direct product, metabelian, supersoluble, monomial
Aliases: C3×Q8×Dic3, C62.207C23, C3⋊3(Q8×C12), (C3×Q8)⋊5C12, C6.58(S3×Q8), C6.16(C6×Q8), C32⋊13(C4×Q8), (C6×Q8).13C6, (C6×Q8).30S3, (Q8×C32)⋊8C4, C4.4(C6×Dic3), C12.14(C2×C12), (C2×C12).331D6, C4⋊Dic3.12C6, (C4×Dic3).4C6, C6.26(C22×C12), C12.39(C2×Dic3), (C6×C12).127C22, C6.62(Q8⋊3S3), (Dic3×C12).13C2, C6.46(C22×Dic3), (C6×Dic3).139C22, C2.3(C3×S3×Q8), (Q8×C3×C6).7C2, C2.7(Dic3×C2×C6), (C2×C4).55(S3×C6), C6.35(C3×C4○D4), C22.26(S3×C2×C6), (C2×Q8).9(C3×S3), (C3×C6).68(C2×Q8), (C3×C12).69(C2×C4), (C2×C12).39(C2×C6), C2.3(C3×Q8⋊3S3), (C3×C4⋊Dic3).19C2, (C2×C6).62(C22×C6), (C3×C6).157(C4○D4), (C2×C6).340(C22×S3), (C3×C6).117(C22×C4), (C2×Dic3).40(C2×C6), SmallGroup(288,716)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×Q8×Dic3
G = < a,b,c,d,e | a3=b4=d6=1, c2=b2, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 266 in 159 conjugacy classes, 102 normal (28 characteristic)
C1, C2, C3, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, Q8, C32, Dic3, Dic3, C12, C12, C2×C6, C2×C6, C42, C4⋊C4, C2×Q8, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, C4×Q8, C3×Dic3, C3×Dic3, C3×C12, C62, C4×Dic3, C4⋊Dic3, C4×C12, C3×C4⋊C4, C6×Q8, C6×Q8, C6×Dic3, C6×Dic3, C6×C12, Q8×C32, Q8×Dic3, Q8×C12, Dic3×C12, C3×C4⋊Dic3, Q8×C3×C6, C3×Q8×Dic3
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, Q8, C23, Dic3, C12, D6, C2×C6, C22×C4, C2×Q8, C4○D4, C3×S3, C2×Dic3, C2×C12, C3×Q8, C22×S3, C22×C6, C4×Q8, C3×Dic3, S3×C6, S3×Q8, Q8⋊3S3, C22×Dic3, C22×C12, C6×Q8, C3×C4○D4, C6×Dic3, S3×C2×C6, Q8×Dic3, Q8×C12, C3×S3×Q8, C3×Q8⋊3S3, Dic3×C2×C6, C3×Q8×Dic3
►On 96 points
Generators in S96
(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 25 16 21)(2 26 17 22)(3 27 18 23)(4 28 13 24)(5 29 14 19)(6 30 15 20)(7 84 93 89)(8 79 94 90)(9 80 95 85)(10 81 96 86)(11 82 91 87)(12 83 92 88)(31 46 41 49)(32 47 42 50)(33 48 37 51)(34 43 38 52)(35 44 39 53)(36 45 40 54)(55 75 66 70)(56 76 61 71)(57 77 62 72)(58 78 63 67)(59 73 64 68)(60 74 65 69)
(1 37 16 33)(2 38 17 34)(3 39 18 35)(4 40 13 36)(5 41 14 31)(6 42 15 32)(7 77 93 72)(8 78 94 67)(9 73 95 68)(10 74 96 69)(11 75 91 70)(12 76 92 71)(19 49 29 46)(20 50 30 47)(21 51 25 48)(22 52 26 43)(23 53 27 44)(24 54 28 45)(55 87 66 82)(56 88 61 83)(57 89 62 84)(58 90 63 79)(59 85 64 80)(60 86 65 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 57 4 60)(2 56 5 59)(3 55 6 58)(7 54 10 51)(8 53 11 50)(9 52 12 49)(13 65 16 62)(14 64 17 61)(15 63 18 66)(19 68 22 71)(20 67 23 70)(21 72 24 69)(25 77 28 74)(26 76 29 73)(27 75 30 78)(31 80 34 83)(32 79 35 82)(33 84 36 81)(37 89 40 86)(38 88 41 85)(39 87 42 90)(43 92 46 95)(44 91 47 94)(45 96 48 93)
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), (1,25,16,21)(2,26,17,22)(3,27,18,23)(4,28,13,24)(5,29,14,19)(6,30,15,20)(7,84,93,89)(8,79,94,90)(9,80,95,85)(10,81,96,86)(11,82,91,87)(12,83,92,88)(31,46,41,49)(32,47,42,50)(33,48,37,51)(34,43,38,52)(35,44,39,53)(36,45,40,54)(55,75,66,70)(56,76,61,71)(57,77,62,72)(58,78,63,67)(59,73,64,68)(60,74,65,69), (1,37,16,33)(2,38,17,34)(3,39,18,35)(4,40,13,36)(5,41,14,31)(6,42,15,32)(7,77,93,72)(8,78,94,67)(9,73,95,68)(10,74,96,69)(11,75,91,70)(12,76,92,71)(19,49,29,46)(20,50,30,47)(21,51,25,48)(22,52,26,43)(23,53,27,44)(24,54,28,45)(55,87,66,82)(56,88,61,83)(57,89,62,84)(58,90,63,79)(59,85,64,80)(60,86,65,81), (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), (1,57,4,60)(2,56,5,59)(3,55,6,58)(7,54,10,51)(8,53,11,50)(9,52,12,49)(13,65,16,62)(14,64,17,61)(15,63,18,66)(19,68,22,71)(20,67,23,70)(21,72,24,69)(25,77,28,74)(26,76,29,73)(27,75,30,78)(31,80,34,83)(32,79,35,82)(33,84,36,81)(37,89,40,86)(38,88,41,85)(39,87,42,90)(43,92,46,95)(44,91,47,94)(45,96,48,93)>;
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), (1,25,16,21)(2,26,17,22)(3,27,18,23)(4,28,13,24)(5,29,14,19)(6,30,15,20)(7,84,93,89)(8,79,94,90)(9,80,95,85)(10,81,96,86)(11,82,91,87)(12,83,92,88)(31,46,41,49)(32,47,42,50)(33,48,37,51)(34,43,38,52)(35,44,39,53)(36,45,40,54)(55,75,66,70)(56,76,61,71)(57,77,62,72)(58,78,63,67)(59,73,64,68)(60,74,65,69), (1,37,16,33)(2,38,17,34)(3,39,18,35)(4,40,13,36)(5,41,14,31)(6,42,15,32)(7,77,93,72)(8,78,94,67)(9,73,95,68)(10,74,96,69)(11,75,91,70)(12,76,92,71)(19,49,29,46)(20,50,30,47)(21,51,25,48)(22,52,26,43)(23,53,27,44)(24,54,28,45)(55,87,66,82)(56,88,61,83)(57,89,62,84)(58,90,63,79)(59,85,64,80)(60,86,65,81), (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), (1,57,4,60)(2,56,5,59)(3,55,6,58)(7,54,10,51)(8,53,11,50)(9,52,12,49)(13,65,16,62)(14,64,17,61)(15,63,18,66)(19,68,22,71)(20,67,23,70)(21,72,24,69)(25,77,28,74)(26,76,29,73)(27,75,30,78)(31,80,34,83)(32,79,35,82)(33,84,36,81)(37,89,40,86)(38,88,41,85)(39,87,42,90)(43,92,46,95)(44,91,47,94)(45,96,48,93) );
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)], [(1,25,16,21),(2,26,17,22),(3,27,18,23),(4,28,13,24),(5,29,14,19),(6,30,15,20),(7,84,93,89),(8,79,94,90),(9,80,95,85),(10,81,96,86),(11,82,91,87),(12,83,92,88),(31,46,41,49),(32,47,42,50),(33,48,37,51),(34,43,38,52),(35,44,39,53),(36,45,40,54),(55,75,66,70),(56,76,61,71),(57,77,62,72),(58,78,63,67),(59,73,64,68),(60,74,65,69)], [(1,37,16,33),(2,38,17,34),(3,39,18,35),(4,40,13,36),(5,41,14,31),(6,42,15,32),(7,77,93,72),(8,78,94,67),(9,73,95,68),(10,74,96,69),(11,75,91,70),(12,76,92,71),(19,49,29,46),(20,50,30,47),(21,51,25,48),(22,52,26,43),(23,53,27,44),(24,54,28,45),(55,87,66,82),(56,88,61,83),(57,89,62,84),(58,90,63,79),(59,85,64,80),(60,86,65,81)], [(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)], [(1,57,4,60),(2,56,5,59),(3,55,6,58),(7,54,10,51),(8,53,11,50),(9,52,12,49),(13,65,16,62),(14,64,17,61),(15,63,18,66),(19,68,22,71),(20,67,23,70),(21,72,24,69),(25,77,28,74),(26,76,29,73),(27,75,30,78),(31,80,34,83),(32,79,35,82),(33,84,36,81),(37,89,40,86),(38,88,41,85),(39,87,42,90),(43,92,46,95),(44,91,47,94),(45,96,48,93)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4P | 6A | ··· | 6F | 6G | ··· | 6O | 12A | ··· | 12L | 12M | ··· | 12T | 12U | ··· | 12AL | 12AM | ··· | 12AX |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 6 | ··· | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 3 | ··· | 3 | 4 | ··· | 4 | 6 | ··· | 6 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | - | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | S3 | Q8 | D6 | Dic3 | C4○D4 | C3×S3 | C3×Q8 | S3×C6 | C3×Dic3 | C3×C4○D4 | S3×Q8 | Q8⋊3S3 | C3×S3×Q8 | C3×Q8⋊3S3 |
| kernel | C3×Q8×Dic3 | Dic3×C12 | C3×C4⋊Dic3 | Q8×C3×C6 | Q8×Dic3 | Q8×C32 | C4×Dic3 | C4⋊Dic3 | C6×Q8 | C3×Q8 | C6×Q8 | C3×Dic3 | C2×C12 | C3×Q8 | C3×C6 | C2×Q8 | Dic3 | C2×C4 | Q8 | C6 | C6 | C6 | C2 | C2 |
| # reps | 1 | 3 | 3 | 1 | 2 | 8 | 6 | 6 | 2 | 16 | 1 | 2 | 3 | 4 | 2 | 2 | 4 | 6 | 8 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of C3×Q8×Dic3 ►in GL4(𝔽13) generated by
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 8 |
| 4 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,0,12,0,0,1,0],[12,0,0,0,0,12,0,0,0,0,5,0,0,0,0,8],[4,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[0,12,0,0,1,0,0,0,0,0,12,0,0,0,0,12] >;
C3×Q8×Dic3 in GAP, Magma, Sage, TeX
C_3\times Q_8\times {\rm Dic}_3 % in TeX
G:=Group("C3xQ8xDic3"); // GroupNames label
G:=SmallGroup(288,716);
// by ID
G=gap.SmallGroup(288,716);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,168,344,555,268,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=d^6=1,c^2=b^2,e^2=d^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations