direct product, metabelian, supersoluble, monomial
Aliases: C3×C42⋊7S3, C122⋊12C2, C12.66D12, C62.166C23, D6⋊C4⋊1C6, C6.4(C6×D4), (C4×C12)⋊11C6, (C4×C12)⋊15S3, C2.6(C6×D12), C4.5(C3×D12), C42⋊10(C3×S3), (C2×Dic6)⋊1C6, (C2×D12).2C6, C6.92(C2×D12), C12.28(C3×D4), (C6×Dic6)⋊25C2, (C6×D12).17C2, (C2×C12).348D6, (C3×C12).130D4, C6.114(C4○D12), (C6×C12).280C22, C32⋊11(C4.4D4), (C6×Dic3).89C22, C6.5(C3×C4○D4), (C3×D6⋊C4)⋊32C2, (C2×C4).65(S3×C6), C2.7(C3×C4○D12), C3⋊1(C3×C4.4D4), C22.37(S3×C2×C6), (C2×C12).88(C2×C6), (C3×C6).175(C2×D4), (S3×C2×C6).53C22, (C3×C6).95(C4○D4), (C22×S3).2(C2×C6), (C2×C6).21(C22×C6), (C2×Dic3).3(C2×C6), (C2×C6).299(C22×S3), SmallGroup(288,646)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C42⋊7S3
G = < a,b,c,d,e | a3=b4=c4=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe=bc2, cd=dc, ece=b2c, ede=d-1 >
Subgroups: 434 in 167 conjugacy classes, 66 normal (26 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C2×D4, C2×Q8, C3×S3, C3×C6, C3×C6, Dic6, D12, C2×Dic3, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×C6, C4.4D4, C3×Dic3, C3×C12, C3×C12, S3×C6, C62, D6⋊C4, C4×C12, C4×C12, C3×C22⋊C4, C2×Dic6, C2×D12, C6×D4, C6×Q8, C3×Dic6, C3×D12, C6×Dic3, C6×C12, C6×C12, S3×C2×C6, C42⋊7S3, C3×C4.4D4, C3×D6⋊C4, C122, C6×Dic6, C6×D12, C3×C42⋊7S3
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C4○D4, C3×S3, D12, C3×D4, C22×S3, C22×C6, C4.4D4, S3×C6, C2×D12, C4○D12, C6×D4, C3×C4○D4, C3×D12, S3×C2×C6, C42⋊7S3, C3×C4.4D4, C6×D12, C3×C4○D12, C3×C42⋊7S3
►On 96 points
Generators in S96
(1 23 9)(2 24 10)(3 21 11)(4 22 12)(5 64 68)(6 61 65)(7 62 66)(8 63 67)(13 47 43)(14 48 44)(15 45 41)(16 46 42)(17 73 39)(18 74 40)(19 75 37)(20 76 38)(25 87 29)(26 88 30)(27 85 31)(28 86 32)(33 50 54)(34 51 55)(35 52 56)(36 49 53)(57 93 89)(58 94 90)(59 95 91)(60 96 92)(69 83 79)(70 84 80)(71 81 77)(72 82 78)
(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 54 43 17)(2 55 44 18)(3 56 41 19)(4 53 42 20)(5 69 87 60)(6 70 88 57)(7 71 85 58)(8 72 86 59)(9 50 47 39)(10 51 48 40)(11 52 45 37)(12 49 46 38)(13 73 23 33)(14 74 24 34)(15 75 21 35)(16 76 22 36)(25 92 68 79)(26 89 65 80)(27 90 66 77)(28 91 67 78)(29 96 64 83)(30 93 61 84)(31 94 62 81)(32 95 63 82)
(1 9 23)(2 10 24)(3 11 21)(4 12 22)(5 64 68)(6 61 65)(7 62 66)(8 63 67)(13 43 47)(14 44 48)(15 41 45)(16 42 46)(17 39 73)(18 40 74)(19 37 75)(20 38 76)(25 87 29)(26 88 30)(27 85 31)(28 86 32)(33 54 50)(34 55 51)(35 56 52)(36 53 49)(57 93 89)(58 94 90)(59 95 91)(60 96 92)(69 83 79)(70 84 80)(71 81 77)(72 82 78)
(1 91)(2 79)(3 89)(4 77)(5 76)(6 33)(7 74)(8 35)(9 95)(10 83)(11 93)(12 81)(13 72)(14 60)(15 70)(16 58)(17 26)(18 66)(19 28)(20 68)(21 57)(22 71)(23 59)(24 69)(25 53)(27 55)(29 49)(30 39)(31 51)(32 37)(34 85)(36 87)(38 64)(40 62)(41 80)(42 90)(43 78)(44 92)(45 84)(46 94)(47 82)(48 96)(50 61)(52 63)(54 65)(56 67)(73 88)(75 86)
