metabelian, supersoluble, monomial
Aliases: C12.17D6, C32⋊9SD16, D4.(C3⋊S3), (C3×D4).5S3, (C3×C6).35D4, C3⋊3(D4.S3), C32⋊4C8⋊4C2, C32⋊4Q8⋊3C2, C6.23(C3⋊D4), (D4×C32).2C2, (C3×C12).13C22, C2.5(C32⋊7D4), C4.2(C2×C3⋊S3), SmallGroup(144,97)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32⋊9SD16
G = < a,b,c,d | a3=b3=c8=d2=1, ab=ba, cac-1=a-1, ad=da, cbc-1=b-1, bd=db, dcd=c3 >
Subgroups: 170 in 60 conjugacy classes, 27 normal (11 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, D4, Q8, C32, Dic3, C12, C2×C6, SD16, C3×C6, C3×C6, C3⋊C8, Dic6, C3×D4, C3⋊Dic3, C3×C12, C62, D4.S3, C32⋊4C8, C32⋊4Q8, D4×C32, C32⋊9SD16
Quotients: C1, C2, C22, S3, D4, D6, SD16, C3⋊S3, C3⋊D4, C2×C3⋊S3, D4.S3, C32⋊7D4, C32⋊9SD16
Character table of C32⋊9SD16
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 8A | 8B | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 2 | 36 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 18 | 18 | 4 | 4 | 4 | 4 | |
| 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 2 | 2 | -2 | -1 | -1 | 2 | -1 | 2 | 0 | -1 | 2 | -1 | -1 | -2 | 1 | 1 | 1 | 1 | 1 | 1 | -2 | 0 | 0 | 2 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ6 | 2 | 2 | -2 | -1 | 2 | -1 | -1 | 2 | 0 | -1 | -1 | -1 | 2 | 1 | 1 | 1 | -2 | 1 | -2 | 1 | 1 | 0 | 0 | -1 | -1 | -1 | 2 | orthogonal lifted from D6 |
| ρ7 | 2 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | 0 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | -1 | 0 | 0 | -1 | 2 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | 2 | 0 | 2 | 2 | 2 | 2 | -2 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | orthogonal lifted from D4 |
| ρ9 | 2 | 2 | 2 | -1 | -1 | 2 | -1 | 2 | 0 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | 2 | -1 | 2 | -1 | -1 | 2 | 0 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | 2 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | -2 | 2 | -1 | -1 | -1 | 2 | 0 | 2 | -1 | -1 | -1 | 1 | 1 | -2 | 1 | 1 | 1 | -2 | 1 | 0 | 0 | -1 | -1 | 2 | -1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 2 | 0 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | -1 | 0 | 0 | -1 | -1 | 2 | -1 | orthogonal lifted from S3 |
| ρ13 | 2 | 2 | -2 | -1 | -1 | -1 | 2 | 2 | 0 | -1 | -1 | 2 | -1 | 1 | -2 | 1 | 1 | -2 | 1 | 1 | 1 | 0 | 0 | -1 | 2 | -1 | -1 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | 0 | -1 | -1 | -1 | 2 | -2 | 0 | -1 | -1 | 2 | -1 | -√-3 | 0 | √-3 | -√-3 | 0 | √-3 | -√-3 | √-3 | 0 | 0 | 1 | -2 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ15 | 2 | 2 | 0 | -1 | -1 | 2 | -1 | -2 | 0 | -1 | 2 | -1 | -1 | 0 | -√-3 | -√-3 | -√-3 | √-3 | √-3 | √-3 | 0 | 0 | 0 | -2 | 1 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ16 | 2 | 2 | 0 | -1 | 2 | -1 | -1 | -2 | 0 | -1 | -1 | -1 | 2 | √-3 | -√-3 | √-3 | 0 | √-3 | 0 | -√-3 | -√-3 | 0 | 0 | 1 | 1 | 1 | -2 | complex lifted from C3⋊D4 |
