direct product, metabelian, supersoluble, monomial
Aliases: C2×C32⋊3Q16, C6⋊2Dic12, C12.26D12, C62.54D4, Dic6.30D6, C3⋊C8.28D6, (C3×C6)⋊3Q16, C32⋊6(C2×Q16), C3⋊3(C2×Dic12), (C3×C12).72D4, (C2×C6).63D12, C6⋊1(C3⋊Q16), C6.73(C2×D12), (C2×C12).124D6, (C2×Dic6).2S3, (C6×Dic6).7C2, C4.8(C3⋊D12), C12.73(C3⋊D4), (C6×C12).84C22, (C3×C12).71C23, C12.128(C22×S3), (C3×Dic6).38C22, C22.22(C3⋊D12), C32⋊4Q8.29C22, C4.58(C2×S32), (C2×C4).68S32, (C2×C3⋊C8).6S3, (C6×C3⋊C8).10C2, C3⋊1(C2×C3⋊Q16), C6.9(C2×C3⋊D4), (C3×C6).75(C2×D4), (C3×C3⋊C8).31C22, C2.13(C2×C3⋊D12), (C2×C6).40(C3⋊D4), (C2×C32⋊4Q8).11C2, SmallGroup(288,483)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C32⋊3Q16
G = < a,b,c,d,e | a2=b3=c3=d8=1, e2=d4, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >
Subgroups: 482 in 139 conjugacy classes, 52 normal (30 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C6, C8, C2×C4, C2×C4, Q8, C32, Dic3, C12, C12, C2×C6, C2×C6, C2×C8, Q16, C2×Q8, C3×C6, C3×C6, C3⋊C8, C24, Dic6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C2×Q16, C3×Dic3, C3⋊Dic3, C3×C12, C62, Dic12, C2×C3⋊C8, C3⋊Q16, C2×C24, C2×Dic6, C2×Dic6, C6×Q8, C3×C3⋊C8, C3×Dic6, C3×Dic6, C6×Dic3, C32⋊4Q8, C32⋊4Q8, C2×C3⋊Dic3, C6×C12, C2×Dic12, C2×C3⋊Q16, C32⋊3Q16, C6×C3⋊C8, C6×Dic6, C2×C32⋊4Q8, C2×C32⋊3Q16
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2×D4, D12, C3⋊D4, C22×S3, C2×Q16, S32, Dic12, C3⋊Q16, C2×D12, C2×C3⋊D4, C3⋊D12, C2×S32, C2×Dic12, C2×C3⋊Q16, C32⋊3Q16, C2×C3⋊D12, C2×C32⋊3Q16
►On 96 points
Generators in S96
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 61)(10 62)(11 63)(12 64)(13 57)(14 58)(15 59)(16 60)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(25 90)(26 91)(27 92)(28 93)(29 94)(30 95)(31 96)(32 89)(33 65)(34 66)(35 67)(36 68)(37 69)(38 70)(39 71)(40 72)(41 82)(42 83)(43 84)(44 85)(45 86)(46 87)(47 88)(48 81)
(1 49 45)(2 46 50)(3 51 47)(4 48 52)(5 53 41)(6 42 54)(7 55 43)(8 44 56)(9 95 71)(10 72 96)(11 89 65)(12 66 90)(13 91 67)(14 68 92)(15 93 69)(16 70 94)(17 86 73)(18 74 87)(19 88 75)(20 76 81)(21 82 77)(22 78 83)(23 84 79)(24 80 85)(25 64 34)(26 35 57)(27 58 36)(28 37 59)(29 60 38)(30 39 61)(31 62 40)(32 33 63)
(1 49 45)(2 50 46)(3 51 47)(4 52 48)(5 53 41)(6 54 42)(7 55 43)(8 56 44)(9 71 95)(10 72 96)(11 65 89)(12 66 90)(13 67 91)(14 68 92)(15 69 93)(16 70 94)(17 86 73)(18 87 74)(19 88 75)(20 81 76)(21 82 77)(22 83 78)(23 84 79)(24 85 80)(25 64 34)(26 57 35)(27 58 36)(28 59 37)(29 60 38)(30 61 39)(31 62 40)(32 63 33)
(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 33 5 37)(2 40 6 36)(3 39 7 35)(4 38 8 34)(9 23 13 19)(10 22 14 18)(11 21 15 17)(12 20 16 24)(25 48 29 44)(26 47 30 43)(27 46 31 42)(28 45 32 41)(49 63 53 59)(50 62 54 58)(51 61 55 57)(52 60 56 64)(65 77 69 73)(66 76 70 80)(67 75 71 79)(68 74 72 78)(81 94 85 90)(82 93 86 89)(83 92 87 96)(84 91 88 95)
G:=sub<Sym(96)| (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,61)(10,62)(11,63)(12,64)(13,57)(14,58)(15,59)(16,60)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,90)(26,91)(27,92)(28,93)(29,94)(30,95)(31,96)(32,89)(33,65)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,72)(41,82)(42,83)(43,84)(44,85)(45,86)(46,87)(47,88)(48,81), (1,49,45)(2,46,50)(3,51,47)(4,48,52)(5,53,41)(6,42,54)(7,55,43)(8,44,56)(9,95,71)(10,72,96)(11,89,65)(12,66,90)(13,91,67)(14,68,92)(15,93,69)(16,70,94)(17,86,73)(18,74,87)(19,88,75)(20,76,81)(21,82,77)(22,78,83)(23,84,79)(24,80,85)(25,64,34)(26,35,57)(27,58,36)(28,37,59)(29,60,38)(30,39,61)(31,62,40)(32,33,63), (1,49,45)(2,50,46)(3,51,47)(4,52,48)(5,53,41)(6,54,42)(7,55,43)(8,56,44)(9,71,95)(10,72,96)(11,65,89)(12,66,90)(13,67,91)(14,68,92)(15,69,93)(16,70,94)(17,86,73)(18,87,74)(19,88,75)(20,81,76)(21,82,77)(22,83,78)(23,84,79)(24,85,80)(25,64,34)(26,57,35)(27,58,36)(28,59,37)(29,60,38)(30,61,39)(31,62,40)(32,63,33), (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), (1,33,5,37)(2,40,6,36)(3,39,7,35)(4,38,8,34)(9,23,13,19)(10,22,14,18)(11,21,15,17)(12,20,16,24)(25,48,29,44)(26,47,30,43)(27,46,31,42)(28,45,32,41)(49,63,53,59)(50,62,54,58)(51,61,55,57)(52,60,56,64)(65,77,69,73)(66,76,70,80)(67,75,71,79)(68,74,72,78)(81,94,85,90)(82,93,86,89)(83,92,87,96)(84,91,88,95)>;
G:=Group( (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,61)(10,62)(11,63)(12,64)(13,57)(14,58)(15,59)(16,60)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,90)(26,91)(27,92)(28,93)(29,94)(30,95)(31,96)(32,89)(33,65)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,72)(41,82)(42,83)(43,84)(44,85)(45,86)(46,87)(47,88)(48,81), (1,49,45)(2,46,50)(3,51,47)(4,48,52)(5,53,41)(6,42,54)(7,55,43)(8,44,56)(9,95,71)(10,72,96)(11,89,65)(12,66,90)(13,91,67)(14,68,92)(15,93,69)(16,70,94)(17,86,73)(18,74,87)(19,88,75)(20,76,81)(21,82,77)(22,78,83)(23,84,79)(24,80,85)(25,64,34)(26,35,57)(27,58,36)(28,37,59)(29,60,38)(30,39,61)(31,62,40)(32,33,63), (1,49,45)(2,50,46)(3,51,47)(4,52,48)(5,53,41)(6,54,42)(7,55,43)(8,56,44)(9,71,95)(10,72,96)(11,65,89)(12,66,90)(13,67,91)(14,68,92)(15,69,93)(16,70,94)(17,86,73)(18,87,74)(19,88,75)(20,81,76)(21,82,77)(22,83,78)(23,84,79)(24,85,80)(25,64,34)(26,57,35)(27,58,36)(28,59,37)(29,60,38)(30,61,39)(31,62,40)(32,63,33), (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), (1,33,5,37)(2,40,6,36)(3,39,7,35)(4,38,8,34)(9,23,13,19)(10,22,14,18)(11,21,15,17)(12,20,16,24)(25,48,29,44)(26,47,30,43)(27,46,31,42)(28,45,32,41)(49,63,53,59)(50,62,54,58)(51,61,55,57)(52,60,56,64)(65,77,69,73)(66,76,70,80)(67,75,71,79)(68,74,72,78)(81,94,85,90)(82,93,86,89)(83,92,87,96)(84,91,88,95) );
G=PermutationGroup([[(1,73),(2,74),(3,75),(4,76),(5,77),(6,78),(7,79),(8,80),(9,61),(10,62),(11,63),(12,64),(13,57),(14,58),(15,59),(16,60),(17,49),(18,50),(19,51),(20,52),(21,53),(22,54),(23,55),(24,56),(25,90),(26,91),(27,92),(28,93),(29,94),(30,95),(31,96),(32,89),(33,65),(34,66),(35,67),(36,68),(37,69),(38,70),(39,71),(40,72),(41,82),(42,83),(43,84),(44,85),(45,86),(46,87),(47,88),(48,81)], [(1,49,45),(2,46,50),(3,51,47),(4,48,52),(5,53,41),(6,42,54),(7,55,43),(8,44,56),(9,95,71),(10,72,96),(11,89,65),(12,66,90),(13,91,67),(14,68,92),(15,93,69),(16,70,94),(17,86,73),(18,74,87),(19,88,75),(20,76,81),(21,82,77),(22,78,83),(23,84,79),(24,80,85),(25,64,34),(26,35,57),(27,58,36),(28,37,59),(29,60,38),(30,39,61),(31,62,40),(32,33,63)], [(1,49,45),(2,50,46),(3,51,47),(4,52,48),(5,53,41),(6,54,42),(7,55,43),(8,56,44),(9,71,95),(10,72,96),(11,65,89),(12,66,90),(13,67,91),(14,68,92),(15,69,93),(16,70,94),(17,86,73),(18,87,74),(19,88,75),(20,81,76),(21,82,77),(22,83,78),(23,84,79),(24,85,80),(25,64,34),(26,57,35),(27,58,36),(28,59,37),(29,60,38),(30,61,39),(31,62,40),(32,63,33)], [(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)], [(1,33,5,37),(2,40,6,36),(3,39,7,35),(4,38,8,34),(9,23,13,19),(10,22,14,18),(11,21,15,17),(12,20,16,24),(25,48,29,44),(26,47,30,43),(27,46,31,42),(28,45,32,41),(49,63,53,59),(50,62,54,58),(51,61,55,57),(52,60,56,64),(65,77,69,73),(66,76,70,80),(67,75,71,79),(68,74,72,78),(81,94,85,90),(82,93,86,89),(83,92,87,96),(84,91,88,95)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | 6H | 6I | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | 12L | 12M | 12N | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 12 | 12 | 36 | 36 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 6 | ··· | 6 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | + | - | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | S3 | S3 | D4 | D4 | D6 | D6 | D6 | Q16 | D12 | C3⋊D4 | D12 | C3⋊D4 | Dic12 | S32 | C3⋊Q16 | C3⋊D12 | C2×S32 | C3⋊D12 | C32⋊3Q16 |
| kernel | C2×C32⋊3Q16 | C32⋊3Q16 | C6×C3⋊C8 | C6×Dic6 | C2×C32⋊4Q8 | C2×C3⋊C8 | C2×Dic6 | C3×C12 | C62 | C3⋊C8 | Dic6 | C2×C12 | C3×C6 | C12 | C12 | C2×C6 | C2×C6 | C6 | C2×C4 | C6 | C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 8 | 1 | 2 | 1 | 1 | 1 | 4 |
Matrix representation of C2×C32⋊3Q16 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 72 | 0 | 0 |
| 1 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 72 |
| 0 | 72 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 23 | 5 |
| 0 | 0 | 68 | 18 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 36 | 11 |
| 0 | 0 | 48 | 37 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,72,72,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,72,0,0,1,72],[0,72,0,0,72,0,0,0,0,0,23,68,0,0,5,18],[72,0,0,0,0,72,0,0,0,0,36,48,0,0,11,37] >;
C2×C32⋊3Q16 in GAP, Magma, Sage, TeX
C_2\times C_3^2\rtimes_3Q_{16} % in TeX
G:=Group("C2xC3^2:3Q16"); // GroupNames label
G:=SmallGroup(288,483);
// by ID
G=gap.SmallGroup(288,483);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,141,176,675,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^8=1,e^2=d^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b^-1,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations