direct product, metabelian, supersoluble, monomial
Aliases: C3×C42⋊2S3, C122⋊14C2, C62.163C23, (C4×C12)⋊3S3, (S3×C12)⋊7C4, (C4×S3)⋊3C12, (C4×C12)⋊15C6, D6⋊C4.7C6, C42⋊5(C3×S3), C4.22(S3×C12), (C4×Dic3)⋊8C6, D6.3(C2×C12), C12.113(C4×S3), C12.27(C2×C12), Dic3⋊C4⋊17C6, (C2×C12).347D6, C6.3(C22×C12), (Dic3×C12)⋊26C2, Dic3.5(C2×C12), C6.112(C4○D12), (C6×C12).345C22, C32⋊11(C42⋊C2), (C6×Dic3).158C22, (S3×C2×C4).7C6, C2.5(S3×C2×C12), C6.102(S3×C2×C4), C6.3(C3×C4○D4), (S3×C2×C12).21C2, (C2×C4).64(S3×C6), C2.2(C3×C4○D12), C22.10(S3×C2×C6), (S3×C6).21(C2×C4), (C2×C12).87(C2×C6), (C3×D6⋊C4).17C2, C3⋊1(C3×C42⋊C2), (S3×C2×C6).86C22, (C3×C12).112(C2×C4), (C3×Dic3⋊C4)⋊39C2, (C3×C6).93(C4○D4), (C2×C6).18(C22×C6), (C3×C6).74(C22×C4), (C22×S3).14(C2×C6), (C2×C6).296(C22×S3), (C3×Dic3).27(C2×C4), (C2×Dic3).16(C2×C6), SmallGroup(288,643)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C42⋊2S3
G = < a,b,c,d,e | a3=b4=c4=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >
Subgroups: 338 in 167 conjugacy classes, 82 normal (30 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, C23, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C3×S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C2×C12, C22×S3, C22×C6, C42⋊C2, C3×Dic3, C3×Dic3, C3×C12, C3×C12, S3×C6, S3×C6, C62, C4×Dic3, Dic3⋊C4, D6⋊C4, C4×C12, C4×C12, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, C22×C12, S3×C12, C6×Dic3, C6×Dic3, C6×C12, C6×C12, S3×C2×C6, C42⋊2S3, C3×C42⋊C2, Dic3×C12, C3×Dic3⋊C4, C3×D6⋊C4, C122, S3×C2×C12, C3×C42⋊2S3
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C23, C12, D6, C2×C6, C22×C4, C4○D4, C3×S3, C4×S3, C2×C12, C22×S3, C22×C6, C42⋊C2, S3×C6, S3×C2×C4, C4○D12, C22×C12, C3×C4○D4, S3×C12, S3×C2×C6, C42⋊2S3, C3×C42⋊C2, S3×C2×C12, C3×C4○D12, C3×C42⋊2S3
►On 96 points
(1 39 5)(2 40 6)(3 37 7)(4 38 8)(9 36 22)(10 33 23)(11 34 24)(12 35 21)(13 45 51)(14 46 52)(15 47 49)(16 48 50)(17 43 29)(18 44 30)(19 41 31)(20 42 32)(25 63 95)(26 64 96)(27 61 93)(28 62 94)(53 78 86)(54 79 87)(55 80 88)(56 77 85)(57 67 89)(58 68 90)(59 65 91)(60 66 92)(69 75 83)(70 76 84)(71 73 81)(72 74 82)
(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 13 29 23)(2 14 30 24)(3 15 31 21)(4 16 32 22)(5 51 43 33)(6 52 44 34)(7 49 41 35)(8 50 42 36)(9 38 48 20)(10 39 45 17)(11 40 46 18)(12 37 47 19)(25 91 83 79)(26 92 84 80)(27 89 81 77)(28 90 82 78)(53 94 68 74)(54 95 65 75)(55 96 66 76)(56 93 67 73)(57 71 85 61)(58 72 86 62)(59 69 87 63)(60 70 88 64)
(1 5 39)(2 6 40)(3 7 37)(4 8 38)(9 22 36)(10 23 33)(11 24 34)(12 21 35)(13 51 45)(14 52 46)(15 49 47)(16 50 48)(17 29 43)(18 30 44)(19 31 41)(20 32 42)(25 63 95)(26 64 96)(27 61 93)(28 62 94)(53 78 86)(54 79 87)(55 80 88)(56 77 85)(57 67 89)(58 68 90)(59 65 91)(60 66 92)(69 75 83)(70 76 84)(71 73 81)(72 74 82)
(1 87)(2 88)(3 85)(4 86)(5 79)(6 80)(7 77)(8 78)(9 76)(10 73)(11 74)(12 75)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 49)(26 50)(27 51)(28 52)(29 59)(30 60)(31 57)(32 58)(33 81)(34 82)(35 83)(36 84)(37 56)(38 53)(39 54)(40 55)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)
G:=sub<Sym(96)| (1,39,5)(2,40,6)(3,37,7)(4,38,8)(9,36,22)(10,33,23)(11,34,24)(12,35,21)(13,45,51)(14,46,52)(15,47,49)(16,48,50)(17,43,29)(18,44,30)(19,41,31)(20,42,32)(25,63,95)(26,64,96)(27,61,93)(28,62,94)(53,78,86)(54,79,87)(55,80,88)(56,77,85)(57,67,89)(58,68,90)(59,65,91)(60,66,92)(69,75,83)(70,76,84)(71,73,81)(72,74,82), (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), (1,13,29,23)(2,14,30,24)(3,15,31,21)(4,16,32,22)(5,51,43,33)(6,52,44,34)(7,49,41,35)(8,50,42,36)(9,38,48,20)(10,39,45,17)(11,40,46,18)(12,37,47,19)(25,91,83,79)(26,92,84,80)(27,89,81,77)(28,90,82,78)(53,94,68,74)(54,95,65,75)(55,96,66,76)(56,93,67,73)(57,71,85,61)(58,72,86,62)(59,69,87,63)(60,70,88,64), (1,5,39)(2,6,40)(3,7,37)(4,8,38)(9,22,36)(10,23,33)(11,24,34)(12,21,35)(13,51,45)(14,52,46)(15,49,47)(16,50,48)(17,29,43)(18,30,44)(19,31,41)(20,32,42)(25,63,95)(26,64,96)(27,61,93)(28,62,94)(53,78,86)(54,79,87)(55,80,88)(56,77,85)(57,67,89)(58,68,90)(59,65,91)(60,66,92)(69,75,83)(70,76,84)(71,73,81)(72,74,82), (1,87)(2,88)(3,85)(4,86)(5,79)(6,80)(7,77)(8,78)(9,76)(10,73)(11,74)(12,75)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,49)(26,50)(27,51)(28,52)(29,59)(30,60)(31,57)(32,58)(33,81)(34,82)(35,83)(36,84)(37,56)(38,53)(39,54)(40,55)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96)>;
G:=Group( (1,39,5)(2,40,6)(3,37,7)(4,38,8)(9,36,22)(10,33,23)(11,34,24)(12,35,21)(13,45,51)(14,46,52)(15,47,49)(16,48,50)(17,43,29)(18,44,30)(19,41,31)(20,42,32)(25,63,95)(26,64,96)(27,61,93)(28,62,94)(53,78,86)(54,79,87)(55,80,88)(56,77,85)(57,67,89)(58,68,90)(59,65,91)(60,66,92)(69,75,83)(70,76,84)(71,73,81)(72,74,82), (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), (1,13,29,23)(2,14,30,24)(3,15,31,21)(4,16,32,22)(5,51,43,33)(6,52,44,34)(7,49,41,35)(8,50,42,36)(9,38,48,20)(10,39,45,17)(11,40,46,18)(12,37,47,19)(25,91,83,79)(26,92,84,80)(27,89,81,77)(28,90,82,78)(53,94,68,74)(54,95,65,75)(55,96,66,76)(56,93,67,73)(57,71,85,61)(58,72,86,62)(59,69,87,63)(60,70,88,64), (1,5,39)(2,6,40)(3,7,37)(4,8,38)(9,22,36)(10,23,33)(11,24,34)(12,21,35)(13,51,45)(14,52,46)(15,49,47)(16,50,48)(17,29,43)(18,30,44)(19,31,41)(20,32,42)(25,63,95)(26,64,96)(27,61,93)(28,62,94)(53,78,86)(54,79,87)(55,80,88)(56,77,85)(57,67,89)(58,68,90)(59,65,91)(60,66,92)(69,75,83)(70,76,84)(71,73,81)(72,74,82), (1,87)(2,88)(3,85)(4,86)(5,79)(6,80)(7,77)(8,78)(9,76)(10,73)(11,74)(12,75)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,49)(26,50)(27,51)(28,52)(29,59)(30,60)(31,57)(32,58)(33,81)(34,82)(35,83)(36,84)(37,56)(38,53)(39,54)(40,55)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96) );
G=PermutationGroup([[(1,39,5),(2,40,6),(3,37,7),(4,38,8),(9,36,22),(10,33,23),(11,34,24),(12,35,21),(13,45,51),(14,46,52),(15,47,49),(16,48,50),(17,43,29),(18,44,30),(19,41,31),(20,42,32),(25,63,95),(26,64,96),(27,61,93),(28,62,94),(53,78,86),(54,79,87),(55,80,88),(56,77,85),(57,67,89),(58,68,90),(59,65,91),(60,66,92),(69,75,83),(70,76,84),(71,73,81),(72,74,82)], [(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)], [(1,13,29,23),(2,14,30,24),(3,15,31,21),(4,16,32,22),(5,51,43,33),(6,52,44,34),(7,49,41,35),(8,50,42,36),(9,38,48,20),(10,39,45,17),(11,40,46,18),(12,37,47,19),(25,91,83,79),(26,92,84,80),(27,89,81,77),(28,90,82,78),(53,94,68,74),(54,95,65,75),(55,96,66,76),(56,93,67,73),(57,71,85,61),(58,72,86,62),(59,69,87,63),(60,70,88,64)], [(1,5,39),(2,6,40),(3,7,37),(4,8,38),(9,22,36),(10,23,33),(11,24,34),(12,21,35),(13,51,45),(14,52,46),(15,49,47),(16,50,48),(17,29,43),(18,30,44),(19,31,41),(20,32,42),(25,63,95),(26,64,96),(27,61,93),(28,62,94),(53,78,86),(54,79,87),(55,80,88),(56,77,85),(57,67,89),(58,68,90),(59,65,91),(60,66,92),(69,75,83),(70,76,84),(71,73,81),(72,74,82)], [(1,87),(2,88),(3,85),(4,86),(5,79),(6,80),(7,77),(8,78),(9,76),(10,73),(11,74),(12,75),(13,61),(14,62),(15,63),(16,64),(17,65),(18,66),(19,67),(20,68),(21,69),(22,70),(23,71),(24,72),(25,49),(26,50),(27,51),(28,52),(29,59),(30,60),(31,57),(32,58),(33,81),(34,82),(35,83),(36,84),(37,56),(38,53),(39,54),(40,55),(41,89),(42,90),(43,91),(44,92),(45,93),(46,94),(47,95),(48,96)]])
108 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 6A | ··· | 6F | 6G | ··· | 6O | 6P | 6Q | 6R | 6S | 12A | ··· | 12H | 12I | ··· | 12AZ | 12BA | ··· | 12BL |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 |
108 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 |
| type | + | + | + | + | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C6 | C12 | S3 | D6 | C4○D4 | C3×S3 | C4×S3 | S3×C6 | C4○D12 | C3×C4○D4 | S3×C12 | C3×C4○D12 |
| kernel | C3×C42⋊2S3 | Dic3×C12 | C3×Dic3⋊C4 | C3×D6⋊C4 | C122 | S3×C2×C12 | C42⋊2S3 | S3×C12 | C4×Dic3 | Dic3⋊C4 | D6⋊C4 | C4×C12 | S3×C2×C4 | C4×S3 | C4×C12 | C2×C12 | C3×C6 | C42 | C12 | C2×C4 | C6 | C6 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 8 | 2 | 4 | 4 | 2 | 2 | 16 | 1 | 3 | 4 | 2 | 4 | 6 | 8 | 8 | 8 | 16 |
Matrix representation of C3×C42⋊2S3 ►in GL4(𝔽13) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 5 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 12 |
| 3 | 11 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 9 |
| 6 | 7 | 0 | 0 |
| 8 | 7 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[12,0,0,0,0,12,0,0,0,0,8,0,0,0,0,8],[5,0,0,0,0,5,0,0,0,0,1,0,0,0,0,12],[3,0,0,0,11,9,0,0,0,0,3,0,0,0,0,9],[6,8,0,0,7,7,0,0,0,0,0,1,0,0,1,0] >;
C3×C42⋊2S3 in GAP, Magma, Sage, TeX
C_3\times C_4^2\rtimes_2S_3
% in TeX
G:=Group("C3xC4^2:2S3"); // GroupNames label
G:=SmallGroup(288,643);
// by ID
G=gap.SmallGroup(288,643);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,344,1094,142,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=c^4=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations