direct product, metabelian, supersoluble, monomial
Aliases: C3×C6.D8, D12⋊3C12, C12.84D12, C62.104D4, C6.7(C3×D8), (C3×D12)⋊5C4, C4.1(S3×C12), C4.9(C3×D12), (C3×C6).29D8, C12.1(C3×D4), C12.49(C4×S3), C12.3(C2×C12), (C2×D12).5C6, (C6×D12).4C2, (C3×C12).38D4, C6.7(C3×SD16), C6.44(D6⋊C4), C6.29(D4⋊S3), (C2×C12).313D6, (C3×C6).26SD16, C32⋊8(D4⋊C4), (C6×C12).41C22, C6.15(Q8⋊2S3), (C2×C3⋊C8)⋊1C6, (C6×C3⋊C8)⋊5C2, (C3×C4⋊C4)⋊1C6, C4⋊C4⋊1(C3×S3), (C3×C4⋊C4)⋊10S3, C2.5(C3×D6⋊C4), C2.2(C3×D4⋊S3), C3⋊1(C3×D4⋊C4), (C32×C4⋊C4)⋊1C2, (C2×C4).34(S3×C6), (C2×C6).39(C3×D4), C6.3(C3×C22⋊C4), (C2×C12).11(C2×C6), (C3×C12).39(C2×C4), C2.2(C3×Q8⋊2S3), C22.14(C3×C3⋊D4), (C2×C6).107(C3⋊D4), (C3×C6).43(C22⋊C4), SmallGroup(288,243)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C6.D8
G = < a,b,c,d | a3=b6=c8=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=b3c-1 >
Subgroups: 338 in 113 conjugacy classes, 46 normal (42 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, C23, C32, C12, C12, D6, C2×C6, C2×C6, C4⋊C4, C2×C8, C2×D4, C3×S3, C3×C6, C3⋊C8, C24, D12, D12, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, D4⋊C4, C3×C12, C3×C12, S3×C6, C62, C2×C3⋊C8, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C2×D12, C6×D4, C3×C3⋊C8, C3×D12, C3×D12, C6×C12, C6×C12, S3×C2×C6, C6.D8, C3×D4⋊C4, C6×C3⋊C8, C32×C4⋊C4, C6×D12, C3×C6.D8
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, C12, D6, C2×C6, C22⋊C4, D8, SD16, C3×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, D4⋊C4, S3×C6, D6⋊C4, D4⋊S3, Q8⋊2S3, C3×C22⋊C4, C3×D8, C3×SD16, S3×C12, C3×D12, C3×C3⋊D4, C6.D8, C3×D4⋊C4, C3×D6⋊C4, C3×D4⋊S3, C3×Q8⋊2S3, C3×C6.D8
►On 96 points
(1 29 94)(2 30 95)(3 31 96)(4 32 89)(5 25 90)(6 26 91)(7 27 92)(8 28 93)(9 71 38)(10 72 39)(11 65 40)(12 66 33)(13 67 34)(14 68 35)(15 69 36)(16 70 37)(17 76 58)(18 77 59)(19 78 60)(20 79 61)(21 80 62)(22 73 63)(23 74 64)(24 75 57)(41 84 55)(42 85 56)(43 86 49)(44 87 50)(45 88 51)(46 81 52)(47 82 53)(48 83 54)
(1 57 29 24 94 75)(2 76 95 17 30 58)(3 59 31 18 96 77)(4 78 89 19 32 60)(5 61 25 20 90 79)(6 80 91 21 26 62)(7 63 27 22 92 73)(8 74 93 23 28 64)(9 83 71 54 38 48)(10 41 39 55 72 84)(11 85 65 56 40 42)(12 43 33 49 66 86)(13 87 67 50 34 44)(14 45 35 51 68 88)(15 81 69 52 36 46)(16 47 37 53 70 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 49)(2 11)(3 55)(4 9)(5 53)(6 15)(7 51)(8 13)(10 18)(12 24)(14 22)(16 20)(17 56)(19 54)(21 52)(23 50)(25 47)(26 69)(27 45)(28 67)(29 43)(30 65)(31 41)(32 71)(33 57)(34 93)(35 63)(36 91)(37 61)(38 89)(39 59)(40 95)(42 76)(44 74)(46 80)(48 78)(58 85)(60 83)(62 81)(64 87)(66 75)(68 73)(70 79)(72 77)(82 90)(84 96)(86 94)(88 92)
G:=sub<Sym(96)| (1,29,94)(2,30,95)(3,31,96)(4,32,89)(5,25,90)(6,26,91)(7,27,92)(8,28,93)(9,71,38)(10,72,39)(11,65,40)(12,66,33)(13,67,34)(14,68,35)(15,69,36)(16,70,37)(17,76,58)(18,77,59)(19,78,60)(20,79,61)(21,80,62)(22,73,63)(23,74,64)(24,75,57)(41,84,55)(42,85,56)(43,86,49)(44,87,50)(45,88,51)(46,81,52)(47,82,53)(48,83,54), (1,57,29,24,94,75)(2,76,95,17,30,58)(3,59,31,18,96,77)(4,78,89,19,32,60)(5,61,25,20,90,79)(6,80,91,21,26,62)(7,63,27,22,92,73)(8,74,93,23,28,64)(9,83,71,54,38,48)(10,41,39,55,72,84)(11,85,65,56,40,42)(12,43,33,49,66,86)(13,87,67,50,34,44)(14,45,35,51,68,88)(15,81,69,52,36,46)(16,47,37,53,70,82), (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), (1,49)(2,11)(3,55)(4,9)(5,53)(6,15)(7,51)(8,13)(10,18)(12,24)(14,22)(16,20)(17,56)(19,54)(21,52)(23,50)(25,47)(26,69)(27,45)(28,67)(29,43)(30,65)(31,41)(32,71)(33,57)(34,93)(35,63)(36,91)(37,61)(38,89)(39,59)(40,95)(42,76)(44,74)(46,80)(48,78)(58,85)(60,83)(62,81)(64,87)(66,75)(68,73)(70,79)(72,77)(82,90)(84,96)(86,94)(88,92)>;
G:=Group( (1,29,94)(2,30,95)(3,31,96)(4,32,89)(5,25,90)(6,26,91)(7,27,92)(8,28,93)(9,71,38)(10,72,39)(11,65,40)(12,66,33)(13,67,34)(14,68,35)(15,69,36)(16,70,37)(17,76,58)(18,77,59)(19,78,60)(20,79,61)(21,80,62)(22,73,63)(23,74,64)(24,75,57)(41,84,55)(42,85,56)(43,86,49)(44,87,50)(45,88,51)(46,81,52)(47,82,53)(48,83,54), (1,57,29,24,94,75)(2,76,95,17,30,58)(3,59,31,18,96,77)(4,78,89,19,32,60)(5,61,25,20,90,79)(6,80,91,21,26,62)(7,63,27,22,92,73)(8,74,93,23,28,64)(9,83,71,54,38,48)(10,41,39,55,72,84)(11,85,65,56,40,42)(12,43,33,49,66,86)(13,87,67,50,34,44)(14,45,35,51,68,88)(15,81,69,52,36,46)(16,47,37,53,70,82), (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), (1,49)(2,11)(3,55)(4,9)(5,53)(6,15)(7,51)(8,13)(10,18)(12,24)(14,22)(16,20)(17,56)(19,54)(21,52)(23,50)(25,47)(26,69)(27,45)(28,67)(29,43)(30,65)(31,41)(32,71)(33,57)(34,93)(35,63)(36,91)(37,61)(38,89)(39,59)(40,95)(42,76)(44,74)(46,80)(48,78)(58,85)(60,83)(62,81)(64,87)(66,75)(68,73)(70,79)(72,77)(82,90)(84,96)(86,94)(88,92) );
G=PermutationGroup([[(1,29,94),(2,30,95),(3,31,96),(4,32,89),(5,25,90),(6,26,91),(7,27,92),(8,28,93),(9,71,38),(10,72,39),(11,65,40),(12,66,33),(13,67,34),(14,68,35),(15,69,36),(16,70,37),(17,76,58),(18,77,59),(19,78,60),(20,79,61),(21,80,62),(22,73,63),(23,74,64),(24,75,57),(41,84,55),(42,85,56),(43,86,49),(44,87,50),(45,88,51),(46,81,52),(47,82,53),(48,83,54)], [(1,57,29,24,94,75),(2,76,95,17,30,58),(3,59,31,18,96,77),(4,78,89,19,32,60),(5,61,25,20,90,79),(6,80,91,21,26,62),(7,63,27,22,92,73),(8,74,93,23,28,64),(9,83,71,54,38,48),(10,41,39,55,72,84),(11,85,65,56,40,42),(12,43,33,49,66,86),(13,87,67,50,34,44),(14,45,35,51,68,88),(15,81,69,52,36,46),(16,47,37,53,70,82)], [(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)], [(1,49),(2,11),(3,55),(4,9),(5,53),(6,15),(7,51),(8,13),(10,18),(12,24),(14,22),(16,20),(17,56),(19,54),(21,52),(23,50),(25,47),(26,69),(27,45),(28,67),(29,43),(30,65),(31,41),(32,71),(33,57),(34,93),(35,63),(36,91),(37,61),(38,89),(39,59),(40,95),(42,76),(44,74),(46,80),(48,78),(58,85),(60,83),(62,81),(64,87),(66,75),(68,73),(70,79),(72,77),(82,90),(84,96),(86,94),(88,92)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | ··· | 6O | 6P | 6Q | 6R | 6S | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12Z | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
72 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 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | S3 | D4 | D4 | D6 | D8 | SD16 | C3×S3 | C4×S3 | D12 | C3×D4 | C3⋊D4 | C3×D4 | S3×C6 | C3×D8 | C3×SD16 | S3×C12 | C3×D12 | C3×C3⋊D4 | D4⋊S3 | Q8⋊2S3 | C3×D4⋊S3 | C3×Q8⋊2S3 |
| kernel | C3×C6.D8 | C6×C3⋊C8 | C32×C4⋊C4 | C6×D12 | C6.D8 | C3×D12 | C2×C3⋊C8 | C3×C4⋊C4 | C2×D12 | D12 | C3×C4⋊C4 | C3×C12 | C62 | C2×C12 | C3×C6 | C3×C6 | C4⋊C4 | C12 | C12 | C12 | C2×C6 | C2×C6 | C2×C4 | C6 | C6 | C4 | C4 | C22 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of C3×C6.D8 ►in GL4(𝔽73) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 9 | 0 | 0 | 0 |
| 0 | 65 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 46 | 0 | 0 |
| 27 | 0 | 0 | 0 |
| 0 | 0 | 16 | 57 |
| 0 | 0 | 16 | 16 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 17 | 2 |
| 0 | 0 | 2 | 56 |
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,65,0,0,0,0,1,0,0,0,0,1],[0,27,0,0,46,0,0,0,0,0,16,16,0,0,57,16],[0,1,0,0,1,0,0,0,0,0,17,2,0,0,2,56] >;
C3×C6.D8 in GAP, Magma, Sage, TeX
C_3\times C_6.D_8
% in TeX
G:=Group("C3xC6.D8"); // GroupNames label
G:=SmallGroup(288,243);
// by ID
G=gap.SmallGroup(288,243);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,365,92,2524,1271,102,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^6=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=b^3*c^-1>;
// generators/relations