direct product, metabelian, supersoluble, monomial
Aliases: C3×Dic6⋊C4, Dic6⋊5C12, C62.178C23, C3⋊2(Q8×C12), C6.9(C6×Q8), C4.4(S3×C12), C6.51(S3×Q8), C32⋊11(C4×Q8), C12.52(C4×S3), (C3×Dic6)⋊9C4, Dic3⋊3(C3×Q8), C12.10(C2×C12), (C3×Dic3)⋊10Q8, (C2×C12).268D6, Dic3⋊C4.4C6, C6.8(C22×C12), (C2×Dic6).7C6, (C4×Dic3).8C6, Dic3.2(C2×C12), (C6×Dic6).13C2, (C6×C12).245C22, (Dic3×C12).19C2, C6.116(D4⋊2S3), (C6×Dic3).159C22, C2.1(C3×S3×Q8), (C3×C4⋊C4).4C6, C4⋊C4.7(C3×S3), C2.10(S3×C2×C12), C6.107(S3×C2×C4), (C3×C4⋊C4).30S3, (C2×C4).41(S3×C6), C6.24(C3×C4○D4), C22.15(S3×C2×C6), (C3×C6).61(C2×Q8), (C2×C12).55(C2×C6), (C3×C12).65(C2×C4), C2.3(C3×D4⋊2S3), (C32×C4⋊C4).7C2, (C3×C6).79(C22×C4), (C2×C6).33(C22×C6), (C3×C6).130(C4○D4), (C3×Dic3⋊C4).12C2, (C2×C6).311(C22×S3), (C2×Dic3).24(C2×C6), (C3×Dic3).18(C2×C4), SmallGroup(288,658)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×Dic6⋊C4
G = < a,b,c,d | a3=b12=d4=1, c2=b6, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b7, cd=dc >
Subgroups: 274 in 153 conjugacy classes, 86 normal (38 characteristic)
C1, C2, C3, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, C2×C4, Q8, C32, Dic3, Dic3, C12, C12, C2×C6, C2×C6, C42, C4⋊C4, C4⋊C4, C2×Q8, C3×C6, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C2×C12, C3×Q8, C4×Q8, C3×Dic3, C3×Dic3, C3×C12, C3×C12, C62, C4×Dic3, C4×Dic3, Dic3⋊C4, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C6×Q8, C3×Dic6, C6×Dic3, C6×Dic3, C6×C12, C6×C12, Dic6⋊C4, Q8×C12, Dic3×C12, Dic3×C12, C3×Dic3⋊C4, C32×C4⋊C4, C6×Dic6, C3×Dic6⋊C4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, Q8, C23, C12, D6, C2×C6, C22×C4, C2×Q8, C4○D4, C3×S3, C4×S3, C2×C12, C3×Q8, C22×S3, C22×C6, C4×Q8, S3×C6, S3×C2×C4, D4⋊2S3, S3×Q8, C22×C12, C6×Q8, C3×C4○D4, S3×C12, S3×C2×C6, Dic6⋊C4, Q8×C12, S3×C2×C12, C3×D4⋊2S3, C3×S3×Q8, C3×Dic6⋊C4
►On 96 points
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 41 45)(38 42 46)(39 43 47)(40 44 48)(49 57 53)(50 58 54)(51 59 55)(52 60 56)(61 65 69)(62 66 70)(63 67 71)(64 68 72)(73 77 81)(74 78 82)(75 79 83)(76 80 84)(85 93 89)(86 94 90)(87 95 91)(88 96 92)
(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 29 7 35)(2 28 8 34)(3 27 9 33)(4 26 10 32)(5 25 11 31)(6 36 12 30)(13 38 19 44)(14 37 20 43)(15 48 21 42)(16 47 22 41)(17 46 23 40)(18 45 24 39)(49 66 55 72)(50 65 56 71)(51 64 57 70)(52 63 58 69)(53 62 59 68)(54 61 60 67)(73 90 79 96)(74 89 80 95)(75 88 81 94)(76 87 82 93)(77 86 83 92)(78 85 84 91)
(1 41 78 68)(2 48 79 63)(3 43 80 70)(4 38 81 65)(5 45 82 72)(6 40 83 67)(7 47 84 62)(8 42 73 69)(9 37 74 64)(10 44 75 71)(11 39 76 66)(12 46 77 61)(13 88 50 32)(14 95 51 27)(15 90 52 34)(16 85 53 29)(17 92 54 36)(18 87 55 31)(19 94 56 26)(20 89 57 33)(21 96 58 28)(22 91 59 35)(23 86 60 30)(24 93 49 25)
G:=sub<Sym(96)| (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,41,45)(38,42,46)(39,43,47)(40,44,48)(49,57,53)(50,58,54)(51,59,55)(52,60,56)(61,65,69)(62,66,70)(63,67,71)(64,68,72)(73,77,81)(74,78,82)(75,79,83)(76,80,84)(85,93,89)(86,94,90)(87,95,91)(88,96,92), (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), (1,29,7,35)(2,28,8,34)(3,27,9,33)(4,26,10,32)(5,25,11,31)(6,36,12,30)(13,38,19,44)(14,37,20,43)(15,48,21,42)(16,47,22,41)(17,46,23,40)(18,45,24,39)(49,66,55,72)(50,65,56,71)(51,64,57,70)(52,63,58,69)(53,62,59,68)(54,61,60,67)(73,90,79,96)(74,89,80,95)(75,88,81,94)(76,87,82,93)(77,86,83,92)(78,85,84,91), (1,41,78,68)(2,48,79,63)(3,43,80,70)(4,38,81,65)(5,45,82,72)(6,40,83,67)(7,47,84,62)(8,42,73,69)(9,37,74,64)(10,44,75,71)(11,39,76,66)(12,46,77,61)(13,88,50,32)(14,95,51,27)(15,90,52,34)(16,85,53,29)(17,92,54,36)(18,87,55,31)(19,94,56,26)(20,89,57,33)(21,96,58,28)(22,91,59,35)(23,86,60,30)(24,93,49,25)>;
G:=Group( (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,41,45)(38,42,46)(39,43,47)(40,44,48)(49,57,53)(50,58,54)(51,59,55)(52,60,56)(61,65,69)(62,66,70)(63,67,71)(64,68,72)(73,77,81)(74,78,82)(75,79,83)(76,80,84)(85,93,89)(86,94,90)(87,95,91)(88,96,92), (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), (1,29,7,35)(2,28,8,34)(3,27,9,33)(4,26,10,32)(5,25,11,31)(6,36,12,30)(13,38,19,44)(14,37,20,43)(15,48,21,42)(16,47,22,41)(17,46,23,40)(18,45,24,39)(49,66,55,72)(50,65,56,71)(51,64,57,70)(52,63,58,69)(53,62,59,68)(54,61,60,67)(73,90,79,96)(74,89,80,95)(75,88,81,94)(76,87,82,93)(77,86,83,92)(78,85,84,91), (1,41,78,68)(2,48,79,63)(3,43,80,70)(4,38,81,65)(5,45,82,72)(6,40,83,67)(7,47,84,62)(8,42,73,69)(9,37,74,64)(10,44,75,71)(11,39,76,66)(12,46,77,61)(13,88,50,32)(14,95,51,27)(15,90,52,34)(16,85,53,29)(17,92,54,36)(18,87,55,31)(19,94,56,26)(20,89,57,33)(21,96,58,28)(22,91,59,35)(23,86,60,30)(24,93,49,25) );
G=PermutationGroup([[(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,21,17),(14,22,18),(15,23,19),(16,24,20),(25,33,29),(26,34,30),(27,35,31),(28,36,32),(37,41,45),(38,42,46),(39,43,47),(40,44,48),(49,57,53),(50,58,54),(51,59,55),(52,60,56),(61,65,69),(62,66,70),(63,67,71),(64,68,72),(73,77,81),(74,78,82),(75,79,83),(76,80,84),(85,93,89),(86,94,90),(87,95,91),(88,96,92)], [(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)], [(1,29,7,35),(2,28,8,34),(3,27,9,33),(4,26,10,32),(5,25,11,31),(6,36,12,30),(13,38,19,44),(14,37,20,43),(15,48,21,42),(16,47,22,41),(17,46,23,40),(18,45,24,39),(49,66,55,72),(50,65,56,71),(51,64,57,70),(52,63,58,69),(53,62,59,68),(54,61,60,67),(73,90,79,96),(74,89,80,95),(75,88,81,94),(76,87,82,93),(77,86,83,92),(78,85,84,91)], [(1,41,78,68),(2,48,79,63),(3,43,80,70),(4,38,81,65),(5,45,82,72),(6,40,83,67),(7,47,84,62),(8,42,73,69),(9,37,74,64),(10,44,75,71),(11,39,76,66),(12,46,77,61),(13,88,50,32),(14,95,51,27),(15,90,52,34),(16,85,53,29),(17,92,54,36),(18,87,55,31),(19,94,56,26),(20,89,57,33),(21,96,58,28),(22,91,59,35),(23,86,60,30),(24,93,49,25)]])
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 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | - | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C12 | S3 | Q8 | D6 | C4○D4 | C3×S3 | C3×Q8 | C4×S3 | S3×C6 | C3×C4○D4 | S3×C12 | D4⋊2S3 | S3×Q8 | C3×D4⋊2S3 | C3×S3×Q8 |
| kernel | C3×Dic6⋊C4 | Dic3×C12 | C3×Dic3⋊C4 | C32×C4⋊C4 | C6×Dic6 | Dic6⋊C4 | C3×Dic6 | C4×Dic3 | Dic3⋊C4 | C3×C4⋊C4 | C2×Dic6 | Dic6 | C3×C4⋊C4 | C3×Dic3 | C2×C12 | C3×C6 | C4⋊C4 | Dic3 | C12 | C2×C4 | C6 | C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 3 | 2 | 1 | 1 | 2 | 8 | 6 | 4 | 2 | 2 | 16 | 1 | 2 | 3 | 2 | 2 | 4 | 4 | 6 | 4 | 8 | 1 | 1 | 2 | 2 |
Matrix representation of C3×Dic6⋊C4 ►in GL5(𝔽13)
| 9 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 9 | 6 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 5 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 8 | 3 | 0 | 0 |
| 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 |
| 0 | 0 | 0 | 5 | 0 |
| 8 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(13))| [9,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,9,0,0,0,0,6,3,0,0,0,0,0,8,0,0,0,0,0,5],[12,0,0,0,0,0,8,5,0,0,0,3,5,0,0,0,0,0,0,5,0,0,0,5,0],[8,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,1,0] >;
C3×Dic6⋊C4 in GAP, Magma, Sage, TeX
C_3\times {\rm Dic}_6\rtimes C_4 % in TeX
G:=Group("C3xDic6:C4"); // GroupNames label
G:=SmallGroup(288,658);
// by ID
G=gap.SmallGroup(288,658);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,176,303,394,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^12=d^4=1,c^2=b^6,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=b^7,c*d=d*c>;
// generators/relations