G:=sub<Sym(96)| (1,23,9)(2,24,10)(3,21,11)(4,22,12)(5,64,68)(6,61,65)(7,62,66)(8,63,67)(13,47,43)(14,48,44)(15,45,41)(16,46,42)(17,73,39)(18,74,40)(19,75,37)(20,76,38)(25,87,29)(26,88,30)(27,85,31)(28,86,32)(33,50,54)(34,51,55)(35,52,56)(36,49,53)(57,93,89)(58,94,90)(59,95,91)(60,96,92)(69,83,79)(70,84,80)(71,81,77)(72,82,78), (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), (1,54,43,17)(2,55,44,18)(3,56,41,19)(4,53,42,20)(5,69,87,60)(6,70,88,57)(7,71,85,58)(8,72,86,59)(9,50,47,39)(10,51,48,40)(11,52,45,37)(12,49,46,38)(13,73,23,33)(14,74,24,34)(15,75,21,35)(16,76,22,36)(25,92,68,79)(26,89,65,80)(27,90,66,77)(28,91,67,78)(29,96,64,83)(30,93,61,84)(31,94,62,81)(32,95,63,82), (1,9,23)(2,10,24)(3,11,21)(4,12,22)(5,64,68)(6,61,65)(7,62,66)(8,63,67)(13,43,47)(14,44,48)(15,41,45)(16,42,46)(17,39,73)(18,40,74)(19,37,75)(20,38,76)(25,87,29)(26,88,30)(27,85,31)(28,86,32)(33,54,50)(34,55,51)(35,56,52)(36,53,49)(57,93,89)(58,94,90)(59,95,91)(60,96,92)(69,83,79)(70,84,80)(71,81,77)(72,82,78), (1,91)(2,79)(3,89)(4,77)(5,76)(6,33)(7,74)(8,35)(9,95)(10,83)(11,93)(12,81)(13,72)(14,60)(15,70)(16,58)(17,26)(18,66)(19,28)(20,68)(21,57)(22,71)(23,59)(24,69)(25,53)(27,55)(29,49)(30,39)(31,51)(32,37)(34,85)(36,87)(38,64)(40,62)(41,80)(42,90)(43,78)(44,92)(45,84)(46,94)(47,82)(48,96)(50,61)(52,63)(54,65)(56,67)(73,88)(75,86)>;
G:=Group( (1,23,9)(2,24,10)(3,21,11)(4,22,12)(5,64,68)(6,61,65)(7,62,66)(8,63,67)(13,47,43)(14,48,44)(15,45,41)(16,46,42)(17,73,39)(18,74,40)(19,75,37)(20,76,38)(25,87,29)(26,88,30)(27,85,31)(28,86,32)(33,50,54)(34,51,55)(35,52,56)(36,49,53)(57,93,89)(58,94,90)(59,95,91)(60,96,92)(69,83,79)(70,84,80)(71,81,77)(72,82,78), (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), (1,54,43,17)(2,55,44,18)(3,56,41,19)(4,53,42,20)(5,69,87,60)(6,70,88,57)(7,71,85,58)(8,72,86,59)(9,50,47,39)(10,51,48,40)(11,52,45,37)(12,49,46,38)(13,73,23,33)(14,74,24,34)(15,75,21,35)(16,76,22,36)(25,92,68,79)(26,89,65,80)(27,90,66,77)(28,91,67,78)(29,96,64,83)(30,93,61,84)(31,94,62,81)(32,95,63,82), (1,9,23)(2,10,24)(3,11,21)(4,12,22)(5,64,68)(6,61,65)(7,62,66)(8,63,67)(13,43,47)(14,44,48)(15,41,45)(16,42,46)(17,39,73)(18,40,74)(19,37,75)(20,38,76)(25,87,29)(26,88,30)(27,85,31)(28,86,32)(33,54,50)(34,55,51)(35,56,52)(36,53,49)(57,93,89)(58,94,90)(59,95,91)(60,96,92)(69,83,79)(70,84,80)(71,81,77)(72,82,78), (1,91)(2,79)(3,89)(4,77)(5,76)(6,33)(7,74)(8,35)(9,95)(10,83)(11,93)(12,81)(13,72)(14,60)(15,70)(16,58)(17,26)(18,66)(19,28)(20,68)(21,57)(22,71)(23,59)(24,69)(25,53)(27,55)(29,49)(30,39)(31,51)(32,37)(34,85)(36,87)(38,64)(40,62)(41,80)(42,90)(43,78)(44,92)(45,84)(46,94)(47,82)(48,96)(50,61)(52,63)(54,65)(56,67)(73,88)(75,86) );
G=PermutationGroup([[(1,23,9),(2,24,10),(3,21,11),(4,22,12),(5,64,68),(6,61,65),(7,62,66),(8,63,67),(13,47,43),(14,48,44),(15,45,41),(16,46,42),(17,73,39),(18,74,40),(19,75,37),(20,76,38),(25,87,29),(26,88,30),(27,85,31),(28,86,32),(33,50,54),(34,51,55),(35,52,56),(36,49,53),(57,93,89),(58,94,90),(59,95,91),(60,96,92),(69,83,79),(70,84,80),(71,81,77),(72,82,78)], [(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)], [(1,54,43,17),(2,55,44,18),(3,56,41,19),(4,53,42,20),(5,69,87,60),(6,70,88,57),(7,71,85,58),(8,72,86,59),(9,50,47,39),(10,51,48,40),(11,52,45,37),(12,49,46,38),(13,73,23,33),(14,74,24,34),(15,75,21,35),(16,76,22,36),(25,92,68,79),(26,89,65,80),(27,90,66,77),(28,91,67,78),(29,96,64,83),(30,93,61,84),(31,94,62,81),(32,95,63,82)], [(1,9,23),(2,10,24),(3,11,21),(4,12,22),(5,64,68),(6,61,65),(7,62,66),(8,63,67),(13,43,47),(14,44,48),(15,41,45),(16,42,46),(17,39,73),(18,40,74),(19,37,75),(20,38,76),(25,87,29),(26,88,30),(27,85,31),(28,86,32),(33,54,50),(34,55,51),(35,56,52),(36,53,49),(57,93,89),(58,94,90),(59,95,91),(60,96,92),(69,83,79),(70,84,80),(71,81,77),(72,82,78)], [(1,91),(2,79),(3,89),(4,77),(5,76),(6,33),(7,74),(8,35),(9,95),(10,83),(11,93),(12,81),(13,72),(14,60),(15,70),(16,58),(17,26),(18,66),(19,28),(20,68),(21,57),(22,71),(23,59),(24,69),(25,53),(27,55),(29,49),(30,39),(31,51),(32,37),(34,85),(36,87),(38,64),(40,62),(41,80),(42,90),(43,78),(44,92),(45,84),(46,94),(47,82),(48,96),(50,61),(52,63),(54,65),(56,67),(73,88),(75,86)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 3E | 4A | ··· | 4F | 4G | 4H | 6A | ··· | 6F | 6G | ··· | 6O | 6P | 6Q | 6R | 6S | 12A | ··· | 12AV | 12AW | 12AX | 12AY | 12AZ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 1 | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | 12 | 1 | ··· | 1 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 12 | 12 | 12 | 12 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | S3 | D4 | D6 | C4○D4 | C3×S3 | D12 | C3×D4 | S3×C6 | C4○D12 | C3×C4○D4 | C3×D12 | C3×C4○D12 |
| kernel | C3×C42⋊7S3 | C3×D6⋊C4 | C122 | C6×Dic6 | C6×D12 | C42⋊7S3 | D6⋊C4 | C4×C12 | C2×Dic6 | C2×D12 | C4×C12 | C3×C12 | C2×C12 | C3×C6 | C42 | C12 | C12 | C2×C4 | C6 | C6 | C4 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 8 | 2 | 2 | 2 | 1 | 2 | 3 | 4 | 2 | 4 | 4 | 6 | 8 | 8 | 8 | 16 |
Matrix representation of C3×C42⋊7S3 ►in GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 3 |
| 6 | 3 | 0 | 0 |
| 10 | 7 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,3,0,0,0,0,3],[8,0,0,0,0,8,0,0,0,0,1,0,0,0,0,12],[0,1,0,0,1,0,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,3],[6,10,0,0,3,7,0,0,0,0,0,1,0,0,1,0] >;
C3×C42⋊7S3 in GAP, Magma, Sage, TeX
C_3\times C_4^2\rtimes_7S_3
% in TeX
G:=Group("C3xC4^2:7S3"); // GroupNames label
G:=SmallGroup(288,646);
// by ID
G=gap.SmallGroup(288,646);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,176,590,268,9414]);
// 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,e*b*e=b*c^2,c*d=d*c,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations