direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C22×SD16, C8⋊3C23, C4.2C24, Q8⋊1C23, D4.1C23, C23.61D4, C4.17(C2×D4), (C2×C4).88D4, (C2×C8)⋊14C22, (C22×C8)⋊10C2, (C22×Q8)⋊8C2, (C2×Q8)⋊13C22, C22.65(C2×D4), C2.24(C22×D4), (C2×C4).136C23, (C2×D4).72C22, (C22×D4).12C2, (C22×C4).130C22, SmallGroup(64,251)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22×SD16
G = < a,b,c,d | a2=b2=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c3 >
Subgroups: 249 in 149 conjugacy classes, 89 normal (9 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C22×C8, C2×SD16, C22×D4, C22×Q8, C22×SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C24, C2×SD16, C22×D4, C22×SD16
Character table of C22×SD16
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 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 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | 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 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 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 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | 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 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 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 | complex lifted from SD16 |
| 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 | complex lifted from SD16 |
| 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 | complex lifted from SD16 |
| 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 | complex lifted from SD16 |
| 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 | complex lifted from SD16 |
| 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 | complex lifted from SD16 |
| 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 | complex lifted from SD16 |
| 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 | complex lifted from SD16 |
►On 32 points
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)(17 30)(18 31)(19 32)(20 25)(21 26)(22 27)(23 28)(24 29)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 25)(7 26)(8 27)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 17)(16 18)
(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 28)(2 31)(3 26)(4 29)(5 32)(6 27)(7 30)(8 25)(9 19)(10 22)(11 17)(12 20)(13 23)(14 18)(15 21)(16 24)
G:=sub<Sym(32)| (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12)(17,30)(18,31)(19,32)(20,25)(21,26)(22,27)(23,28)(24,29), (1,28)(2,29)(3,30)(4,31)(5,32)(6,25)(7,26)(8,27)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,17)(16,18), (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), (1,28)(2,31)(3,26)(4,29)(5,32)(6,27)(7,30)(8,25)(9,19)(10,22)(11,17)(12,20)(13,23)(14,18)(15,21)(16,24)>;
G:=Group( (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12)(17,30)(18,31)(19,32)(20,25)(21,26)(22,27)(23,28)(24,29), (1,28)(2,29)(3,30)(4,31)(5,32)(6,25)(7,26)(8,27)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,17)(16,18), (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), (1,28)(2,31)(3,26)(4,29)(5,32)(6,27)(7,30)(8,25)(9,19)(10,22)(11,17)(12,20)(13,23)(14,18)(15,21)(16,24) );
G=PermutationGroup([[(1,13),(2,14),(3,15),(4,16),(5,9),(6,10),(7,11),(8,12),(17,30),(18,31),(19,32),(20,25),(21,26),(22,27),(23,28),(24,29)], [(1,28),(2,29),(3,30),(4,31),(5,32),(6,25),(7,26),(8,27),(9,19),(10,20),(11,21),(12,22),(13,23),(14,24),(15,17),(16,18)], [(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)], [(1,28),(2,31),(3,26),(4,29),(5,32),(6,27),(7,30),(8,25),(9,19),(10,22),(11,17),(12,20),(13,23),(14,18),(15,21),(16,24)]])
C22×SD16 is a maximal subgroup of
(C2×SD16)⋊14C4 (C2×SD16)⋊15C4 (C2×C4)⋊9SD16 C8⋊(C22⋊C4) M4(2).31D4 C23⋊3SD16 (C22×D8).C2 (C2×C4)⋊3SD16 (C2×C8)⋊20D4 (C2×C8).41D4 M4(2).5D4 C4⋊C4.97D4 (C2×C4)⋊5SD16 C42.278C23 D4.(C2×D4) (C2×Q8)⋊16D4 C42.15C23 C42.16C23 (C2×C8)⋊11D4 M4(2)⋊10D4 SD16⋊D4 SD16⋊6D4 SD16⋊10D4
C22×SD16 is a maximal quotient of
C42.222D4 C42.223D4 C42.365D4 C23⋊4SD16 C42.264D4 C42.266D4 C42.279D4 C42.281D4 C42.294D4 D4⋊7SD16 D4⋊8SD16 D4⋊9SD16 Q8⋊7SD16 Q8⋊8SD16 Q8⋊9SD16
Matrix representation of C22×SD16 ►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 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 5 |
| 0 | 0 | 12 | 12 |
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,1,0,0,0,0,12,12,0,0,5,12],[16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,16] >;
C22×SD16 in GAP, Magma, Sage, TeX
C_2^2\times {\rm SD}_{16} % in TeX
G:=Group("C2^2xSD16"); // GroupNames label
G:=SmallGroup(64,251);
// by ID
G=gap.SmallGroup(64,251);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,-2,192,217,1444,730,88]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^3>;
// generators/relations
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