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, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C8, SD16, C2×D4, C2×Q8, C3×S3, C3×C6, C3×C6, C24, C24, Dic6, Dic6, D12, D12, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×C6, C2×SD16, C3×Dic3, C3×C12, S3×C6, C62, C24⋊C2, C2×C24, C2×C24, C3×SD16, C2×Dic6, C2×D12, C6×D4, C6×Q8, C3×C24, C3×Dic6, C3×Dic6, C3×D12, C3×D12, C6×Dic3, C6×C12, S3×C2×C6, C2×C24⋊C2, C6×SD16, C3×C24⋊C2, C6×C24, C6×Dic6, C6×D12, C6×C24⋊C2
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, SD16, C2×D4, C3×S3, D12, C3×D4, C22×S3, C22×C6, C2×SD16, S3×C6, C24⋊C2, C3×SD16, C2×D12, C6×D4, C3×D12, S3×C2×C6, C2×C24⋊C2, C6×SD16, C3×C24⋊C2, C6×D12, C6×C24⋊C2
►On 96 points
(1 68 9 52 17 60)(2 69 10 53 18 61)(3 70 11 54 19 62)(4 71 12 55 20 63)(5 72 13 56 21 64)(6 49 14 57 22 65)(7 50 15 58 23 66)(8 51 16 59 24 67)(25 76 41 92 33 84)(26 77 42 93 34 85)(27 78 43 94 35 86)(28 79 44 95 36 87)(29 80 45 96 37 88)(30 81 46 73 38 89)(31 82 47 74 39 90)(32 83 48 75 40 91)
(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 41)(2 28)(3 39)(4 26)(5 37)(6 48)(7 35)(8 46)(9 33)(10 44)(11 31)(12 42)(13 29)(14 40)(15 27)(16 38)(17 25)(18 36)(19 47)(20 34)(21 45)(22 32)(23 43)(24 30)(49 75)(50 86)(51 73)(52 84)(53 95)(54 82)(55 93)(56 80)(57 91)(58 78)(59 89)(60 76)(61 87)(62 74)(63 85)(64 96)(65 83)(66 94)(67 81)(68 92)(69 79)(70 90)(71 77)(72 88)
G:=sub<Sym(96)| (1,68,9,52,17,60)(2,69,10,53,18,61)(3,70,11,54,19,62)(4,71,12,55,20,63)(5,72,13,56,21,64)(6,49,14,57,22,65)(7,50,15,58,23,66)(8,51,16,59,24,67)(25,76,41,92,33,84)(26,77,42,93,34,85)(27,78,43,94,35,86)(28,79,44,95,36,87)(29,80,45,96,37,88)(30,81,46,73,38,89)(31,82,47,74,39,90)(32,83,48,75,40,91), (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), (1,41)(2,28)(3,39)(4,26)(5,37)(6,48)(7,35)(8,46)(9,33)(10,44)(11,31)(12,42)(13,29)(14,40)(15,27)(16,38)(17,25)(18,36)(19,47)(20,34)(21,45)(22,32)(23,43)(24,30)(49,75)(50,86)(51,73)(52,84)(53,95)(54,82)(55,93)(56,80)(57,91)(58,78)(59,89)(60,76)(61,87)(62,74)(63,85)(64,96)(65,83)(66,94)(67,81)(68,92)(69,79)(70,90)(71,77)(72,88)>;
G:=Group( (1,68,9,52,17,60)(2,69,10,53,18,61)(3,70,11,54,19,62)(4,71,12,55,20,63)(5,72,13,56,21,64)(6,49,14,57,22,65)(7,50,15,58,23,66)(8,51,16,59,24,67)(25,76,41,92,33,84)(26,77,42,93,34,85)(27,78,43,94,35,86)(28,79,44,95,36,87)(29,80,45,96,37,88)(30,81,46,73,38,89)(31,82,47,74,39,90)(32,83,48,75,40,91), (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), (1,41)(2,28)(3,39)(4,26)(5,37)(6,48)(7,35)(8,46)(9,33)(10,44)(11,31)(12,42)(13,29)(14,40)(15,27)(16,38)(17,25)(18,36)(19,47)(20,34)(21,45)(22,32)(23,43)(24,30)(49,75)(50,86)(51,73)(52,84)(53,95)(54,82)(55,93)(56,80)(57,91)(58,78)(59,89)(60,76)(61,87)(62,74)(63,85)(64,96)(65,83)(66,94)(67,81)(68,92)(69,79)(70,90)(71,77)(72,88) );
G=PermutationGroup([[(1,68,9,52,17,60),(2,69,10,53,18,61),(3,70,11,54,19,62),(4,71,12,55,20,63),(5,72,13,56,21,64),(6,49,14,57,22,65),(7,50,15,58,23,66),(8,51,16,59,24,67),(25,76,41,92,33,84),(26,77,42,93,34,85),(27,78,43,94,35,86),(28,79,44,95,36,87),(29,80,45,96,37,88),(30,81,46,73,38,89),(31,82,47,74,39,90),(32,83,48,75,40,91)], [(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)], [(1,41),(2,28),(3,39),(4,26),(5,37),(6,48),(7,35),(8,46),(9,33),(10,44),(11,31),(12,42),(13,29),(14,40),(15,27),(16,38),(17,25),(18,36),(19,47),(20,34),(21,45),(22,32),(23,43),(24,30),(49,75),(50,86),(51,73),(52,84),(53,95),(54,82),(55,93),(56,80),(57,91),(58,78),(59,89),(60,76),(61,87),(62,74),(63,85),(64,96),(65,83),(66,94),(67,81),(68,92),(69,79),(70,90),(71,77),(72,88)]])
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