of C2.Q32 ►in GL3(𝔽17) generated by
| 16 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 13 | 0 | 0 |
| 0 | 14 | 0 |
| 0 | 0 | 11 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 16 | 0 |
G:=sub<GL(3,GF(17))| [16,0,0,0,1,0,0,0,1],[13,0,0,0,14,0,0,0,11],[1,0,0,0,0,16,0,1,0] >;
C2.Q32 in GAP, Magma, Sage, TeX
C_2.Q_{32} % in TeX
G:=Group("C2.Q32"); // GroupNames label
G:=SmallGroup(64,39);
// by ID
G=gap.SmallGroup(64,39);
# by ID
G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,199,362,188,230,1444,730,88]);
// 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=a*b^-1>;
// generators/relations
Export
Subgroup lattice of C2.Q32 in TeX
Character table of C2.Q32 in TeX