| ρ17 | 2 | 2 | 0 | -1 | 2 | -1 | -1 | -2 | 0 | -1 | -1 | -1 | 2 | -√-3 | √-3 | -√-3 | 0 | -√-3 | 0 | √-3 | √-3 | 0 | 0 | 1 | 1 | 1 | -2 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | 0 | -1 | -1 | -1 | 2 | -2 | 0 | -1 | -1 | 2 | -1 | √-3 | 0 | -√-3 | √-3 | 0 | -√-3 | √-3 | -√-3 | 0 | 0 | 1 | -2 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ19 | 2 | 2 | 0 | -1 | -1 | 2 | -1 | -2 | 0 | -1 | 2 | -1 | -1 | 0 | √-3 | √-3 | √-3 | -√-3 | -√-3 | -√-3 | 0 | 0 | 0 | -2 | 1 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ20 | 2 | 2 | 0 | 2 | -1 | -1 | -1 | -2 | 0 | 2 | -1 | -1 | -1 | -√-3 | -√-3 | 0 | √-3 | √-3 | -√-3 | 0 | √-3 | 0 | 0 | 1 | 1 | -2 | 1 | complex lifted from C3⋊D4 |
| ρ21 | 2 | 2 | 0 | 2 | -1 | -1 | -1 | -2 | 0 | 2 | -1 | -1 | -1 | √-3 | √-3 | 0 | -√-3 | -√-3 | √-3 | 0 | -√-3 | 0 | 0 | 1 | 1 | -2 | 1 | complex lifted from C3⋊D4 |
| ρ22 | 2 | -2 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | -√-2 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ23 | 2 | -2 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | √-2 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ24 | 4 | -4 | 0 | 4 | -2 | -2 | -2 | 0 | 0 | -4 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4.S3, Schur index 2 |
| ρ25 | 4 | -4 | 0 | -2 | -2 | 4 | -2 | 0 | 0 | 2 | -4 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4.S3, Schur index 2 |
| ρ26 | 4 | -4 | 0 | -2 | -2 | -2 | 4 | 0 | 0 | 2 | 2 | -4 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4.S3, Schur index 2 |
| ρ27 | 4 | -4 | 0 | -2 | 4 | -2 | -2 | 0 | 0 | 2 | 2 | 2 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4.S3, Schur index 2 |
►On 72 points
Generators in S72
(1 61 31)(2 32 62)(3 63 25)(4 26 64)(5 57 27)(6 28 58)(7 59 29)(8 30 60)(9 52 33)(10 34 53)(11 54 35)(12 36 55)(13 56 37)(14 38 49)(15 50 39)(16 40 51)(17 45 71)(18 72 46)(19 47 65)(20 66 48)(21 41 67)(22 68 42)(23 43 69)(24 70 44)
(1 15 45)(2 46 16)(3 9 47)(4 48 10)(5 11 41)(6 42 12)(7 13 43)(8 44 14)(17 31 39)(18 40 32)(19 25 33)(20 34 26)(21 27 35)(22 36 28)(23 29 37)(24 38 30)(49 60 70)(50 71 61)(51 62 72)(52 65 63)(53 64 66)(54 67 57)(55 58 68)(56 69 59)
(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)(65 66 67 68 69 70 71 72)
(2 4)(3 7)(6 8)(9 13)(10 16)(12 14)(18 20)(19 23)(22 24)(25 29)(26 32)(28 30)(33 37)(34 40)(36 38)(42 44)(43 47)(46 48)(49 55)(51 53)(52 56)(58 60)(59 63)(62 64)(65 69)(66 72)(68 70)
G:=sub<Sym(72)| (1,61,31)(2,32,62)(3,63,25)(4,26,64)(5,57,27)(6,28,58)(7,59,29)(8,30,60)(9,52,33)(10,34,53)(11,54,35)(12,36,55)(13,56,37)(14,38,49)(15,50,39)(16,40,51)(17,45,71)(18,72,46)(19,47,65)(20,66,48)(21,41,67)(22,68,42)(23,43,69)(24,70,44), (1,15,45)(2,46,16)(3,9,47)(4,48,10)(5,11,41)(6,42,12)(7,13,43)(8,44,14)(17,31,39)(18,40,32)(19,25,33)(20,34,26)(21,27,35)(22,36,28)(23,29,37)(24,38,30)(49,60,70)(50,71,61)(51,62,72)(52,65,63)(53,64,66)(54,67,57)(55,58,68)(56,69,59), (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)(65,66,67,68,69,70,71,72), (2,4)(3,7)(6,8)(9,13)(10,16)(12,14)(18,20)(19,23)(22,24)(25,29)(26,32)(28,30)(33,37)(34,40)(36,38)(42,44)(43,47)(46,48)(49,55)(51,53)(52,56)(58,60)(59,63)(62,64)(65,69)(66,72)(68,70)>;
