metabelian, supersoluble, monomial
Aliases: C32⋊7D8, C12.16D6, D4⋊(C3⋊S3), (C3×D4)⋊1S3, C3⋊3(D4⋊S3), (C3×C6).34D4, C12⋊S3⋊3C2, C32⋊4C8⋊3C2, (D4×C32)⋊2C2, C6.22(C3⋊D4), (C3×C12).12C22, C2.4(C32⋊7D4), C4.1(C2×C3⋊S3), SmallGroup(144,96)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32⋊7D8
G = < a,b,c,d | a3=b3=c8=d2=1, ab=ba, cac-1=dad=a-1, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 258 in 66 conjugacy classes, 27 normal (11 characteristic)
C1, C2, C2, C3, C4, C22, S3, C6, C6, C8, D4, D4, C32, C12, D6, C2×C6, D8, C3⋊S3, C3×C6, C3×C6, C3⋊C8, D12, C3×D4, C3×C12, C2×C3⋊S3, C62, D4⋊S3, C32⋊4C8, C12⋊S3, D4×C32, C32⋊7D8
Quotients: C1, C2, C22, S3, D4, D6, D8, C3⋊S3, C3⋊D4, C2×C3⋊S3, D4⋊S3, C32⋊7D4, C32⋊7D8
Character table of C32⋊7D8
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 8A | 8B | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 4 | 36 | 2 | 2 | 2 | 2 | 2 | 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 | 0 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | -2 | 1 | 1 | 1 | 1 | 1 | 1 | -2 | 0 | 0 | 2 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ6 | 2 | 2 | 2 | 0 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | -1 | 0 | 0 | -1 | 2 | -1 | -1 | orthogonal lifted from S3 |
| ρ7 | 2 | 2 | -2 | 0 | 2 | -1 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | 1 | 1 | 1 | -2 | 1 | -2 | 1 | 1 | 0 | 0 | -1 | -1 | -1 | 2 | orthogonal lifted from D6 |
| ρ8 | 2 | 2 | 2 | 0 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ9 | 2 | 2 | -2 | 0 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | 2 | 1 | -2 | 1 | 1 | -2 | 1 | 1 | 1 | 0 | 0 | -1 | 2 | -1 | -1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | -2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | -2 | 0 | -1 | -1 | -1 | 2 | 2 | 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 | 0 | -1 | -1 | -1 | 2 | 2 | 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 | 0 | 2 | -1 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | 2 | orthogonal lifted from S3 |
| ρ14 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ15 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ16 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | -2 | -1 | 2 | -1 | -1 | -√-3 | √-3 | -√-3 | 0 | -√-3 | 0 | √-3 | √-3 | 0 | 0 | 1 | 1 | 1 | -2 | complex lifted from C3⋊D4 |
| ρ17 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | -2 | -1 | 2 | -1 | -1 | √-3 | -√-3 | √-3 | 0 | √-3 | 0 | -√-3 | -√-3 | 0 | 0 | 1 | 1 | 1 | -2 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | -1 | -2 | -1 | -1 | 2 | -1 | 0 | -√-3 | -√-3 | -√-3 | √-3 | √-3 | √-3 | 0 | 0 | 0 | -2 | 1 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ19 | 2 | 2 | 0 | 0 | -1 | -1 | 2 | -1 | -2 | -1 | -1 | -1 | 2 | √-3 | 0 | -√-3 | √-3 | 0 | -√-3 | √-3 | -√-3 | 0 | 0 | 1 | -2 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ20 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | -2 | 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 | 0 | -1 | 2 | -1 | -1 | -2 | -1 | -1 | 2 | -1 | 0 | √-3 | √-3 | √-3 | -√-3 | -√-3 | -√-3 | 0 | 0 | 0 | -2 | 1 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ22 | 2 | 2 | 0 | 0 | -1 | -1 | 2 | -1 | -2 | -1 | -1 | -1 | 2 | -√-3 | 0 | √-3 | -√-3 | 0 | √-3 | -√-3 | √-3 | 0 | 0 | 1 | -2 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ23 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | -2 | 2 | -1 | -1 | -1 | -√-3 | -√-3 | 0 | √-3 | √-3 | -√-3 | 0 | √-3 | 0 | 0 | 1 | 1 | -2 | 1 | complex lifted from C3⋊D4 |
| ρ24 | 4 | -4 | 0 | 0 | -2 | 4 | -2 | -2 | 0 | 2 | 2 | -4 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4⋊S3, Schur index 2 |
| ρ25 | 4 | -4 | 0 | 0 | -2 | -2 | -2 | 4 | 0 | -4 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4⋊S3, Schur index 2 |
| ρ26 | 4 | -4 | 0 | 0 | 4 | -2 | -2 | -2 | 0 | 2 | -4 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4⋊S3, Schur index 2 |
| ρ27 | 4 | -4 | 0 | 0 | -2 | -2 | 4 | -2 | 0 | 2 | 2 | 2 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4⋊S3, Schur index 2 |
►On 72 points
Generators in S72
(1 9 47)(2 48 10)(3 11 41)(4 42 12)(5 13 43)(6 44 14)(7 15 45)(8 46 16)(17 55 72)(18 65 56)(19 49 66)(20 67 50)(21 51 68)(22 69 52)(23 53 70)(24 71 54)(25 58 34)(26 35 59)(27 60 36)(28 37 61)(29 62 38)(30 39 63)(31 64 40)(32 33 57)
(1 20 25)(2 26 21)(3 22 27)(4 28 23)(5 24 29)(6 30 17)(7 18 31)(8 32 19)(9 67 58)(10 59 68)(11 69 60)(12 61 70)(13 71 62)(14 63 72)(15 65 64)(16 57 66)(33 49 46)(34 47 50)(35 51 48)(36 41 52)(37 53 42)(38 43 54)(39 55 44)(40 45 56)
(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 8)(3 7)(4 6)(9 47)(10 46)(11 45)(12 44)(13 43)(14 42)(15 41)(16 48)(17 28)(18 27)(19 26)(20 25)(21 32)(22 31)(23 30)(24 29)(33 68)(34 67)(35 66)(36 65)(37 72)(38 71)(39 70)(40 69)(49 59)(50 58)(51 57)(52 64)(53 63)(54 62)(55 61)(56 60)
G:=sub<Sym(72)| (1,9,47)(2,48,10)(3,11,41)(4,42,12)(5,13,43)(6,44,14)(7,15,45)(8,46,16)(17,55,72)(18,65,56)(19,49,66)(20,67,50)(21,51,68)(22,69,52)(23,53,70)(24,71,54)(25,58,34)(26,35,59)(27,60,36)(28,37,61)(29,62,38)(30,39,63)(31,64,40)(32,33,57), (1,20,25)(2,26,21)(3,22,27)(4,28,23)(5,24,29)(6,30,17)(7,18,31)(8,32,19)(9,67,58)(10,59,68)(11,69,60)(12,61,70)(13,71,62)(14,63,72)(15,65,64)(16,57,66)(33,49,46)(34,47,50)(35,51,48)(36,41,52)(37,53,42)(38,43,54)(39,55,44)(40,45,56), (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), (2,8)(3,7)(4,6)(9,47)(10,46)(11,45)(12,44)(13,43)(14,42)(15,41)(16,48)(17,28)(18,27)(19,26)(20,25)(21,32)(22,31)(23,30)(24,29)(33,68)(34,67)(35,66)(36,65)(37,72)(38,71)(39,70)(40,69)(49,59)(50,58)(51,57)(52,64)(53,63)(54,62)(55,61)(56,60)>;
