direct product, metabelian, supersoluble, monomial
Aliases: C6×Q8⋊3S3, C62.271C23, Q8⋊7(S3×C6), D12⋊9(C2×C6), (C3×Q8)⋊24D6, (C6×Q8)⋊10C6, (C6×Q8)⋊17S3, (C2×D12)⋊12C6, (C6×D12)⋊19C2, C6.9(C23×C6), (C2×C12).336D6, C6.77(S3×C23), (C3×C6).46C24, D6.4(C22×C6), (S3×C12)⋊21C22, (C3×D12)⋊35C22, (S3×C6).31C23, C12.23(C22×C6), (C6×C12).165C22, C12.174(C22×S3), (C3×C12).124C23, Dic3.9(C22×C6), (Q8×C32)⋊18C22, (C3×Dic3).37C23, (C6×Dic3).168C22, (S3×C2×C4)⋊5C6, C3⋊3(C6×C4○D4), C6⋊3(C3×C4○D4), C4.23(S3×C2×C6), (Q8×C3×C6)⋊10C2, (S3×C2×C12)⋊13C2, (C4×S3)⋊5(C2×C6), (C3×Q8)⋊8(C2×C6), (C2×C4).62(S3×C6), (C2×Q8)⋊10(C3×S3), C32⋊17(C2×C4○D4), C2.10(S3×C22×C6), C22.32(S3×C2×C6), (C3×C6)⋊11(C4○D4), (C2×C12).47(C2×C6), (S3×C2×C6).111C22, (C2×C6).72(C22×C6), (C22×S3).31(C2×C6), (C2×C6).349(C22×S3), (C2×Dic3).51(C2×C6), SmallGroup(288,996)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×Q8⋊3S3
G = < a,b,c,d,e | a6=b4=d3=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ce=ec, ede=d-1 >
Subgroups: 714 in 347 conjugacy classes, 178 normal (20 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C32, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C22×C4, C2×D4, C2×Q8, C4○D4, C3×S3, C3×C6, C3×C6, C4×S3, D12, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×Q8, C3×Q8, C22×S3, C22×C6, C2×C4○D4, C3×Dic3, C3×C12, S3×C6, S3×C6, C62, S3×C2×C4, C2×D12, Q8⋊3S3, C22×C12, C6×D4, C6×Q8, C6×Q8, C3×C4○D4, S3×C12, C3×D12, C6×Dic3, C6×C12, Q8×C32, S3×C2×C6, C2×Q8⋊3S3, C6×C4○D4, S3×C2×C12, C6×D12, C3×Q8⋊3S3, Q8×C3×C6, C6×Q8⋊3S3
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C4○D4, C24, C3×S3, C22×S3, C22×C6, C2×C4○D4, S3×C6, Q8⋊3S3, C3×C4○D4, S3×C23, C23×C6, S3×C2×C6, C2×Q8⋊3S3, C6×C4○D4, C3×Q8⋊3S3, S3×C22×C6, C6×Q8⋊3S3
►On 96 points
Generators in S96
(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 22 41 33)(2 23 42 34)(3 24 37 35)(4 19 38 36)(5 20 39 31)(6 21 40 32)(7 46 51 13)(8 47 52 14)(9 48 53 15)(10 43 54 16)(11 44 49 17)(12 45 50 18)(25 94 77 61)(26 95 78 62)(27 96 73 63)(28 91 74 64)(29 92 75 65)(30 93 76 66)(55 80 88 69)(56 81 89 70)(57 82 90 71)(58 83 85 72)(59 84 86 67)(60 79 87 68)
(1 47 41 14)(2 48 42 15)(3 43 37 16)(4 44 38 17)(5 45 39 18)(6 46 40 13)(7 32 51 21)(8 33 52 22)(9 34 53 23)(10 35 54 24)(11 36 49 19)(12 31 50 20)(25 69 77 80)(26 70 78 81)(27 71 73 82)(28 72 74 83)(29 67 75 84)(30 68 76 79)(55 94 88 61)(56 95 89 62)(57 96 90 63)(58 91 85 64)(59 92 86 65)(60 93 87 66)
(1 5 3)(2 6 4)(7 11 9)(8 12 10)(13 17 15)(14 18 16)(19 23 21)(20 24 22)(25 27 29)(26 28 30)(31 35 33)(32 36 34)(37 41 39)(38 42 40)(43 47 45)(44 48 46)(49 53 51)(50 54 52)(55 57 59)(56 58 60)(61 63 65)(62 64 66)(67 69 71)(68 70 72)(73 75 77)(74 76 78)(79 81 83)(80 82 84)(85 87 89)(86 88 90)(91 93 95)(92 94 96)
(1 59)(2 60)(3 55)(4 56)(5 57)(6 58)(7 74)(8 75)(9 76)(10 77)(11 78)(12 73)(13 64)(14 65)(15 66)(16 61)(17 62)(18 63)(19 70)(20 71)(21 72)(22 67)(23 68)(24 69)(25 54)(26 49)(27 50)(28 51)(29 52)(30 53)(31 82)(32 83)(33 84)(34 79)(35 80)(36 81)(37 88)(38 89)(39 90)(40 85)(41 86)(42 87)(43 94)(44 95)(45 96)(46 91)(47 92)(48 93)
G:=sub<Sym(96)| (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), (1,22,41,33)(2,23,42,34)(3,24,37,35)(4,19,38,36)(5,20,39,31)(6,21,40,32)(7,46,51,13)(8,47,52,14)(9,48,53,15)(10,43,54,16)(11,44,49,17)(12,45,50,18)(25,94,77,61)(26,95,78,62)(27,96,73,63)(28,91,74,64)(29,92,75,65)(30,93,76,66)(55,80,88,69)(56,81,89,70)(57,82,90,71)(58,83,85,72)(59,84,86,67)(60,79,87,68), (1,47,41,14)(2,48,42,15)(3,43,37,16)(4,44,38,17)(5,45,39,18)(6,46,40,13)(7,32,51,21)(8,33,52,22)(9,34,53,23)(10,35,54,24)(11,36,49,19)(12,31,50,20)(25,69,77,80)(26,70,78,81)(27,71,73,82)(28,72,74,83)(29,67,75,84)(30,68,76,79)(55,94,88,61)(56,95,89,62)(57,96,90,63)(58,91,85,64)(59,92,86,65)(60,93,87,66), (1,5,3)(2,6,4)(7,11,9)(8,12,10)(13,17,15)(14,18,16)(19,23,21)(20,24,22)(25,27,29)(26,28,30)(31,35,33)(32,36,34)(37,41,39)(38,42,40)(43,47,45)(44,48,46)(49,53,51)(50,54,52)(55,57,59)(56,58,60)(61,63,65)(62,64,66)(67,69,71)(68,70,72)(73,75,77)(74,76,78)(79,81,83)(80,82,84)(85,87,89)(86,88,90)(91,93,95)(92,94,96), (1,59)(2,60)(3,55)(4,56)(5,57)(6,58)(7,74)(8,75)(9,76)(10,77)(11,78)(12,73)(13,64)(14,65)(15,66)(16,61)(17,62)(18,63)(19,70)(20,71)(21,72)(22,67)(23,68)(24,69)(25,54)(26,49)(27,50)(28,51)(29,52)(30,53)(31,82)(32,83)(33,84)(34,79)(35,80)(36,81)(37,88)(38,89)(39,90)(40,85)(41,86)(42,87)(43,94)(44,95)(45,96)(46,91)(47,92)(48,93)>;
G:=Group( (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), (1,22,41,33)(2,23,42,34)(3,24,37,35)(4,19,38,36)(5,20,39,31)(6,21,40,32)(7,46,51,13)(8,47,52,14)(9,48,53,15)(10,43,54,16)(11,44,49,17)(12,45,50,18)(25,94,77,61)(26,95,78,62)(27,96,73,63)(28,91,74,64)(29,92,75,65)(30,93,76,66)(55,80,88,69)(56,81,89,70)(57,82,90,71)(58,83,85,72)(59,84,86,67)(60,79,87,68), (1,47,41,14)(2,48,42,15)(3,43,37,16)(4,44,38,17)(5,45,39,18)(6,46,40,13)(7,32,51,21)(8,33,52,22)(9,34,53,23)(10,35,54,24)(11,36,49,19)(12,31,50,20)(25,69,77,80)(26,70,78,81)(27,71,73,82)(28,72,74,83)(29,67,75,84)(30,68,76,79)(55,94,88,61)(56,95,89,62)(57,96,90,63)(58,91,85,64)(59,92,86,65)(60,93,87,66), (1,5,3)(2,6,4)(7,11,9)(8,12,10)(13,17,15)(14,18,16)(19,23,21)(20,24,22)(25,27,29)(26,28,30)(31,35,33)(32,36,34)(37,41,39)(38,42,40)(43,47,45)(44,48,46)(49,53,51)(50,54,52)(55,57,59)(56,58,60)(61,63,65)(62,64,66)(67,69,71)(68,70,72)(73,75,77)(74,76,78)(79,81,83)(80,82,84)(85,87,89)(86,88,90)(91,93,95)(92,94,96), (1,59)(2,60)(3,55)(4,56)(5,57)(6,58)(7,74)(8,75)(9,76)(10,77)(11,78)(12,73)(13,64)(14,65)(15,66)(16,61)(17,62)(18,63)(19,70)(20,71)(21,72)(22,67)(23,68)(24,69)(25,54)(26,49)(27,50)(28,51)(29,52)(30,53)(31,82)(32,83)(33,84)(34,79)(35,80)(36,81)(37,88)(38,89)(39,90)(40,85)(41,86)(42,87)(43,94)(44,95)(45,96)(46,91)(47,92)(48,93) );
G=PermutationGroup([[(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)], [(1,22,41,33),(2,23,42,34),(3,24,37,35),(4,19,38,36),(5,20,39,31),(6,21,40,32),(7,46,51,13),(8,47,52,14),(9,48,53,15),(10,43,54,16),(11,44,49,17),(12,45,50,18),(25,94,77,61),(26,95,78,62),(27,96,73,63),(28,91,74,64),(29,92,75,65),(30,93,76,66),(55,80,88,69),(56,81,89,70),(57,82,90,71),(58,83,85,72),(59,84,86,67),(60,79,87,68)], [(1,47,41,14),(2,48,42,15),(3,43,37,16),(4,44,38,17),(5,45,39,18),(6,46,40,13),(7,32,51,21),(8,33,52,22),(9,34,53,23),(10,35,54,24),(11,36,49,19),(12,31,50,20),(25,69,77,80),(26,70,78,81),(27,71,73,82),(28,72,74,83),(29,67,75,84),(30,68,76,79),(55,94,88,61),(56,95,89,62),(57,96,90,63),(58,91,85,64),(59,92,86,65),(60,93,87,66)], [(1,5,3),(2,6,4),(7,11,9),(8,12,10),(13,17,15),(14,18,16),(19,23,21),(20,24,22),(25,27,29),(26,28,30),(31,35,33),(32,36,34),(37,41,39),(38,42,40),(43,47,45),(44,48,46),(49,53,51),(50,54,52),(55,57,59),(56,58,60),(61,63,65),(62,64,66),(67,69,71),(68,70,72),(73,75,77),(74,76,78),(79,81,83),(80,82,84),(85,87,89),(86,88,90),(91,93,95),(92,94,96)], [(1,59),(2,60),(3,55),(4,56),(5,57),(6,58),(7,74),(8,75),(9,76),(10,77),(11,78),(12,73),(13,64),(14,65),(15,66),(16,61),(17,62),(18,63),(19,70),(20,71),(21,72),(22,67),(23,68),(24,69),(25,54),(26,49),(27,50),(28,51),(29,52),(30,53),(31,82),(32,83),(33,84),(34,79),(35,80),(36,81),(37,88),(38,89),(39,90),(40,85),(41,86),(42,87),(43,94),(44,95),(45,96),(46,91),(47,92),(48,93)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 3A | 3B | 3C | 3D | 3E | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 6A | ··· | 6F | 6G | ··· | 6O | 6P | ··· | 6AA | 12A | ··· | 12L | 12M | ··· | 12T | 12U | ··· | 12AL |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 6 | ··· | 6 | 1 | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 3 | ··· | 3 | 4 | ··· | 4 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | S3 | D6 | D6 | C4○D4 | C3×S3 | S3×C6 | S3×C6 | C3×C4○D4 | Q8⋊3S3 | C3×Q8⋊3S3 |
| kernel | C6×Q8⋊3S3 | S3×C2×C12 | C6×D12 | C3×Q8⋊3S3 | Q8×C3×C6 | C2×Q8⋊3S3 | S3×C2×C4 | C2×D12 | Q8⋊3S3 | C6×Q8 | C6×Q8 | C2×C12 | C3×Q8 | C3×C6 | C2×Q8 | C2×C4 | Q8 | C6 | C6 | C2 |
| # reps | 1 | 3 | 3 | 8 | 1 | 2 | 6 | 6 | 16 | 2 | 1 | 3 | 4 | 4 | 2 | 6 | 8 | 8 | 2 | 4 |
Matrix representation of C6×Q8⋊3S3 ►in GL5(𝔽13)
| 4 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 12 | 12 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 6 | 5 |
| 0 | 0 | 0 | 3 | 7 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 12 |
| 0 | 0 | 0 | 2 | 9 |
G:=sub<GL(5,GF(13))| [4,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,12,0,0,0,2,12],[12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,6,3,0,0,0,5,7],[1,0,0,0,0,0,9,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,4,2,0,0,0,12,9] >;
C6×Q8⋊3S3 in GAP, Magma, Sage, TeX
C_6\times Q_8\rtimes_3S_3
% in TeX
G:=Group("C6xQ8:3S3"); // GroupNames label
G:=SmallGroup(288,996);
// by ID
G=gap.SmallGroup(288,996);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,268,1571,409,192,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^4=d^3=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e=b^-1,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations