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 [×2], C3 [×4], C4, C22 [×2], S3 [×4], C6 [×4], C6 [×4], C8, D4, D4, C32, C12 [×4], D6 [×4], C2×C6 [×4], D8, C3⋊S3, C3×C6, C3×C6, C3⋊C8 [×4], D12 [×4], C3×D4 [×4], C3×C12, C2×C3⋊S3, C62, D4⋊S3 [×4], C32⋊4C8, C12⋊S3, D4×C32, C32⋊7D8
Quotients: C1, C2 [×3], C22, S3 [×4], D4, D6 [×4], D8, C3⋊S3, C3⋊D4 [×4], C2×C3⋊S3, D4⋊S3 [×4], 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 45 29)(2 30 46)(3 47 31)(4 32 48)(5 41 25)(6 26 42)(7 43 27)(8 28 44)(9 20 38)(10 39 21)(11 22 40)(12 33 23)(13 24 34)(14 35 17)(15 18 36)(16 37 19)(49 61 70)(50 71 62)(51 63 72)(52 65 64)(53 57 66)(54 67 58)(55 59 68)(56 69 60)
(1 57 13)(2 14 58)(3 59 15)(4 16 60)(5 61 9)(6 10 62)(7 63 11)(8 12 64)(17 67 46)(18 47 68)(19 69 48)(20 41 70)(21 71 42)(22 43 72)(23 65 44)(24 45 66)(25 49 38)(26 39 50)(27 51 40)(28 33 52)(29 53 34)(30 35 54)(31 55 36)(32 37 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 61)(10 60)(11 59)(12 58)(13 57)(14 64)(15 63)(16 62)(17 52)(18 51)(19 50)(20 49)(21 56)(22 55)(23 54)(24 53)(25 41)(26 48)(27 47)(28 46)(29 45)(30 44)(31 43)(32 42)(33 67)(34 66)(35 65)(36 72)(37 71)(38 70)(39 69)(40 68)
G:=sub<Sym(72)| (1,45,29)(2,30,46)(3,47,31)(4,32,48)(5,41,25)(6,26,42)(7,43,27)(8,28,44)(9,20,38)(10,39,21)(11,22,40)(12,33,23)(13,24,34)(14,35,17)(15,18,36)(16,37,19)(49,61,70)(50,71,62)(51,63,72)(52,65,64)(53,57,66)(54,67,58)(55,59,68)(56,69,60), (1,57,13)(2,14,58)(3,59,15)(4,16,60)(5,61,9)(6,10,62)(7,63,11)(8,12,64)(17,67,46)(18,47,68)(19,69,48)(20,41,70)(21,71,42)(22,43,72)(23,65,44)(24,45,66)(25,49,38)(26,39,50)(27,51,40)(28,33,52)(29,53,34)(30,35,54)(31,55,36)(32,37,56), (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)(9,61)(10,60)(11,59)(12,58)(13,57)(14,64)(15,63)(16,62)(17,52)(18,51)(19,50)(20,49)(21,56)(22,55)(23,54)(24,53)(25,41)(26,48)(27,47)(28,46)(29,45)(30,44)(31,43)(32,42)(33,67)(34,66)(35,65)(36,72)(37,71)(38,70)(39,69)(40,68)>;
G:=Group( (1,45,29)(2,30,46)(3,47,31)(4,32,48)(5,41,25)(6,26,42)(7,43,27)(8,28,44)(9,20,38)(10,39,21)(11,22,40)(12,33,23)(13,24,34)(14,35,17)(15,18,36)(16,37,19)(49,61,70)(50,71,62)(51,63,72)(52,65,64)(53,57,66)(54,67,58)(55,59,68)(56,69,60), (1,57,13)(2,14,58)(3,59,15)(4,16,60)(5,61,9)(6,10,62)(7,63,11)(8,12,64)(17,67,46)(18,47,68)(19,69,48)(20,41,70)(21,71,42)(22,43,72)(23,65,44)(24,45,66)(25,49,38)(26,39,50)(27,51,40)(28,33,52)(29,53,34)(30,35,54)(31,55,36)(32,37,56), (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)(9,61)(10,60)(11,59)(12,58)(13,57)(14,64)(15,63)(16,62)(17,52)(18,51)(19,50)(20,49)(21,56)(22,55)(23,54)(24,53)(25,41)(26,48)(27,47)(28,46)(29,45)(30,44)(31,43)(32,42)(33,67)(34,66)(35,65)(36,72)(37,71)(38,70)(39,69)(40,68) );
G=PermutationGroup([(1,45,29),(2,30,46),(3,47,31),(4,32,48),(5,41,25),(6,26,42),(7,43,27),(8,28,44),(9,20,38),(10,39,21),(11,22,40),(12,33,23),(13,24,34),(14,35,17),(15,18,36),(16,37,19),(49,61,70),(50,71,62),(51,63,72),(52,65,64),(53,57,66),(54,67,58),(55,59,68),(56,69,60)], [(1,57,13),(2,14,58),(3,59,15),(4,16,60),(5,61,9),(6,10,62),(7,63,11),(8,12,64),(17,67,46),(18,47,68),(19,69,48),(20,41,70),(21,71,42),(22,43,72),(23,65,44),(24,45,66),(25,49,38),(26,39,50),(27,51,40),(28,33,52),(29,53,34),(30,35,54),(31,55,36),(32,37,56)], [(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),(9,61),(10,60),(11,59),(12,58),(13,57),(14,64),(15,63),(16,62),(17,52),(18,51),(19,50),(20,49),(21,56),(22,55),(23,54),(24,53),(25,41),(26,48),(27,47),(28,46),(29,45),(30,44),(31,43),(32,42),(33,67),(34,66),(35,65),(36,72),(37,71),(38,70),(39,69),(40,68)])
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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