G:=Group( (1,61,31)(2,32,62)(3,63,25)(4,26,64)(5,57,27)(6,28,58)(7,59,29)(8,30,60)(9,52,33)(10,34,53)(11,54,35)(12,36,55)(13,56,37)(14,38,49)(15,50,39)(16,40,51)(17,45,71)(18,72,46)(19,47,65)(20,66,48)(21,41,67)(22,68,42)(23,43,69)(24,70,44), (1,15,45)(2,46,16)(3,9,47)(4,48,10)(5,11,41)(6,42,12)(7,13,43)(8,44,14)(17,31,39)(18,40,32)(19,25,33)(20,34,26)(21,27,35)(22,36,28)(23,29,37)(24,38,30)(49,60,70)(50,71,61)(51,62,72)(52,65,63)(53,64,66)(54,67,57)(55,58,68)(56,69,59), (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)(65,66,67,68,69,70,71,72), (2,4)(3,7)(6,8)(9,13)(10,16)(12,14)(18,20)(19,23)(22,24)(25,29)(26,32)(28,30)(33,37)(34,40)(36,38)(42,44)(43,47)(46,48)(49,55)(51,53)(52,56)(58,60)(59,63)(62,64)(65,69)(66,72)(68,70) );
G=PermutationGroup([[(1,61,31),(2,32,62),(3,63,25),(4,26,64),(5,57,27),(6,28,58),(7,59,29),(8,30,60),(9,52,33),(10,34,53),(11,54,35),(12,36,55),(13,56,37),(14,38,49),(15,50,39),(16,40,51),(17,45,71),(18,72,46),(19,47,65),(20,66,48),(21,41,67),(22,68,42),(23,43,69),(24,70,44)], [(1,15,45),(2,46,16),(3,9,47),(4,48,10),(5,11,41),(6,42,12),(7,13,43),(8,44,14),(17,31,39),(18,40,32),(19,25,33),(20,34,26),(21,27,35),(22,36,28),(23,29,37),(24,38,30),(49,60,70),(50,71,61),(51,62,72),(52,65,63),(53,64,66),(54,67,57),(55,58,68),(56,69,59)], [(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),(65,66,67,68,69,70,71,72)], [(2,4),(3,7),(6,8),(9,13),(10,16),(12,14),(18,20),(19,23),(22,24),(25,29),(26,32),(28,30),(33,37),(34,40),(36,38),(42,44),(43,47),(46,48),(49,55),(51,53),(52,56),(58,60),(59,63),(62,64),(65,69),(66,72),(68,70)]])
C32⋊9SD16 is a maximal subgroup of
S3×D4.S3 Dic6.19D6 D12⋊9D6 D12.22D6 C24⋊8D6 C24.26D6 SD16×C3⋊S3 C24.32D6 C62.131D4 C62.74D4 C62.75D4 He3⋊8SD16 C36.17D6 C33⋊12SD16 C33⋊14SD16 C33⋊24SD16
C32⋊9SD16 is a maximal quotient of
C12.10Dic6 C62.114D4 C62.116D4 C36.17D6 He3⋊9SD16 C33⋊12SD16 C33⋊14SD16 C33⋊24SD16
Matrix representation of C32⋊9SD16 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 64 |
| 0 | 69 | 0 | 0 | 0 | 0 |
| 18 | 61 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 3 | 0 | 0 |
| 0 | 0 | 45 | 42 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 3 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,64,0,0,0,0,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,8,0,0,0,0,0,0,64],[0,18,0,0,0,0,69,61,0,0,0,0,0,0,31,45,0,0,0,0,3,42,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[1,3,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72] >;
C32⋊9SD16 in GAP, Magma, Sage, TeX
C_3^2\rtimes_9{\rm SD}_{16} % in TeX
G:=Group("C3^2:9SD16"); // GroupNames label
G:=SmallGroup(144,97);
// by ID
G=gap.SmallGroup(144,97);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,73,218,116,50,964,3461]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^8=d^2=1,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d=c^3>;
// generators/relations
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