metabelian, supersoluble, monomial
Aliases: C32⋊2Q32, C24.17D6, Dic12.2S3, C8.15S32, (C3×C6).11D8, C3⋊2(C3⋊Q32), (C3×C12).26D4, C6.10(D4⋊S3), C4.3(D6⋊S3), C12.23(C3⋊D4), C24.S3.1C2, (C3×C24).11C22, (C3×Dic12).2C2, C2.5(C32⋊2D8), SmallGroup(288,198)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32⋊2Q32
G = < a,b,c,d | a3=b3=c16=1, d2=c8, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, bd=db, dcd-1=c-1 >
Smallest permutation representation of C32⋊2Q32
►On 96 points
Generators in S96
(1 22 67)(2 68 23)(3 24 69)(4 70 25)(5 26 71)(6 72 27)(7 28 73)(8 74 29)(9 30 75)(10 76 31)(11 32 77)(12 78 17)(13 18 79)(14 80 19)(15 20 65)(16 66 21)(33 54 87)(34 88 55)(35 56 89)(36 90 57)(37 58 91)(38 92 59)(39 60 93)(40 94 61)(41 62 95)(42 96 63)(43 64 81)(44 82 49)(45 50 83)(46 84 51)(47 52 85)(48 86 53)
(1 67 22)(2 23 68)(3 69 24)(4 25 70)(5 71 26)(6 27 72)(7 73 28)(8 29 74)(9 75 30)(10 31 76)(11 77 32)(12 17 78)(13 79 18)(14 19 80)(15 65 20)(16 21 66)(33 54 87)(34 88 55)(35 56 89)(36 90 57)(37 58 91)(38 92 59)(39 60 93)(40 94 61)(41 62 95)(42 96 63)(43 64 81)(44 82 49)(45 50 83)(46 84 51)(47 52 85)(48 86 53)
(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 40 9 48)(2 39 10 47)(3 38 11 46)(4 37 12 45)(5 36 13 44)(6 35 14 43)(7 34 15 42)(8 33 16 41)(17 50 25 58)(18 49 26 57)(19 64 27 56)(20 63 28 55)(21 62 29 54)(22 61 30 53)(23 60 31 52)(24 59 32 51)(65 96 73 88)(66 95 74 87)(67 94 75 86)(68 93 76 85)(69 92 77 84)(70 91 78 83)(71 90 79 82)(72 89 80 81)
G:=sub<Sym(96)| (1,22,67)(2,68,23)(3,24,69)(4,70,25)(5,26,71)(6,72,27)(7,28,73)(8,74,29)(9,30,75)(10,76,31)(11,32,77)(12,78,17)(13,18,79)(14,80,19)(15,20,65)(16,66,21)(33,54,87)(34,88,55)(35,56,89)(36,90,57)(37,58,91)(38,92,59)(39,60,93)(40,94,61)(41,62,95)(42,96,63)(43,64,81)(44,82,49)(45,50,83)(46,84,51)(47,52,85)(48,86,53), (1,67,22)(2,23,68)(3,69,24)(4,25,70)(5,71,26)(6,27,72)(7,73,28)(8,29,74)(9,75,30)(10,31,76)(11,77,32)(12,17,78)(13,79,18)(14,19,80)(15,65,20)(16,21,66)(33,54,87)(34,88,55)(35,56,89)(36,90,57)(37,58,91)(38,92,59)(39,60,93)(40,94,61)(41,62,95)(42,96,63)(43,64,81)(44,82,49)(45,50,83)(46,84,51)(47,52,85)(48,86,53), (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), (1,40,9,48)(2,39,10,47)(3,38,11,46)(4,37,12,45)(5,36,13,44)(6,35,14,43)(7,34,15,42)(8,33,16,41)(17,50,25,58)(18,49,26,57)(19,64,27,56)(20,63,28,55)(21,62,29,54)(22,61,30,53)(23,60,31,52)(24,59,32,51)(65,96,73,88)(66,95,74,87)(67,94,75,86)(68,93,76,85)(69,92,77,84)(70,91,78,83)(71,90,79,82)(72,89,80,81)>;
G:=Group( (1,22,67)(2,68,23)(3,24,69)(4,70,25)(5,26,71)(6,72,27)(7,28,73)(8,74,29)(9,30,75)(10,76,31)(11,32,77)(12,78,17)(13,18,79)(14,80,19)(15,20,65)(16,66,21)(33,54,87)(34,88,55)(35,56,89)(36,90,57)(37,58,91)(38,92,59)(39,60,93)(40,94,61)(41,62,95)(42,96,63)(43,64,81)(44,82,49)(45,50,83)(46,84,51)(47,52,85)(48,86,53), (1,67,22)(2,23,68)(3,69,24)(4,25,70)(5,71,26)(6,27,72)(7,73,28)(8,29,74)(9,75,30)(10,31,76)(11,77,32)(12,17,78)(13,79,18)(14,19,80)(15,65,20)(16,21,66)(33,54,87)(34,88,55)(35,56,89)(36,90,57)(37,58,91)(38,92,59)(39,60,93)(40,94,61)(41,62,95)(42,96,63)(43,64,81)(44,82,49)(45,50,83)(46,84,51)(47,52,85)(48,86,53), (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), (1,40,9,48)(2,39,10,47)(3,38,11,46)(4,37,12,45)(5,36,13,44)(6,35,14,43)(7,34,15,42)(8,33,16,41)(17,50,25,58)(18,49,26,57)(19,64,27,56)(20,63,28,55)(21,62,29,54)(22,61,30,53)(23,60,31,52)(24,59,32,51)(65,96,73,88)(66,95,74,87)(67,94,75,86)(68,93,76,85)(69,92,77,84)(70,91,78,83)(71,90,79,82)(72,89,80,81) );
G=PermutationGroup([[(1,22,67),(2,68,23),(3,24,69),(4,70,25),(5,26,71),(6,72,27),(7,28,73),(8,74,29),(9,30,75),(10,76,31),(11,32,77),(12,78,17),(13,18,79),(14,80,19),(15,20,65),(16,66,21),(33,54,87),(34,88,55),(35,56,89),(36,90,57),(37,58,91),(38,92,59),(39,60,93),(40,94,61),(41,62,95),(42,96,63),(43,64,81),(44,82,49),(45,50,83),(46,84,51),(47,52,85),(48,86,53)], [(1,67,22),(2,23,68),(3,69,24),(4,25,70),(5,71,26),(6,27,72),(7,73,28),(8,29,74),(9,75,30),(10,31,76),(11,77,32),(12,17,78),(13,79,18),(14,19,80),(15,65,20),(16,21,66),(33,54,87),(34,88,55),(35,56,89),(36,90,57),(37,58,91),(38,92,59),(39,60,93),(40,94,61),(41,62,95),(42,96,63),(43,64,81),(44,82,49),(45,50,83),(46,84,51),(47,52,85),(48,86,53)], [(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)], [(1,40,9,48),(2,39,10,47),(3,38,11,46),(4,37,12,45),(5,36,13,44),(6,35,14,43),(7,34,15,42),(8,33,16,41),(17,50,25,58),(18,49,26,57),(19,64,27,56),(20,63,28,55),(21,62,29,54),(22,61,30,53),(23,60,31,52),(24,59,32,51),(65,96,73,88),(66,95,74,87),(67,94,75,86),(68,93,76,85),(69,92,77,84),(70,91,78,83),(71,90,79,82),(72,89,80,81)]])
33 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 4A | 4B | 4C | 6A | 6B | 6C | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 16A | 16B | 16C | 16D | 24A | ··· | 24H |
| order | 1 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 16 | 16 | 16 | 16 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 2 | 4 | 2 | 24 | 24 | 2 | 2 | 4 | 2 | 2 | 4 | 4 | 4 | 4 | 24 | 24 | 24 | 24 | 18 | 18 | 18 | 18 | 4 | ··· | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | - | - | |||
| image | C1 | C2 | C2 | S3 | D4 | D6 | D8 | C3⋊D4 | Q32 | S32 | D4⋊S3 | D6⋊S3 | C3⋊Q32 | C32⋊2D8 | C32⋊2Q32 |
| kernel | C32⋊2Q32 | C24.S3 | C3×Dic12 | Dic12 | C3×C12 | C24 | C3×C6 | C12 | C32 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 4 | 4 | 1 | 2 | 1 | 4 | 2 | 4 |
Matrix representation of C32⋊2Q32 ►in GL6(𝔽97)
| 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 | 0 | 96 |
| 0 | 0 | 0 | 0 | 1 | 96 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 96 | 0 | 0 |
| 0 | 0 | 1 | 96 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 73 | 93 | 0 | 0 | 0 | 0 |
| 2 | 69 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 57 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 96 | 0 | 0 | 0 |
| 0 | 0 | 0 | 96 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(97))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,96,96],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,96,96,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[73,2,0,0,0,0,93,69,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,57,0,0,0,0,17,0,0,0,0,0,0,0,96,0,0,0,0,0,0,96,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C32⋊2Q32 in GAP, Magma, Sage, TeX
C_3^2\rtimes_2Q_{32} % in TeX
G:=Group("C3^2:2Q32"); // GroupNames label
G:=SmallGroup(288,198);
// by ID
G=gap.SmallGroup(288,198);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,85,120,254,135,142,675,346,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^16=1,d^2=c^8,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
Export