direct product, metabelian, supersoluble, monomial
Aliases: C6×C24⋊C2, C24⋊28D6, C12.67D12, C62.85D4, C8⋊8(S3×C6), (C2×C24)⋊7C6, C24⋊9(C2×C6), C6.9(C6×D4), (C6×C24)⋊12C2, (C2×C24)⋊13S3, C6⋊1(C3×SD16), C3⋊1(C6×SD16), (C3×C6)⋊6SD16, C4.6(C3×D12), (C2×Dic6)⋊5C6, Dic6⋊3(C2×C6), D12.6(C2×C6), (C2×D12).4C6, C6.97(C2×D12), C12.29(C3×D4), C2.11(C6×D12), (C2×C6).73D12, (C3×C24)⋊27C22, (C6×Dic6)⋊29C2, (C6×D12).19C2, (C2×C12).441D6, (C3×C12).131D4, C32⋊13(C2×SD16), C12.28(C22×C6), C22.12(C3×D12), (C6×C12).321C22, (C3×C12).160C23, C12.215(C22×S3), (C3×Dic6)⋊36C22, (C3×D12).45C22, (C2×C8)⋊5(C3×S3), C4.26(S3×C2×C6), (C2×C4).78(S3×C6), (C2×C6).20(C3×D4), (C3×C6).179(C2×D4), (C2×C12).106(C2×C6), SmallGroup(288,673)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×C24⋊C2
G = < a,b,c | a6=b24=c2=1, ab=ba, ac=ca, cbc=b11 >
Subgroups: 410 in 147 conjugacy classes, 66 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], S3 [×2], C6 [×2], C6 [×4], C6 [×5], C8 [×2], C2×C4, C2×C4, D4 [×3], Q8 [×3], C23, C32, Dic3 [×2], C12 [×4], C12 [×4], D6 [×4], C2×C6 [×2], C2×C6 [×5], C2×C8, SD16 [×4], C2×D4, C2×Q8, C3×S3 [×2], C3×C6, C3×C6 [×2], C24 [×4], C24 [×2], Dic6 [×2], Dic6, D12 [×2], D12, C2×Dic3, C2×C12 [×2], C2×C12 [×2], C3×D4 [×3], C3×Q8 [×3], C22×S3, C22×C6, C2×SD16, C3×Dic3 [×2], C3×C12 [×2], S3×C6 [×4], C62, C24⋊C2 [×4], C2×C24 [×2], C2×C24, C3×SD16 [×4], C2×Dic6, C2×D12, C6×D4, C6×Q8, C3×C24 [×2], C3×Dic6 [×2], C3×Dic6, C3×D12 [×2], C3×D12, C6×Dic3, C6×C12, S3×C2×C6, C2×C24⋊C2, C6×SD16, C3×C24⋊C2 [×4], C6×C24, C6×Dic6, C6×D12, C6×C24⋊C2
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], SD16 [×2], C2×D4, C3×S3, D12 [×2], C3×D4 [×2], C22×S3, C22×C6, C2×SD16, S3×C6 [×3], C24⋊C2 [×2], C3×SD16 [×2], C2×D12, C6×D4, C3×D12 [×2], S3×C2×C6, C2×C24⋊C2, C6×SD16, C3×C24⋊C2 [×2], C6×D12, C6×C24⋊C2
►On 96 points
(1 28 9 36 17 44)(2 29 10 37 18 45)(3 30 11 38 19 46)(4 31 12 39 20 47)(5 32 13 40 21 48)(6 33 14 41 22 25)(7 34 15 42 23 26)(8 35 16 43 24 27)(49 96 65 88 57 80)(50 73 66 89 58 81)(51 74 67 90 59 82)(52 75 68 91 60 83)(53 76 69 92 61 84)(54 77 70 93 62 85)(55 78 71 94 63 86)(56 79 72 95 64 87)
(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 94)(2 81)(3 92)(4 79)(5 90)(6 77)(7 88)(8 75)(9 86)(10 73)(11 84)(12 95)(13 82)(14 93)(15 80)(16 91)(17 78)(18 89)(19 76)(20 87)(21 74)(22 85)(23 96)(24 83)(25 54)(26 65)(27 52)(28 63)(29 50)(30 61)(31 72)(32 59)(33 70)(34 57)(35 68)(36 55)(37 66)(38 53)(39 64)(40 51)(41 62)(42 49)(43 60)(44 71)(45 58)(46 69)(47 56)(48 67)
G:=sub<Sym(96)| (1,28,9,36,17,44)(2,29,10,37,18,45)(3,30,11,38,19,46)(4,31,12,39,20,47)(5,32,13,40,21,48)(6,33,14,41,22,25)(7,34,15,42,23,26)(8,35,16,43,24,27)(49,96,65,88,57,80)(50,73,66,89,58,81)(51,74,67,90,59,82)(52,75,68,91,60,83)(53,76,69,92,61,84)(54,77,70,93,62,85)(55,78,71,94,63,86)(56,79,72,95,64,87), (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), (1,94)(2,81)(3,92)(4,79)(5,90)(6,77)(7,88)(8,75)(9,86)(10,73)(11,84)(12,95)(13,82)(14,93)(15,80)(16,91)(17,78)(18,89)(19,76)(20,87)(21,74)(22,85)(23,96)(24,83)(25,54)(26,65)(27,52)(28,63)(29,50)(30,61)(31,72)(32,59)(33,70)(34,57)(35,68)(36,55)(37,66)(38,53)(39,64)(40,51)(41,62)(42,49)(43,60)(44,71)(45,58)(46,69)(47,56)(48,67)>;
