direct product, metabelian, supersoluble, monomial
Aliases: C3×Dic3⋊5D4, D12⋊5C12, C62.184C23, C12⋊8(C4×S3), C4⋊1(S3×C12), C3⋊3(D4×C12), C12⋊2(C2×C12), D6⋊C4⋊12C6, (C3×D12)⋊9C4, D6⋊3(C2×C12), C6.24(C6×D4), C32⋊20(C4×D4), Dic3⋊5(C3×D4), (C4×Dic3)⋊3C6, (C2×D12).7C6, C6.183(S3×D4), (C3×Dic3)⋊20D4, (C6×D12).12C2, (C2×C12).271D6, (Dic3×C12)⋊13C2, C6.11(C22×C12), (C6×C12).249C22, C6.58(Q8⋊3S3), (C6×Dic3).127C22, (C3×C4⋊C4)⋊4C6, C4⋊C4⋊8(C3×S3), C2.4(C3×S3×D4), (S3×C2×C4)⋊12C6, (C3×C4⋊C4)⋊17S3, (S3×C2×C12)⋊26C2, C2.13(S3×C2×C12), C6.110(S3×C2×C4), (S3×C6)⋊16(C2×C4), (C3×C12)⋊10(C2×C4), (C3×D6⋊C4)⋊28C2, (C32×C4⋊C4)⋊5C2, (C2×C4).43(S3×C6), C6.33(C3×C4○D4), C22.18(S3×C2×C6), (C2×C12).58(C2×C6), (C3×C6).212(C2×D4), (S3×C2×C6).93C22, C2.2(C3×Q8⋊3S3), (C3×C6).82(C22×C4), (C2×C6).39(C22×C6), (C3×C6).155(C4○D4), (C22×S3).20(C2×C6), (C2×C6).317(C22×S3), (C2×Dic3).48(C2×C6), SmallGroup(288,664)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×Dic3⋊5D4
G = < a,b,c,d,e | a3=b6=d4=e2=1, c2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ce=ec, ede=d-1 >
Subgroups: 466 in 201 conjugacy classes, 86 normal (38 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, C23, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×S3, C3×C6, C4×S3, D12, C2×Dic3, C2×C12, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C4×D4, C3×Dic3, C3×Dic3, C3×C12, C3×C12, S3×C6, S3×C6, C62, C4×Dic3, D6⋊C4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C22×C12, C6×D4, S3×C12, C3×D12, C6×Dic3, C6×C12, C6×C12, S3×C2×C6, Dic3⋊5D4, D4×C12, Dic3×C12, C3×D6⋊C4, C32×C4⋊C4, S3×C2×C12, C6×D12, C3×Dic3⋊5D4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, C23, C12, D6, C2×C6, C22×C4, C2×D4, C4○D4, C3×S3, C4×S3, C2×C12, C3×D4, C22×S3, C22×C6, C4×D4, S3×C6, S3×C2×C4, S3×D4, Q8⋊3S3, C22×C12, C6×D4, C3×C4○D4, S3×C12, S3×C2×C6, Dic3⋊5D4, D4×C12, S3×C2×C12, C3×S3×D4, C3×Q8⋊3S3, C3×Dic3⋊5D4
►On 96 points
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(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 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 69 4 72)(2 68 5 71)(3 67 6 70)(7 36 10 33)(8 35 11 32)(9 34 12 31)(13 77 16 74)(14 76 17 73)(15 75 18 78)(19 59 22 56)(20 58 23 55)(21 57 24 60)(25 62 28 65)(26 61 29 64)(27 66 30 63)(37 91 40 94)(38 96 41 93)(39 95 42 92)(43 87 46 90)(44 86 47 89)(45 85 48 88)(49 83 52 80)(50 82 53 79)(51 81 54 84)
(1 41 17 36)(2 42 18 31)(3 37 13 32)(4 38 14 33)(5 39 15 34)(6 40 16 35)(7 72 96 76)(8 67 91 77)(9 68 92 78)(10 69 93 73)(11 70 94 74)(12 71 95 75)(19 49 30 43)(20 50 25 44)(21 51 26 45)(22 52 27 46)(23 53 28 47)(24 54 29 48)(55 79 65 89)(56 80 66 90)(57 81 61 85)(58 82 62 86)(59 83 63 87)(60 84 64 88)
(1 88)(2 87)(3 86)(4 85)(5 90)(6 89)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 82)(14 81)(15 80)(16 79)(17 84)(18 83)(25 94)(26 93)(27 92)(28 91)(29 96)(30 95)(31 59)(32 58)(33 57)(34 56)(35 55)(36 60)(37 62)(38 61)(39 66)(40 65)(41 64)(42 63)(43 71)(44 70)(45 69)(46 68)(47 67)(48 72)(49 75)(50 74)(51 73)(52 78)(53 77)(54 76)
G:=sub<Sym(96)| (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,23,21)(20,24,22)(25,29,27)(26,30,28)(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), (1,2,3,4,5,6)(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), (1,69,4,72)(2,68,5,71)(3,67,6,70)(7,36,10,33)(8,35,11,32)(9,34,12,31)(13,77,16,74)(14,76,17,73)(15,75,18,78)(19,59,22,56)(20,58,23,55)(21,57,24,60)(25,62,28,65)(26,61,29,64)(27,66,30,63)(37,91,40,94)(38,96,41,93)(39,95,42,92)(43,87,46,90)(44,86,47,89)(45,85,48,88)(49,83,52,80)(50,82,53,79)(51,81,54,84), (1,41,17,36)(2,42,18,31)(3,37,13,32)(4,38,14,33)(5,39,15,34)(6,40,16,35)(7,72,96,76)(8,67,91,77)(9,68,92,78)(10,69,93,73)(11,70,94,74)(12,71,95,75)(19,49,30,43)(20,50,25,44)(21,51,26,45)(22,52,27,46)(23,53,28,47)(24,54,29,48)(55,79,65,89)(56,80,66,90)(57,81,61,85)(58,82,62,86)(59,83,63,87)(60,84,64,88), (1,88)(2,87)(3,86)(4,85)(5,90)(6,89)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,82)(14,81)(15,80)(16,79)(17,84)(18,83)(25,94)(26,93)(27,92)(28,91)(29,96)(30,95)(31,59)(32,58)(33,57)(34,56)(35,55)(36,60)(37,62)(38,61)(39,66)(40,65)(41,64)(42,63)(43,71)(44,70)(45,69)(46,68)(47,67)(48,72)(49,75)(50,74)(51,73)(52,78)(53,77)(54,76)>;
G:=Group( (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,23,21)(20,24,22)(25,29,27)(26,30,28)(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), (1,2,3,4,5,6)(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), (1,69,4,72)(2,68,5,71)(3,67,6,70)(7,36,10,33)(8,35,11,32)(9,34,12,31)(13,77,16,74)(14,76,17,73)(15,75,18,78)(19,59,22,56)(20,58,23,55)(21,57,24,60)(25,62,28,65)(26,61,29,64)(27,66,30,63)(37,91,40,94)(38,96,41,93)(39,95,42,92)(43,87,46,90)(44,86,47,89)(45,85,48,88)(49,83,52,80)(50,82,53,79)(51,81,54,84), (1,41,17,36)(2,42,18,31)(3,37,13,32)(4,38,14,33)(5,39,15,34)(6,40,16,35)(7,72,96,76)(8,67,91,77)(9,68,92,78)(10,69,93,73)(11,70,94,74)(12,71,95,75)(19,49,30,43)(20,50,25,44)(21,51,26,45)(22,52,27,46)(23,53,28,47)(24,54,29,48)(55,79,65,89)(56,80,66,90)(57,81,61,85)(58,82,62,86)(59,83,63,87)(60,84,64,88), (1,88)(2,87)(3,86)(4,85)(5,90)(6,89)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,82)(14,81)(15,80)(16,79)(17,84)(18,83)(25,94)(26,93)(27,92)(28,91)(29,96)(30,95)(31,59)(32,58)(33,57)(34,56)(35,55)(36,60)(37,62)(38,61)(39,66)(40,65)(41,64)(42,63)(43,71)(44,70)(45,69)(46,68)(47,67)(48,72)(49,75)(50,74)(51,73)(52,78)(53,77)(54,76) );
