direct product, metacyclic, supersoluble, monomial
Aliases: C3×D6.C8, C48⋊6C6, C48⋊7S3, D6.C24, C24.98D6, Dic3.C24, C32⋊7M5(2), C3⋊C16⋊4C6, C16⋊3(C3×S3), C3⋊C8.2C12, (C3×C48)⋊10C2, (S3×C6).3C8, (S3×C8).2C6, C8.20(S3×C6), C6.23(S3×C8), C2.3(S3×C24), C6.2(C2×C24), (S3×C24).5C2, (S3×C12).7C4, (C4×S3).2C12, C4.17(S3×C12), C24.33(C2×C6), C3⋊1(C3×M5(2)), C12.108(C4×S3), C12.22(C2×C12), (C3×Dic3).3C8, (C3×C24).65C22, (C3×C3⋊C8).5C4, (C3×C3⋊C16)⋊11C2, (C3×C6).28(C2×C8), (C3×C12).102(C2×C4), SmallGroup(288,232)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D6.C8
G = < a,b,c,d | a3=b6=c2=1, d8=b3, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b3c >
Subgroups: 102 in 57 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, C32, Dic3, C12, C12, D6, C2×C6, C16, C16, C2×C8, C3×S3, C3×C6, C3⋊C8, C24, C24, C4×S3, C2×C12, M5(2), C3×Dic3, C3×C12, S3×C6, C3⋊C16, C48, C48, S3×C8, C2×C24, C3×C3⋊C8, C3×C24, S3×C12, D6.C8, C3×M5(2), C3×C3⋊C16, C3×C48, S3×C24, C3×D6.C8
Quotients: C1, C2, C3, C4, C22, S3, C6, C8, C2×C4, C12, D6, C2×C6, C2×C8, C3×S3, C24, C4×S3, C2×C12, M5(2), S3×C6, S3×C8, C2×C24, S3×C12, D6.C8, C3×M5(2), S3×C24, C3×D6.C8
►On 96 points
Generators in S96
(1 64 68)(2 49 69)(3 50 70)(4 51 71)(5 52 72)(6 53 73)(7 54 74)(8 55 75)(9 56 76)(10 57 77)(11 58 78)(12 59 79)(13 60 80)(14 61 65)(15 62 66)(16 63 67)(17 92 38)(18 93 39)(19 94 40)(20 95 41)(21 96 42)(22 81 43)(23 82 44)(24 83 45)(25 84 46)(26 85 47)(27 86 48)(28 87 33)(29 88 34)(30 89 35)(31 90 36)(32 91 37)
(1 76 64 9 68 56)(2 77 49 10 69 57)(3 78 50 11 70 58)(4 79 51 12 71 59)(5 80 52 13 72 60)(6 65 53 14 73 61)(7 66 54 15 74 62)(8 67 55 16 75 63)(17 84 38 25 92 46)(18 85 39 26 93 47)(19 86 40 27 94 48)(20 87 41 28 95 33)(21 88 42 29 96 34)(22 89 43 30 81 35)(23 90 44 31 82 36)(24 91 45 32 83 37)
(1 86)(2 95)(3 88)(4 81)(5 90)(6 83)(7 92)(8 85)(9 94)(10 87)(11 96)(12 89)(13 82)(14 91)(15 84)(16 93)(17 74)(18 67)(19 76)(20 69)(21 78)(22 71)(23 80)(24 73)(25 66)(26 75)(27 68)(28 77)(29 70)(30 79)(31 72)(32 65)(33 57)(34 50)(35 59)(36 52)(37 61)(38 54)(39 63)(40 56)(41 49)(42 58)(43 51)(44 60)(45 53)(46 62)(47 55)(48 64)
(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)
G:=sub<Sym(96)| (1,64,68)(2,49,69)(3,50,70)(4,51,71)(5,52,72)(6,53,73)(7,54,74)(8,55,75)(9,56,76)(10,57,77)(11,58,78)(12,59,79)(13,60,80)(14,61,65)(15,62,66)(16,63,67)(17,92,38)(18,93,39)(19,94,40)(20,95,41)(21,96,42)(22,81,43)(23,82,44)(24,83,45)(25,84,46)(26,85,47)(27,86,48)(28,87,33)(29,88,34)(30,89,35)(31,90,36)(32,91,37), (1,76,64,9,68,56)(2,77,49,10,69,57)(3,78,50,11,70,58)(4,79,51,12,71,59)(5,80,52,13,72,60)(6,65,53,14,73,61)(7,66,54,15,74,62)(8,67,55,16,75,63)(17,84,38,25,92,46)(18,85,39,26,93,47)(19,86,40,27,94,48)(20,87,41,28,95,33)(21,88,42,29,96,34)(22,89,43,30,81,35)(23,90,44,31,82,36)(24,91,45,32,83,37), (1,86)(2,95)(3,88)(4,81)(5,90)(6,83)(7,92)(8,85)(9,94)(10,87)(11,96)(12,89)(13,82)(14,91)(15,84)(16,93)(17,74)(18,67)(19,76)(20,69)(21,78)(22,71)(23,80)(24,73)(25,66)(26,75)(27,68)(28,77)(29,70)(30,79)(31,72)(32,65)(33,57)(34,50)(35,59)(36,52)(37,61)(38,54)(39,63)(40,56)(41,49)(42,58)(43,51)(44,60)(45,53)(46,62)(47,55)(48,64), (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)>;