G:=Group( (1,9,47)(2,48,10)(3,11,41)(4,42,12)(5,13,43)(6,44,14)(7,15,45)(8,46,16)(17,55,72)(18,65,56)(19,49,66)(20,67,50)(21,51,68)(22,69,52)(23,53,70)(24,71,54)(25,58,34)(26,35,59)(27,60,36)(28,37,61)(29,62,38)(30,39,63)(31,64,40)(32,33,57), (1,20,25)(2,26,21)(3,22,27)(4,28,23)(5,24,29)(6,30,17)(7,18,31)(8,32,19)(9,67,58)(10,59,68)(11,69,60)(12,61,70)(13,71,62)(14,63,72)(15,65,64)(16,57,66)(33,49,46)(34,47,50)(35,51,48)(36,41,52)(37,53,42)(38,43,54)(39,55,44)(40,45,56), (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), (2,8)(3,7)(4,6)(9,47)(10,46)(11,45)(12,44)(13,43)(14,42)(15,41)(16,48)(17,28)(18,27)(19,26)(20,25)(21,32)(22,31)(23,30)(24,29)(33,68)(34,67)(35,66)(36,65)(37,72)(38,71)(39,70)(40,69)(49,59)(50,58)(51,57)(52,64)(53,63)(54,62)(55,61)(56,60) );
G=PermutationGroup([[(1,9,47),(2,48,10),(3,11,41),(4,42,12),(5,13,43),(6,44,14),(7,15,45),(8,46,16),(17,55,72),(18,65,56),(19,49,66),(20,67,50),(21,51,68),(22,69,52),(23,53,70),(24,71,54),(25,58,34),(26,35,59),(27,60,36),(28,37,61),(29,62,38),(30,39,63),(31,64,40),(32,33,57)], [(1,20,25),(2,26,21),(3,22,27),(4,28,23),(5,24,29),(6,30,17),(7,18,31),(8,32,19),(9,67,58),(10,59,68),(11,69,60),(12,61,70),(13,71,62),(14,63,72),(15,65,64),(16,57,66),(33,49,46),(34,47,50),(35,51,48),(36,41,52),(37,53,42),(38,43,54),(39,55,44),(40,45,56)], [(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)], [(2,8),(3,7),(4,6),(9,47),(10,46),(11,45),(12,44),(13,43),(14,42),(15,41),(16,48),(17,28),(18,27),(19,26),(20,25),(21,32),(22,31),(23,30),(24,29),(33,68),(34,67),(35,66),(36,65),(37,72),(38,71),(39,70),(40,69),(49,59),(50,58),(51,57),(52,64),(53,63),(54,62),(55,61),(56,60)]])
C32⋊7D8 is a maximal subgroup of
S3×D4⋊S3 Dic6⋊3D6 D12.7D6 Dic6.20D6 D8×C3⋊S3 C24⋊8D6 C24⋊7D6 C24.40D6 C62.131D4 C62.73D4 C62.74D4 He3⋊6D8 C36.18D6 C33⋊6D8 C33⋊7D8 C33⋊15D8
C32⋊7D8 is a maximal quotient of
C12.9Dic6 C62.113D4 C32⋊7D16 C32⋊8SD32 C32⋊10SD32 C32⋊7Q32 C62.116D4 C36.18D6 He3⋊7D8 C33⋊6D8 C33⋊7D8 C33⋊15D8
Matrix representation of C32⋊7D8 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 72 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 60 | 43 | 0 | 0 | 0 | 0 |
| 30 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 57 | 57 |
| 0 | 0 | 0 | 0 | 16 | 57 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,30,0,0,0,0,43,13,0,0,0,0,0,0,72,0,0,0,0,0,1,1,0,0,0,0,0,0,57,16,0,0,0,0,57,57],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,1,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C32⋊7D8 in GAP, Magma, Sage, TeX
C_3^2\rtimes_7D_8
% in TeX
G:=Group("C3^2:7D8"); // GroupNames label
G:=SmallGroup(144,96);
// by ID
G=gap.SmallGroup(144,96);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,73,218,116,50,964,3461]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^8=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
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