G=PermutationGroup([[(1,5,3),(2,6,4),(7,9,11),(8,10,12),(13,17,15),(14,18,16),(19,23,21),(20,24,22),(25,29,27),(26,30,28),(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)], [(1,2,3,4,5,6),(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)], [(1,69,4,72),(2,68,5,71),(3,67,6,70),(7,36,10,33),(8,35,11,32),(9,34,12,31),(13,77,16,74),(14,76,17,73),(15,75,18,78),(19,59,22,56),(20,58,23,55),(21,57,24,60),(25,62,28,65),(26,61,29,64),(27,66,30,63),(37,91,40,94),(38,96,41,93),(39,95,42,92),(43,87,46,90),(44,86,47,89),(45,85,48,88),(49,83,52,80),(50,82,53,79),(51,81,54,84)], [(1,41,17,36),(2,42,18,31),(3,37,13,32),(4,38,14,33),(5,39,15,34),(6,40,16,35),(7,72,96,76),(8,67,91,77),(9,68,92,78),(10,69,93,73),(11,70,94,74),(12,71,95,75),(19,49,30,43),(20,50,25,44),(21,51,26,45),(22,52,27,46),(23,53,28,47),(24,54,29,48),(55,79,65,89),(56,80,66,90),(57,81,61,85),(58,82,62,86),(59,83,63,87),(60,84,64,88)], [(1,88),(2,87),(3,86),(4,85),(5,90),(6,89),(7,24),(8,23),(9,22),(10,21),(11,20),(12,19),(13,82),(14,81),(15,80),(16,79),(17,84),(18,83),(25,94),(26,93),(27,92),(28,91),(29,96),(30,95),(31,59),(32,58),(33,57),(34,56),(35,55),(36,60),(37,62),(38,61),(39,66),(40,65),(41,64),(42,63),(43,71),(44,70),(45,69),(46,68),(47,67),(48,72),(49,75),(50,74),(51,73),(52,78),(53,77),(54,76)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 3E | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | ··· | 6F | 6G | ··· | 6O | 6P | ··· | 6W | 12A | ··· | 12L | 12M | ··· | 12T | 12U | ··· | 12AL | 12AM | 12AN | 12AO | 12AP |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 1 | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 3 | ··· | 3 | 4 | ··· | 4 | 6 | 6 | 6 | 6 |
90 irreducible representations
| dim | 1 | 1 | 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 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C6 | C12 | S3 | D4 | D6 | C4○D4 | C3×S3 | C3×D4 | C4×S3 | S3×C6 | C3×C4○D4 | S3×C12 | S3×D4 | Q8⋊3S3 | C3×S3×D4 | C3×Q8⋊3S3 |
| kernel | C3×Dic3⋊5D4 | Dic3×C12 | C3×D6⋊C4 | C32×C4⋊C4 | S3×C2×C12 | C6×D12 | Dic3⋊5D4 | C3×D12 | C4×Dic3 | D6⋊C4 | C3×C4⋊C4 | S3×C2×C4 | C2×D12 | D12 | C3×C4⋊C4 | C3×Dic3 | C2×C12 | C3×C6 | C4⋊C4 | Dic3 | C12 | C2×C4 | C6 | C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 8 | 2 | 4 | 2 | 4 | 2 | 16 | 1 | 2 | 3 | 2 | 2 | 4 | 4 | 6 | 4 | 8 | 1 | 1 | 2 | 2 |
Matrix representation of C3×Dic3⋊5D4 ►in GL5(𝔽13)
| 3 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 11 |
| 0 | 0 | 0 | 1 | 12 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 11 |
| 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(5,GF(13))| [3,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,3,0,0,0,0,0,9,0,0,0,0,0,1,0,0,0,0,0,1],[5,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,1,0,0,0,11,12],[12,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,11,12] >;
C3×Dic3⋊5D4 in GAP, Magma, Sage, TeX
C_3\times {\rm Dic}_3\rtimes_5D_4 % in TeX
G:=Group("C3xDic3:5D4"); // GroupNames label
G:=SmallGroup(288,664);
// by ID
G=gap.SmallGroup(288,664);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,344,555,142,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^6=d^4=e^2=1,c^2=b^3,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