G:=Group( (1,64,68)(2,49,69)(3,50,70)(4,51,71)(5,52,72)(6,53,73)(7,54,74)(8,55,75)(9,56,76)(10,57,77)(11,58,78)(12,59,79)(13,60,80)(14,61,65)(15,62,66)(16,63,67)(17,92,38)(18,93,39)(19,94,40)(20,95,41)(21,96,42)(22,81,43)(23,82,44)(24,83,45)(25,84,46)(26,85,47)(27,86,48)(28,87,33)(29,88,34)(30,89,35)(31,90,36)(32,91,37), (1,76,64,9,68,56)(2,77,49,10,69,57)(3,78,50,11,70,58)(4,79,51,12,71,59)(5,80,52,13,72,60)(6,65,53,14,73,61)(7,66,54,15,74,62)(8,67,55,16,75,63)(17,84,38,25,92,46)(18,85,39,26,93,47)(19,86,40,27,94,48)(20,87,41,28,95,33)(21,88,42,29,96,34)(22,89,43,30,81,35)(23,90,44,31,82,36)(24,91,45,32,83,37), (1,86)(2,95)(3,88)(4,81)(5,90)(6,83)(7,92)(8,85)(9,94)(10,87)(11,96)(12,89)(13,82)(14,91)(15,84)(16,93)(17,74)(18,67)(19,76)(20,69)(21,78)(22,71)(23,80)(24,73)(25,66)(26,75)(27,68)(28,77)(29,70)(30,79)(31,72)(32,65)(33,57)(34,50)(35,59)(36,52)(37,61)(38,54)(39,63)(40,56)(41,49)(42,58)(43,51)(44,60)(45,53)(46,62)(47,55)(48,64), (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) );
G=PermutationGroup([[(1,64,68),(2,49,69),(3,50,70),(4,51,71),(5,52,72),(6,53,73),(7,54,74),(8,55,75),(9,56,76),(10,57,77),(11,58,78),(12,59,79),(13,60,80),(14,61,65),(15,62,66),(16,63,67),(17,92,38),(18,93,39),(19,94,40),(20,95,41),(21,96,42),(22,81,43),(23,82,44),(24,83,45),(25,84,46),(26,85,47),(27,86,48),(28,87,33),(29,88,34),(30,89,35),(31,90,36),(32,91,37)], [(1,76,64,9,68,56),(2,77,49,10,69,57),(3,78,50,11,70,58),(4,79,51,12,71,59),(5,80,52,13,72,60),(6,65,53,14,73,61),(7,66,54,15,74,62),(8,67,55,16,75,63),(17,84,38,25,92,46),(18,85,39,26,93,47),(19,86,40,27,94,48),(20,87,41,28,95,33),(21,88,42,29,96,34),(22,89,43,30,81,35),(23,90,44,31,82,36),(24,91,45,32,83,37)], [(1,86),(2,95),(3,88),(4,81),(5,90),(6,83),(7,92),(8,85),(9,94),(10,87),(11,96),(12,89),(13,82),(14,91),(15,84),(16,93),(17,74),(18,67),(19,76),(20,69),(21,78),(22,71),(23,80),(24,73),(25,66),(26,75),(27,68),(28,77),(29,70),(30,79),(31,72),(32,65),(33,57),(34,50),(35,59),(36,52),(37,61),(38,54),(39,63),(40,56),(41,49),(42,58),(43,51),(44,60),(45,53),(46,62),(47,55),(48,64)], [(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)]])
108 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 8E | 8F | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | 12L | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 24A | ··· | 24H | 24I | ··· | 24T | 24U | 24V | 24W | 24X | 48A | ··· | 48AF | 48AG | ··· | 48AN |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 24 | ··· | 24 | 24 | ··· | 24 | 24 | 24 | 24 | 24 | 48 | ··· | 48 | 48 | ··· | 48 |
| size | 1 | 1 | 6 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 6 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 1 | 1 | 1 | 1 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 6 | 6 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | 6 | 6 | 6 | 2 | ··· | 2 | 6 | ··· | 6 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 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 | C3 | C4 | C4 | C6 | C6 | C6 | C8 | C8 | C12 | C12 | C24 | C24 | S3 | D6 | C3×S3 | C4×S3 | M5(2) | S3×C6 | S3×C8 | S3×C12 | D6.C8 | C3×M5(2) | S3×C24 | C3×D6.C8 |
| kernel | C3×D6.C8 | C3×C3⋊C16 | C3×C48 | S3×C24 | D6.C8 | C3×C3⋊C8 | S3×C12 | C3⋊C16 | C48 | S3×C8 | C3×Dic3 | S3×C6 | C3⋊C8 | C4×S3 | Dic3 | D6 | C48 | C24 | C16 | C12 | C32 | C8 | C6 | C4 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 8 | 8 | 16 |
Matrix representation of C3×D6.C8 ►in GL2(𝔽97) generated by
| 61 | 0 |
| 0 | 61 |
| 62 | 0 |
| 0 | 36 |
| 0 | 36 |
| 62 | 0 |
| 12 | 0 |
| 0 | 85 |
G:=sub<GL(2,GF(97))| [61,0,0,61],[62,0,0,36],[0,62,36,0],[12,0,0,85] >;
C3×D6.C8 in GAP, Magma, Sage, TeX
C_3\times D_6.C_8
% in TeX
G:=Group("C3xD6.C8"); // GroupNames label
G:=SmallGroup(288,232);
// by ID
G=gap.SmallGroup(288,232);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,701,92,80,102,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^6=c^2=1,d^8=b^3,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^3*c>;
// generators/relations