G:=Group( (1,28,9,36,17,44)(2,29,10,37,18,45)(3,30,11,38,19,46)(4,31,12,39,20,47)(5,32,13,40,21,48)(6,33,14,41,22,25)(7,34,15,42,23,26)(8,35,16,43,24,27)(49,96,65,88,57,80)(50,73,66,89,58,81)(51,74,67,90,59,82)(52,75,68,91,60,83)(53,76,69,92,61,84)(54,77,70,93,62,85)(55,78,71,94,63,86)(56,79,72,95,64,87), (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), (1,94)(2,81)(3,92)(4,79)(5,90)(6,77)(7,88)(8,75)(9,86)(10,73)(11,84)(12,95)(13,82)(14,93)(15,80)(16,91)(17,78)(18,89)(19,76)(20,87)(21,74)(22,85)(23,96)(24,83)(25,54)(26,65)(27,52)(28,63)(29,50)(30,61)(31,72)(32,59)(33,70)(34,57)(35,68)(36,55)(37,66)(38,53)(39,64)(40,51)(41,62)(42,49)(43,60)(44,71)(45,58)(46,69)(47,56)(48,67) );
G=PermutationGroup([(1,28,9,36,17,44),(2,29,10,37,18,45),(3,30,11,38,19,46),(4,31,12,39,20,47),(5,32,13,40,21,48),(6,33,14,41,22,25),(7,34,15,42,23,26),(8,35,16,43,24,27),(49,96,65,88,57,80),(50,73,66,89,58,81),(51,74,67,90,59,82),(52,75,68,91,60,83),(53,76,69,92,61,84),(54,77,70,93,62,85),(55,78,71,94,63,86),(56,79,72,95,64,87)], [(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)], [(1,94),(2,81),(3,92),(4,79),(5,90),(6,77),(7,88),(8,75),(9,86),(10,73),(11,84),(12,95),(13,82),(14,93),(15,80),(16,91),(17,78),(18,89),(19,76),(20,87),(21,74),(22,85),(23,96),(24,83),(25,54),(26,65),(27,52),(28,63),(29,50),(30,61),(31,72),(32,59),(33,70),(34,57),(35,68),(36,55),(37,66),(38,53),(39,64),(40,51),(41,62),(42,49),(43,60),(44,71),(45,58),(46,69),(47,56),(48,67)])
90 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 | ··· | 12P | 12Q | 12R | 12S | 12T | 24A | ··· | 24AF |
| 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 | 12 | 12 | 1 | ··· | 1 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 2 | ··· | 2 |
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 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | S3 | D4 | D4 | D6 | D6 | SD16 | C3×S3 | D12 | C3×D4 | D12 | C3×D4 | S3×C6 | S3×C6 | C24⋊C2 | C3×SD16 | C3×D12 | C3×D12 | C3×C24⋊C2 |
| kernel | C6×C24⋊C2 | C3×C24⋊C2 | C6×C24 | C6×Dic6 | C6×D12 | C2×C24⋊C2 | C24⋊C2 | C2×C24 | C2×Dic6 | C2×D12 | C2×C24 | C3×C12 | C62 | C24 | C2×C12 | C3×C6 | C2×C8 | C12 | C12 | C2×C6 | C2×C6 | C8 | C2×C4 | C6 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 8 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 1 | 4 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 8 | 8 | 4 | 4 | 16 |
Matrix representation of C6×C24⋊C2 ►in GL4(𝔽73) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 9 | 0 | 0 | 0 |
| 56 | 65 | 0 | 0 |
| 0 | 0 | 56 | 40 |
| 0 | 0 | 0 | 43 |
| 10 | 10 | 0 | 0 |
| 12 | 63 | 0 | 0 |
| 0 | 0 | 44 | 47 |
| 0 | 0 | 66 | 29 |
G:=sub<GL(4,GF(73))| [9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[9,56,0,0,0,65,0,0,0,0,56,0,0,0,40,43],[10,12,0,0,10,63,0,0,0,0,44,66,0,0,47,29] >;
C6×C24⋊C2 in GAP, Magma, Sage, TeX
C_6\times C_{24}\rtimes C_2 % in TeX
G:=Group("C6xC24:C2"); // GroupNames label
G:=SmallGroup(288,673);
// by ID
G=gap.SmallGroup(288,673);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,590,142,2524,102,9414]);
// Polycyclic
G:=Group<a,b,c|a^6=b^24=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^11>;
// generators/relations