direct product, metabelian, supersoluble, monomial
Aliases: C3×D6⋊3Q8, C62.208C23, D6⋊3(C3×Q8), (C6×Q8)⋊7C6, D6⋊C4.6C6, (S3×C6)⋊10Q8, (C6×Q8)⋊14S3, C6.57(C6×D4), C6.59(S3×Q8), C6.17(C6×Q8), C4⋊Dic3⋊15C6, C12.22(C3×D4), (C3×C12).92D4, Dic3⋊C4⋊16C6, (C2×C12).245D6, C32⋊24(C22⋊Q8), C12.105(C3⋊D4), (C6×C12).128C22, C6.63(Q8⋊3S3), (C6×Dic3).103C22, (Q8×C3×C6)⋊3C2, C2.9(C3×S3×Q8), (S3×C2×C4).4C6, (C2×Q8)⋊7(C3×S3), C3⋊5(C3×C22⋊Q8), (S3×C2×C12).13C2, (C2×C4).56(S3×C6), C6.36(C3×C4○D4), C4.18(C3×C3⋊D4), C2.21(C6×C3⋊D4), C22.64(S3×C2×C6), (C3×C6).69(C2×Q8), (C2×C12).73(C2×C6), (C3×D6⋊C4).16C2, (C3×C4⋊Dic3)⋊24C2, (C3×C6).265(C2×D4), C6.158(C2×C3⋊D4), C2.8(C3×Q8⋊3S3), (C3×Dic3⋊C4)⋊38C2, (S3×C2×C6).101C22, (C2×C6).63(C22×C6), (C3×C6).158(C4○D4), (C22×S3).28(C2×C6), (C2×C6).341(C22×S3), (C2×Dic3).41(C2×C6), SmallGroup(288,717)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D6⋊3Q8
G = < a,b,c,d,e | a3=b6=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=b3c, ce=ec, ede-1=d-1 >
Subgroups: 362 in 167 conjugacy classes, 70 normal (38 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, Q8, C23, C32, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C3×S3, C3×C6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C2×C12, C3×Q8, C22×S3, C22×C6, C22⋊Q8, C3×Dic3, C3×C12, C3×C12, S3×C6, S3×C6, C62, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, C22×C12, C6×Q8, C6×Q8, S3×C12, C6×Dic3, C6×Dic3, C6×C12, C6×C12, Q8×C32, S3×C2×C6, D6⋊3Q8, C3×C22⋊Q8, C3×Dic3⋊C4, C3×C4⋊Dic3, C3×D6⋊C4, S3×C2×C12, Q8×C3×C6, C3×D6⋊3Q8
Quotients: C1, C2, C3, C22, S3, C6, D4, Q8, C23, D6, C2×C6, C2×D4, C2×Q8, C4○D4, C3×S3, C3⋊D4, C3×D4, C3×Q8, C22×S3, C22×C6, C22⋊Q8, S3×C6, S3×Q8, Q8⋊3S3, C2×C3⋊D4, C6×D4, C6×Q8, C3×C4○D4, C3×C3⋊D4, S3×C2×C6, D6⋊3Q8, C3×C22⋊Q8, C3×S3×Q8, C3×Q8⋊3S3, C6×C3⋊D4, C3×D6⋊3Q8
►On 96 points
(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 21 23)(20 22 24)(25 27 29)(26 28 30)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 45 47)(44 46 48)(49 51 53)(50 52 54)(55 59 57)(56 60 58)(61 65 63)(62 66 64)(67 71 69)(68 72 70)(73 77 75)(74 78 76)(79 83 81)(80 84 82)(85 89 87)(86 90 88)(91 95 93)(92 96 94)
(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 64)(2 63)(3 62)(4 61)(5 66)(6 65)(7 46)(8 45)(9 44)(10 43)(11 48)(12 47)(13 60)(14 59)(15 58)(16 57)(17 56)(18 55)(19 77)(20 76)(21 75)(22 74)(23 73)(24 78)(25 72)(26 71)(27 70)(28 69)(29 68)(30 67)(31 89)(32 88)(33 87)(34 86)(35 85)(36 90)(37 84)(38 83)(39 82)(40 81)(41 80)(42 79)(49 95)(50 94)(51 93)(52 92)(53 91)(54 96)
(1 28 16 20)(2 29 17 21)(3 30 18 22)(4 25 13 23)(5 26 14 24)(6 27 15 19)(7 82 91 89)(8 83 92 90)(9 84 93 85)(10 79 94 86)(11 80 95 87)(12 81 96 88)(31 43 39 50)(32 44 40 51)(33 45 41 52)(34 46 42 53)(35 47 37 54)(36 48 38 49)(55 77 62 70)(56 78 63 71)(57 73 64 72)(58 74 65 67)(59 75 66 68)(60 76 61 69)
(1 40 16 32)(2 41 17 33)(3 42 18 34)(4 37 13 35)(5 38 14 36)(6 39 15 31)(7 74 91 67)(8 75 92 68)(9 76 93 69)(10 77 94 70)(11 78 95 71)(12 73 96 72)(19 50 27 43)(20 51 28 44)(21 52 29 45)(22 53 30 46)(23 54 25 47)(24 49 26 48)(55 86 62 79)(56 87 63 80)(57 88 64 81)(58 89 65 82)(59 90 66 83)(60 85 61 84)
G:=sub<Sym(96)| (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,15,17)(14,16,18)(19,21,23)(20,22,24)(25,27,29)(26,28,30)(31,33,35)(32,34,36)(37,39,41)(38,40,42)(43,45,47)(44,46,48)(49,51,53)(50,52,54)(55,59,57)(56,60,58)(61,65,63)(62,66,64)(67,71,69)(68,72,70)(73,77,75)(74,78,76)(79,83,81)(80,84,82)(85,89,87)(86,90,88)(91,95,93)(92,96,94), (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), (1,64)(2,63)(3,62)(4,61)(5,66)(6,65)(7,46)(8,45)(9,44)(10,43)(11,48)(12,47)(13,60)(14,59)(15,58)(16,57)(17,56)(18,55)(19,77)(20,76)(21,75)(22,74)(23,73)(24,78)(25,72)(26,71)(27,70)(28,69)(29,68)(30,67)(31,89)(32,88)(33,87)(34,86)(35,85)(36,90)(37,84)(38,83)(39,82)(40,81)(41,80)(42,79)(49,95)(50,94)(51,93)(52,92)(53,91)(54,96), (1,28,16,20)(2,29,17,21)(3,30,18,22)(4,25,13,23)(5,26,14,24)(6,27,15,19)(7,82,91,89)(8,83,92,90)(9,84,93,85)(10,79,94,86)(11,80,95,87)(12,81,96,88)(31,43,39,50)(32,44,40,51)(33,45,41,52)(34,46,42,53)(35,47,37,54)(36,48,38,49)(55,77,62,70)(56,78,63,71)(57,73,64,72)(58,74,65,67)(59,75,66,68)(60,76,61,69), (1,40,16,32)(2,41,17,33)(3,42,18,34)(4,37,13,35)(5,38,14,36)(6,39,15,31)(7,74,91,67)(8,75,92,68)(9,76,93,69)(10,77,94,70)(11,78,95,71)(12,73,96,72)(19,50,27,43)(20,51,28,44)(21,52,29,45)(22,53,30,46)(23,54,25,47)(24,49,26,48)(55,86,62,79)(56,87,63,80)(57,88,64,81)(58,89,65,82)(59,90,66,83)(60,85,61,84)>;
G:=Group( (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,15,17)(14,16,18)(19,21,23)(20,22,24)(25,27,29)(26,28,30)(31,33,35)(32,34,36)(37,39,41)(38,40,42)(43,45,47)(44,46,48)(49,51,53)(50,52,54)(55,59,57)(56,60,58)(61,65,63)(62,66,64)(67,71,69)(68,72,70)(73,77,75)(74,78,76)(79,83,81)(80,84,82)(85,89,87)(86,90,88)(91,95,93)(92,96,94), (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), (1,64)(2,63)(3,62)(4,61)(5,66)(6,65)(7,46)(8,45)(9,44)(10,43)(11,48)(12,47)(13,60)(14,59)(15,58)(16,57)(17,56)(18,55)(19,77)(20,76)(21,75)(22,74)(23,73)(24,78)(25,72)(26,71)(27,70)(28,69)(29,68)(30,67)(31,89)(32,88)(33,87)(34,86)(35,85)(36,90)(37,84)(38,83)(39,82)(40,81)(41,80)(42,79)(49,95)(50,94)(51,93)(52,92)(53,91)(54,96), (1,28,16,20)(2,29,17,21)(3,30,18,22)(4,25,13,23)(5,26,14,24)(6,27,15,19)(7,82,91,89)(8,83,92,90)(9,84,93,85)(10,79,94,86)(11,80,95,87)(12,81,96,88)(31,43,39,50)(32,44,40,51)(33,45,41,52)(34,46,42,53)(35,47,37,54)(36,48,38,49)(55,77,62,70)(56,78,63,71)(57,73,64,72)(58,74,65,67)(59,75,66,68)(60,76,61,69), (1,40,16,32)(2,41,17,33)(3,42,18,34)(4,37,13,35)(5,38,14,36)(6,39,15,31)(7,74,91,67)(8,75,92,68)(9,76,93,69)(10,77,94,70)(11,78,95,71)(12,73,96,72)(19,50,27,43)(20,51,28,44)(21,52,29,45)(22,53,30,46)(23,54,25,47)(24,49,26,48)(55,86,62,79)(56,87,63,80)(57,88,64,81)(58,89,65,82)(59,90,66,83)(60,85,61,84) );
G=PermutationGroup([[(1,3,5),(2,4,6),(7,11,9),(8,12,10),(13,15,17),(14,16,18),(19,21,23),(20,22,24),(25,27,29),(26,28,30),(31,33,35),(32,34,36),(37,39,41),(38,40,42),(43,45,47),(44,46,48),(49,51,53),(50,52,54),(55,59,57),(56,60,58),(61,65,63),(62,66,64),(67,71,69),(68,72,70),(73,77,75),(74,78,76),(79,83,81),(80,84,82),(85,89,87),(86,90,88),(91,95,93),(92,96,94)], [(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)], [(1,64),(2,63),(3,62),(4,61),(5,66),(6,65),(7,46),(8,45),(9,44),(10,43),(11,48),(12,47),(13,60),(14,59),(15,58),(16,57),(17,56),(18,55),(19,77),(20,76),(21,75),(22,74),(23,73),(24,78),(25,72),(26,71),(27,70),(28,69),(29,68),(30,67),(31,89),(32,88),(33,87),(34,86),(35,85),(36,90),(37,84),(38,83),(39,82),(40,81),(41,80),(42,79),(49,95),(50,94),(51,93),(52,92),(53,91),(54,96)], [(1,28,16,20),(2,29,17,21),(3,30,18,22),(4,25,13,23),(5,26,14,24),(6,27,15,19),(7,82,91,89),(8,83,92,90),(9,84,93,85),(10,79,94,86),(11,80,95,87),(12,81,96,88),(31,43,39,50),(32,44,40,51),(33,45,41,52),(34,46,42,53),(35,47,37,54),(36,48,38,49),(55,77,62,70),(56,78,63,71),(57,73,64,72),(58,74,65,67),(59,75,66,68),(60,76,61,69)], [(1,40,16,32),(2,41,17,33),(3,42,18,34),(4,37,13,35),(5,38,14,36),(6,39,15,31),(7,74,91,67),(8,75,92,68),(9,76,93,69),(10,77,94,70),(11,78,95,71),(12,73,96,72),(19,50,27,43),(20,51,28,44),(21,52,29,45),(22,53,30,46),(23,54,25,47),(24,49,26,48),(55,86,62,79),(56,87,63,80),(57,88,64,81),(58,89,65,82),(59,90,66,83),(60,85,61,84)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 6G | ··· | 6O | 6P | 6Q | 6R | 6S | 12A | 12B | 12C | 12D | 12E | ··· | 12Z | 12AA | 12AB | 12AC | 12AD | 12AE | 12AF | 12AG | 12AH |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 12 | 12 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | - | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | S3 | D4 | Q8 | D6 | C4○D4 | C3×S3 | C3⋊D4 | C3×D4 | C3×Q8 | S3×C6 | C3×C4○D4 | C3×C3⋊D4 | S3×Q8 | Q8⋊3S3 | C3×S3×Q8 | C3×Q8⋊3S3 |
| kernel | C3×D6⋊3Q8 | C3×Dic3⋊C4 | C3×C4⋊Dic3 | C3×D6⋊C4 | S3×C2×C12 | Q8×C3×C6 | D6⋊3Q8 | Dic3⋊C4 | C4⋊Dic3 | D6⋊C4 | S3×C2×C4 | C6×Q8 | C6×Q8 | C3×C12 | S3×C6 | C2×C12 | C3×C6 | C2×Q8 | C12 | C12 | D6 | C2×C4 | C6 | C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 4 | 2 | 4 | 2 | 2 | 1 | 2 | 2 | 3 | 2 | 2 | 4 | 4 | 4 | 6 | 4 | 8 | 1 | 1 | 2 | 2 |
Matrix representation of C3×D6⋊3Q8 ►in GL4(𝔽13) generated by
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 10 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 8 | 1 |
| 12 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 3 |
| 0 | 0 | 8 | 1 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 12 | 8 |
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,10,0,0,0,0,12,0,0,0,0,12],[0,4,0,0,10,0,0,0,0,0,12,8,0,0,0,1],[12,0,0,0,0,1,0,0,0,0,12,8,0,0,3,1],[12,0,0,0,0,12,0,0,0,0,5,12,0,0,0,8] >;
C3×D6⋊3Q8 in GAP, Magma, Sage, TeX
C_3\times D_6\rtimes_3Q_8
% in TeX
G:=Group("C3xD6:3Q8"); // GroupNames label
G:=SmallGroup(288,717);
// by ID
G=gap.SmallGroup(288,717);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,344,590,555,268,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^6=c^2=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b^3*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations