metabelian, supersoluble, monomial
Aliases: D24.1S3, C6.13D24, C24.16D6, C32:3SD32, C12.10D12, C8.5S32, C3:C16:2S3, (C3xC6).9D8, C6.2(D4:S3), C3:1(D8.S3), (C3xD24).3C2, C3:2(C48:C2), (C3xC12).24D4, C32:5Q16:4C2, (C3xC24).9C22, C4.2(C3:D12), C2.5(C3:D24), C12.67(C3:D4), (C3xC3:C16):2C2, SmallGroup(288,196)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32:3SD32
G = < a,b,c,d | a3=b3=c16=d2=1, ab=ba, cac-1=a-1, ad=da, bc=cb, dbd=b-1, dcd=c7 >
Subgroups: 322 in 61 conjugacy classes, 22 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, D4, Q8, C32, Dic3, C12, C12, D6, C2xC6, C16, D8, Q16, C3xS3, C3xC6, C24, C24, Dic6, D12, C3xD4, SD32, C3:Dic3, C3xC12, S3xC6, C3:C16, C48, D24, Dic12, C3xD8, C3xC24, C3xD12, C32:4Q8, C48:C2, D8.S3, C3xC3:C16, C3xD24, C32:5Q16, C32:3SD32
Quotients: C1, C2, C22, S3, D4, D6, D8, D12, C3:D4, SD32, S32, D24, D4:S3, C3:D12, C48:C2, D8.S3, C3:D24, C32:3SD32
►On 96 points
Generators in S96
(1 38 68)(2 69 39)(3 40 70)(4 71 41)(5 42 72)(6 73 43)(7 44 74)(8 75 45)(9 46 76)(10 77 47)(11 48 78)(12 79 33)(13 34 80)(14 65 35)(15 36 66)(16 67 37)(17 54 82)(18 83 55)(19 56 84)(20 85 57)(21 58 86)(22 87 59)(23 60 88)(24 89 61)(25 62 90)(26 91 63)(27 64 92)(28 93 49)(29 50 94)(30 95 51)(31 52 96)(32 81 53)
(1 68 38)(2 69 39)(3 70 40)(4 71 41)(5 72 42)(6 73 43)(7 74 44)(8 75 45)(9 76 46)(10 77 47)(11 78 48)(12 79 33)(13 80 34)(14 65 35)(15 66 36)(16 67 37)(17 54 82)(18 55 83)(19 56 84)(20 57 85)(21 58 86)(22 59 87)(23 60 88)(24 61 89)(25 62 90)(26 63 91)(27 64 92)(28 49 93)(29 50 94)(30 51 95)(31 52 96)(32 53 81)
(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 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)
(1 27)(2 18)(3 25)(4 32)(5 23)(6 30)(7 21)(8 28)(9 19)(10 26)(11 17)(12 24)(13 31)(14 22)(15 29)(16 20)(33 61)(34 52)(35 59)(36 50)(37 57)(38 64)(39 55)(40 62)(41 53)(42 60)(43 51)(44 58)(45 49)(46 56)(47 63)(48 54)(65 87)(66 94)(67 85)(68 92)(69 83)(70 90)(71 81)(72 88)(73 95)(74 86)(75 93)(76 84)(77 91)(78 82)(79 89)(80 96)
G:=sub<Sym(96)| (1,38,68)(2,69,39)(3,40,70)(4,71,41)(5,42,72)(6,73,43)(7,44,74)(8,75,45)(9,46,76)(10,77,47)(11,48,78)(12,79,33)(13,34,80)(14,65,35)(15,36,66)(16,67,37)(17,54,82)(18,83,55)(19,56,84)(20,85,57)(21,58,86)(22,87,59)(23,60,88)(24,89,61)(25,62,90)(26,91,63)(27,64,92)(28,93,49)(29,50,94)(30,95,51)(31,52,96)(32,81,53), (1,68,38)(2,69,39)(3,70,40)(4,71,41)(5,72,42)(6,73,43)(7,74,44)(8,75,45)(9,76,46)(10,77,47)(11,78,48)(12,79,33)(13,80,34)(14,65,35)(15,66,36)(16,67,37)(17,54,82)(18,55,83)(19,56,84)(20,57,85)(21,58,86)(22,59,87)(23,60,88)(24,61,89)(25,62,90)(26,63,91)(27,64,92)(28,49,93)(29,50,94)(30,51,95)(31,52,96)(32,53,81), (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)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,27)(2,18)(3,25)(4,32)(5,23)(6,30)(7,21)(8,28)(9,19)(10,26)(11,17)(12,24)(13,31)(14,22)(15,29)(16,20)(33,61)(34,52)(35,59)(36,50)(37,57)(38,64)(39,55)(40,62)(41,53)(42,60)(43,51)(44,58)(45,49)(46,56)(47,63)(48,54)(65,87)(66,94)(67,85)(68,92)(69,83)(70,90)(71,81)(72,88)(73,95)(74,86)(75,93)(76,84)(77,91)(78,82)(79,89)(80,96)>;
G:=Group( (1,38,68)(2,69,39)(3,40,70)(4,71,41)(5,42,72)(6,73,43)(7,44,74)(8,75,45)(9,46,76)(10,77,47)(11,48,78)(12,79,33)(13,34,80)(14,65,35)(15,36,66)(16,67,37)(17,54,82)(18,83,55)(19,56,84)(20,85,57)(21,58,86)(22,87,59)(23,60,88)(24,89,61)(25,62,90)(26,91,63)(27,64,92)(28,93,49)(29,50,94)(30,95,51)(31,52,96)(32,81,53), (1,68,38)(2,69,39)(3,70,40)(4,71,41)(5,72,42)(6,73,43)(7,74,44)(8,75,45)(9,76,46)(10,77,47)(11,78,48)(12,79,33)(13,80,34)(14,65,35)(15,66,36)(16,67,37)(17,54,82)(18,55,83)(19,56,84)(20,57,85)(21,58,86)(22,59,87)(23,60,88)(24,61,89)(25,62,90)(26,63,91)(27,64,92)(28,49,93)(29,50,94)(30,51,95)(31,52,96)(32,53,81), (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)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,27)(2,18)(3,25)(4,32)(5,23)(6,30)(7,21)(8,28)(9,19)(10,26)(11,17)(12,24)(13,31)(14,22)(15,29)(16,20)(33,61)(34,52)(35,59)(36,50)(37,57)(38,64)(39,55)(40,62)(41,53)(42,60)(43,51)(44,58)(45,49)(46,56)(47,63)(48,54)(65,87)(66,94)(67,85)(68,92)(69,83)(70,90)(71,81)(72,88)(73,95)(74,86)(75,93)(76,84)(77,91)(78,82)(79,89)(80,96) );
G=PermutationGroup([[(1,38,68),(2,69,39),(3,40,70),(4,71,41),(5,42,72),(6,73,43),(7,44,74),(8,75,45),(9,46,76),(10,77,47),(11,48,78),(12,79,33),(13,34,80),(14,65,35),(15,36,66),(16,67,37),(17,54,82),(18,83,55),(19,56,84),(20,85,57),(21,58,86),(22,87,59),(23,60,88),(24,89,61),(25,62,90),(26,91,63),(27,64,92),(28,93,49),(29,50,94),(30,95,51),(31,52,96),(32,81,53)], [(1,68,38),(2,69,39),(3,70,40),(4,71,41),(5,72,42),(6,73,43),(7,74,44),(8,75,45),(9,76,46),(10,77,47),(11,78,48),(12,79,33),(13,80,34),(14,65,35),(15,66,36),(16,67,37),(17,54,82),(18,55,83),(19,56,84),(20,57,85),(21,58,86),(22,59,87),(23,60,88),(24,61,89),(25,62,90),(26,63,91),(27,64,92),(28,49,93),(29,50,94),(30,51,95),(31,52,96),(32,53,81)], [(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),(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)], [(1,27),(2,18),(3,25),(4,32),(5,23),(6,30),(7,21),(8,28),(9,19),(10,26),(11,17),(12,24),(13,31),(14,22),(15,29),(16,20),(33,61),(34,52),(35,59),(36,50),(37,57),(38,64),(39,55),(40,62),(41,53),(42,60),(43,51),(44,58),(45,49),(46,56),(47,63),(48,54),(65,87),(66,94),(67,85),(68,92),(69,83),(70,90),(71,81),(72,88),(73,95),(74,86),(75,93),(76,84),(77,91),(78,82),(79,89),(80,96)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 16A | 16B | 16C | 16D | 24A | 24B | 24C | 24D | 24E | ··· | 24J | 48A | ··· | 48H |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 16 | 16 | 16 | 16 | 24 | 24 | 24 | 24 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 24 | 2 | 2 | 4 | 2 | 72 | 2 | 2 | 4 | 24 | 24 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |||
| image | C1 | C2 | C2 | C2 | S3 | S3 | D4 | D6 | D8 | D12 | C3:D4 | SD32 | D24 | C48:C2 | S32 | D4:S3 | C3:D12 | D8.S3 | C3:D24 | C32:3SD32 |
| kernel | C32:3SD32 | C3xC3:C16 | C3xD24 | C32:5Q16 | C3:C16 | D24 | C3xC12 | C24 | C3xC6 | C12 | C12 | C32 | C6 | C3 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 1 | 1 | 1 | 2 | 2 | 4 |
Matrix representation of C32:3SD32 ►in GL4(F97) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 96 |
| 0 | 0 | 1 | 96 |
| 0 | 96 | 0 | 0 |
| 1 | 96 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 5 | 67 | 0 | 0 |
| 30 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 16 | 79 | 0 | 0 |
| 95 | 81 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(97))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,96,96],[0,1,0,0,96,96,0,0,0,0,1,0,0,0,0,1],[5,30,0,0,67,72,0,0,0,0,0,1,0,0,1,0],[16,95,0,0,79,81,0,0,0,0,1,0,0,0,0,1] >;
C32:3SD32 in GAP, Magma, Sage, TeX
C_3^2\rtimes_3{\rm SD}_{32} % in TeX
G:=Group("C3^2:3SD32"); // GroupNames label
G:=SmallGroup(288,196);
// by ID
G=gap.SmallGroup(288,196);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,85,92,590,58,675,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^16=d^2=1,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^7>;
// generators/relations