direct product, metabelian, supersoluble, monomial, A-group
Aliases: C2×C6×C3⋊C8, C62⋊8C8, (C2×C6)⋊5C24, C6⋊2(C2×C24), (C6×C12).32C4, C3⋊2(C22×C24), C32⋊9(C22×C8), C12.47(C2×C12), (C2×C12).17C12, (C2×C12).462D6, (C2×C62).12C4, C4.13(C6×Dic3), (C22×C6).15C12, (C22×C12).25C6, C6.20(C22×C12), C12.40(C22×C6), (C22×C12).44S3, C62.107(C2×C4), (C2×C12).32Dic3, C12.70(C2×Dic3), C23.6(C3×Dic3), (C3×C12).172C23, C12.228(C22×S3), (C6×C12).349C22, C6.40(C22×Dic3), C22.12(C6×Dic3), (C22×C6).19Dic3, (C3×C6)⋊8(C2×C8), C4.40(S3×C2×C6), (C2×C6×C12).20C2, C2.1(Dic3×C2×C6), (C2×C4).100(S3×C6), (C2×C6).41(C2×C12), (C2×C4).9(C3×Dic3), (C2×C12).130(C2×C6), (C3×C12).135(C2×C4), (C22×C4).13(C3×S3), (C2×C6).61(C2×Dic3), (C3×C6).112(C22×C4), SmallGroup(288,691)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C2×C6×C3⋊C8 |
Generators and relations for C2×C6×C3⋊C8
G = < a,b,c,d | a2=b6=c3=d8=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 250 in 179 conjugacy classes, 130 normal (22 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C6, C8, C2×C4, C23, C32, C12, C12, C12, C2×C6, C2×C6, C2×C8, C22×C4, C3×C6, C3×C6, C3⋊C8, C24, C2×C12, C2×C12, C22×C6, C22×C6, C22×C8, C3×C12, C3×C12, C62, C2×C3⋊C8, C2×C24, C22×C12, C22×C12, C3×C3⋊C8, C6×C12, C2×C62, C22×C3⋊C8, C22×C24, C6×C3⋊C8, C2×C6×C12, C2×C6×C3⋊C8
Quotients: C1, C2, C3, C4, C22, S3, C6, C8, C2×C4, C23, Dic3, C12, D6, C2×C6, C2×C8, C22×C4, C3×S3, C3⋊C8, C24, C2×Dic3, C2×C12, C22×S3, C22×C6, C22×C8, C3×Dic3, S3×C6, C2×C3⋊C8, C2×C24, C22×Dic3, C22×C12, C3×C3⋊C8, C6×Dic3, S3×C2×C6, C22×C3⋊C8, C22×C24, C6×C3⋊C8, Dic3×C2×C6, C2×C6×C3⋊C8
►On 96 points
(1 21)(2 22)(3 23)(4 24)(5 17)(6 18)(7 19)(8 20)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(25 91)(26 92)(27 93)(28 94)(29 95)(30 96)(31 89)(32 90)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 73)(40 74)(49 69)(50 70)(51 71)(52 72)(53 65)(54 66)(55 67)(56 68)(57 85)(58 86)(59 87)(60 88)(61 81)(62 82)(63 83)(64 84)
(1 85 79 11 29 51)(2 86 80 12 30 52)(3 87 73 13 31 53)(4 88 74 14 32 54)(5 81 75 15 25 55)(6 82 76 16 26 56)(7 83 77 9 27 49)(8 84 78 10 28 50)(17 61 33 47 91 67)(18 62 34 48 92 68)(19 63 35 41 93 69)(20 64 36 42 94 70)(21 57 37 43 95 71)(22 58 38 44 96 72)(23 59 39 45 89 65)(24 60 40 46 90 66)
(1 79 29)(2 30 80)(3 73 31)(4 32 74)(5 75 25)(6 26 76)(7 77 27)(8 28 78)(9 49 83)(10 84 50)(11 51 85)(12 86 52)(13 53 87)(14 88 54)(15 55 81)(16 82 56)(17 33 91)(18 92 34)(19 35 93)(20 94 36)(21 37 95)(22 96 38)(23 39 89)(24 90 40)(41 69 63)(42 64 70)(43 71 57)(44 58 72)(45 65 59)(46 60 66)(47 67 61)(48 62 68)
(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)| (1,21)(2,22)(3,23)(4,24)(5,17)(6,18)(7,19)(8,20)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(25,91)(26,92)(27,93)(28,94)(29,95)(30,96)(31,89)(32,90)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,73)(40,74)(49,69)(50,70)(51,71)(52,72)(53,65)(54,66)(55,67)(56,68)(57,85)(58,86)(59,87)(60,88)(61,81)(62,82)(63,83)(64,84), (1,85,79,11,29,51)(2,86,80,12,30,52)(3,87,73,13,31,53)(4,88,74,14,32,54)(5,81,75,15,25,55)(6,82,76,16,26,56)(7,83,77,9,27,49)(8,84,78,10,28,50)(17,61,33,47,91,67)(18,62,34,48,92,68)(19,63,35,41,93,69)(20,64,36,42,94,70)(21,57,37,43,95,71)(22,58,38,44,96,72)(23,59,39,45,89,65)(24,60,40,46,90,66), (1,79,29)(2,30,80)(3,73,31)(4,32,74)(5,75,25)(6,26,76)(7,77,27)(8,28,78)(9,49,83)(10,84,50)(11,51,85)(12,86,52)(13,53,87)(14,88,54)(15,55,81)(16,82,56)(17,33,91)(18,92,34)(19,35,93)(20,94,36)(21,37,95)(22,96,38)(23,39,89)(24,90,40)(41,69,63)(42,64,70)(43,71,57)(44,58,72)(45,65,59)(46,60,66)(47,67,61)(48,62,68), (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( (1,21)(2,22)(3,23)(4,24)(5,17)(6,18)(7,19)(8,20)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(25,91)(26,92)(27,93)(28,94)(29,95)(30,96)(31,89)(32,90)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,73)(40,74)(49,69)(50,70)(51,71)(52,72)(53,65)(54,66)(55,67)(56,68)(57,85)(58,86)(59,87)(60,88)(61,81)(62,82)(63,83)(64,84), (1,85,79,11,29,51)(2,86,80,12,30,52)(3,87,73,13,31,53)(4,88,74,14,32,54)(5,81,75,15,25,55)(6,82,76,16,26,56)(7,83,77,9,27,49)(8,84,78,10,28,50)(17,61,33,47,91,67)(18,62,34,48,92,68)(19,63,35,41,93,69)(20,64,36,42,94,70)(21,57,37,43,95,71)(22,58,38,44,96,72)(23,59,39,45,89,65)(24,60,40,46,90,66), (1,79,29)(2,30,80)(3,73,31)(4,32,74)(5,75,25)(6,26,76)(7,77,27)(8,28,78)(9,49,83)(10,84,50)(11,51,85)(12,86,52)(13,53,87)(14,88,54)(15,55,81)(16,82,56)(17,33,91)(18,92,34)(19,35,93)(20,94,36)(21,37,95)(22,96,38)(23,39,89)(24,90,40)(41,69,63)(42,64,70)(43,71,57)(44,58,72)(45,65,59)(46,60,66)(47,67,61)(48,62,68), (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([[(1,21),(2,22),(3,23),(4,24),(5,17),(6,18),(7,19),(8,20),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(25,91),(26,92),(27,93),(28,94),(29,95),(30,96),(31,89),(32,90),(33,75),(34,76),(35,77),(36,78),(37,79),(38,80),(39,73),(40,74),(49,69),(50,70),(51,71),(52,72),(53,65),(54,66),(55,67),(56,68),(57,85),(58,86),(59,87),(60,88),(61,81),(62,82),(63,83),(64,84)], [(1,85,79,11,29,51),(2,86,80,12,30,52),(3,87,73,13,31,53),(4,88,74,14,32,54),(5,81,75,15,25,55),(6,82,76,16,26,56),(7,83,77,9,27,49),(8,84,78,10,28,50),(17,61,33,47,91,67),(18,62,34,48,92,68),(19,63,35,41,93,69),(20,64,36,42,94,70),(21,57,37,43,95,71),(22,58,38,44,96,72),(23,59,39,45,89,65),(24,60,40,46,90,66)], [(1,79,29),(2,30,80),(3,73,31),(4,32,74),(5,75,25),(6,26,76),(7,77,27),(8,28,78),(9,49,83),(10,84,50),(11,51,85),(12,86,52),(13,53,87),(14,88,54),(15,55,81),(16,82,56),(17,33,91),(18,92,34),(19,35,93),(20,94,36),(21,37,95),(22,96,38),(23,39,89),(24,90,40),(41,69,63),(42,64,70),(43,71,57),(44,58,72),(45,65,59),(46,60,66),(47,67,61),(48,62,68)], [(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)]])
144 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | 3B | 3C | 3D | 3E | 4A | ··· | 4H | 6A | ··· | 6N | 6O | ··· | 6AI | 8A | ··· | 8P | 12A | ··· | 12P | 12Q | ··· | 12AN | 24A | ··· | 24AF |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
144 irreducible representations
| dim | 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 | C3 | C4 | C4 | C6 | C6 | C8 | C12 | C12 | C24 | S3 | Dic3 | D6 | Dic3 | C3×S3 | C3⋊C8 | C3×Dic3 | S3×C6 | C3×Dic3 | C3×C3⋊C8 |
| kernel | C2×C6×C3⋊C8 | C6×C3⋊C8 | C2×C6×C12 | C22×C3⋊C8 | C6×C12 | C2×C62 | C2×C3⋊C8 | C22×C12 | C62 | C2×C12 | C22×C6 | C2×C6 | C22×C12 | C2×C12 | C2×C12 | C22×C6 | C22×C4 | C2×C6 | C2×C4 | C2×C4 | C23 | C22 |
| # reps | 1 | 6 | 1 | 2 | 6 | 2 | 12 | 2 | 16 | 12 | 4 | 32 | 1 | 3 | 3 | 1 | 2 | 8 | 6 | 6 | 2 | 16 |
Matrix representation of C2×C6×C3⋊C8 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 65 | 0 |
| 0 | 0 | 0 | 65 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 8 |
| 51 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 72 | 0 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,9,0,0,0,0,65,0,0,0,0,65],[1,0,0,0,0,1,0,0,0,0,64,0,0,0,0,8],[51,0,0,0,0,46,0,0,0,0,0,72,0,0,72,0] >;
C2×C6×C3⋊C8 in GAP, Magma, Sage, TeX
C_2\times C_6\times C_3\rtimes C_8
% in TeX
G:=Group("C2xC6xC3:C8"); // GroupNames label
G:=SmallGroup(288,691);
// by ID
G=gap.SmallGroup(288,691);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,168,102,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^6=c^3=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations