direct product, metacyclic, supersoluble, monomial
Aliases: C3×C24⋊C4, C24⋊5C12, C24⋊7Dic3, C3⋊C8⋊4C12, (C3×C24)⋊13C4, C6.4(C4×C12), C8⋊3(C3×Dic3), C4.21(S3×C12), (C2×C24).24C6, (C6×C24).25C2, (C2×C24).36S3, C12.112(C4×S3), C12.26(C2×C12), C32⋊7(C8⋊C4), (C2×C12).455D6, C62.69(C2×C4), (C3×C6).14C42, (C4×Dic3).6C6, (C6×Dic3).6C4, C6.20(C4×Dic3), C2.4(Dic3×C12), C4.12(C6×Dic3), C6.15(C8⋊S3), C6.2(C3×M4(2)), C22.10(S3×C12), (C2×Dic3).3C12, C12.69(C2×Dic3), (C3×C6).13M4(2), (C6×C12).333C22, (Dic3×C12).16C2, (C3×C3⋊C8)⋊10C4, C3⋊2(C3×C8⋊C4), (C6×C3⋊C8).22C2, (C2×C3⋊C8).10C6, (C2×C8).8(C3×S3), C2.2(C3×C8⋊S3), (C2×C4).92(S3×C6), (C2×C6).80(C4×S3), (C2×C6).14(C2×C12), (C2×C12).122(C2×C6), (C3×C12).106(C2×C4), SmallGroup(288,249)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C24⋊C4
G = < a,b,c | a3=b24=c4=1, ab=ba, ac=ca, cbc-1=b5 >
Subgroups: 154 in 91 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C6, C8, C8, C2×C4, C2×C4, C32, Dic3, C12, C12, C2×C6, C2×C6, C42, C2×C8, C2×C8, C3×C6, C3×C6, C3⋊C8, C24, C24, C2×Dic3, C2×C12, C2×C12, C8⋊C4, C3×Dic3, C3×C12, C62, C2×C3⋊C8, C4×Dic3, C4×C12, C2×C24, C2×C24, C3×C3⋊C8, C3×C24, C6×Dic3, C6×C12, C24⋊C4, C3×C8⋊C4, C6×C3⋊C8, Dic3×C12, C6×C24, C3×C24⋊C4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, Dic3, C12, D6, C2×C6, C42, M4(2), C3×S3, C4×S3, C2×Dic3, C2×C12, C8⋊C4, C3×Dic3, S3×C6, C8⋊S3, C4×Dic3, C4×C12, C3×M4(2), S3×C12, C6×Dic3, C24⋊C4, C3×C8⋊C4, C3×C8⋊S3, Dic3×C12, C3×C24⋊C4
►On 96 points
(1 9 17)(2 10 18)(3 11 19)(4 12 20)(5 13 21)(6 14 22)(7 15 23)(8 16 24)(25 41 33)(26 42 34)(27 43 35)(28 44 36)(29 45 37)(30 46 38)(31 47 39)(32 48 40)(49 65 57)(50 66 58)(51 67 59)(52 68 60)(53 69 61)(54 70 62)(55 71 63)(56 72 64)(73 81 89)(74 82 90)(75 83 91)(76 84 92)(77 85 93)(78 86 94)(79 87 95)(80 88 96)
(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 67 93 37)(2 72 94 42)(3 53 95 47)(4 58 96 28)(5 63 73 33)(6 68 74 38)(7 49 75 43)(8 54 76 48)(9 59 77 29)(10 64 78 34)(11 69 79 39)(12 50 80 44)(13 55 81 25)(14 60 82 30)(15 65 83 35)(16 70 84 40)(17 51 85 45)(18 56 86 26)(19 61 87 31)(20 66 88 36)(21 71 89 41)(22 52 90 46)(23 57 91 27)(24 62 92 32)
G:=sub<Sym(96)| (1,9,17)(2,10,18)(3,11,19)(4,12,20)(5,13,21)(6,14,22)(7,15,23)(8,16,24)(25,41,33)(26,42,34)(27,43,35)(28,44,36)(29,45,37)(30,46,38)(31,47,39)(32,48,40)(49,65,57)(50,66,58)(51,67,59)(52,68,60)(53,69,61)(54,70,62)(55,71,63)(56,72,64)(73,81,89)(74,82,90)(75,83,91)(76,84,92)(77,85,93)(78,86,94)(79,87,95)(80,88,96), (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), (1,67,93,37)(2,72,94,42)(3,53,95,47)(4,58,96,28)(5,63,73,33)(6,68,74,38)(7,49,75,43)(8,54,76,48)(9,59,77,29)(10,64,78,34)(11,69,79,39)(12,50,80,44)(13,55,81,25)(14,60,82,30)(15,65,83,35)(16,70,84,40)(17,51,85,45)(18,56,86,26)(19,61,87,31)(20,66,88,36)(21,71,89,41)(22,52,90,46)(23,57,91,27)(24,62,92,32)>;
G:=Group( (1,9,17)(2,10,18)(3,11,19)(4,12,20)(5,13,21)(6,14,22)(7,15,23)(8,16,24)(25,41,33)(26,42,34)(27,43,35)(28,44,36)(29,45,37)(30,46,38)(31,47,39)(32,48,40)(49,65,57)(50,66,58)(51,67,59)(52,68,60)(53,69,61)(54,70,62)(55,71,63)(56,72,64)(73,81,89)(74,82,90)(75,83,91)(76,84,92)(77,85,93)(78,86,94)(79,87,95)(80,88,96), (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), (1,67,93,37)(2,72,94,42)(3,53,95,47)(4,58,96,28)(5,63,73,33)(6,68,74,38)(7,49,75,43)(8,54,76,48)(9,59,77,29)(10,64,78,34)(11,69,79,39)(12,50,80,44)(13,55,81,25)(14,60,82,30)(15,65,83,35)(16,70,84,40)(17,51,85,45)(18,56,86,26)(19,61,87,31)(20,66,88,36)(21,71,89,41)(22,52,90,46)(23,57,91,27)(24,62,92,32) );
G=PermutationGroup([[(1,9,17),(2,10,18),(3,11,19),(4,12,20),(5,13,21),(6,14,22),(7,15,23),(8,16,24),(25,41,33),(26,42,34),(27,43,35),(28,44,36),(29,45,37),(30,46,38),(31,47,39),(32,48,40),(49,65,57),(50,66,58),(51,67,59),(52,68,60),(53,69,61),(54,70,62),(55,71,63),(56,72,64),(73,81,89),(74,82,90),(75,83,91),(76,84,92),(77,85,93),(78,86,94),(79,87,95),(80,88,96)], [(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)], [(1,67,93,37),(2,72,94,42),(3,53,95,47),(4,58,96,28),(5,63,73,33),(6,68,74,38),(7,49,75,43),(8,54,76,48),(9,59,77,29),(10,64,78,34),(11,69,79,39),(12,50,80,44),(13,55,81,25),(14,60,82,30),(15,65,83,35),(16,70,84,40),(17,51,85,45),(18,56,86,26),(19,61,87,31),(20,66,88,36),(21,71,89,41),(22,52,90,46),(23,57,91,27),(24,62,92,32)]])
108 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 6G | ··· | 6O | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | ··· | 12H | 12I | ··· | 12T | 12U | ··· | 12AB | 24A | ··· | 24AF | 24AG | ··· | 24AN |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 6 | ··· | 6 |
108 irreducible representations
| dim | 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 | 2 | 2 |
| type | + | + | + | + | + | - | + | |||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | C12 | S3 | Dic3 | D6 | M4(2) | C3×S3 | C4×S3 | C4×S3 | C3×Dic3 | S3×C6 | C8⋊S3 | C3×M4(2) | S3×C12 | S3×C12 | C3×C8⋊S3 |
| kernel | C3×C24⋊C4 | C6×C3⋊C8 | Dic3×C12 | C6×C24 | C24⋊C4 | C3×C3⋊C8 | C3×C24 | C6×Dic3 | C2×C3⋊C8 | C4×Dic3 | C2×C24 | C3⋊C8 | C24 | C2×Dic3 | C2×C24 | C24 | C2×C12 | C3×C6 | C2×C8 | C12 | C2×C6 | C8 | C2×C4 | C6 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 8 | 8 | 8 | 1 | 2 | 1 | 4 | 2 | 2 | 2 | 4 | 2 | 8 | 8 | 4 | 4 | 16 |
Matrix representation of C3×C24⋊C4 ►in GL4(𝔽73) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 64 |
| 64 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 7 | 1 |
| 0 | 0 | 0 | 17 |
| 0 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 1 | 29 |
| 0 | 0 | 10 | 72 |
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,64,0,0,0,0,64],[64,0,0,0,0,8,0,0,0,0,7,0,0,0,1,17],[0,72,0,0,1,0,0,0,0,0,1,10,0,0,29,72] >;
C3×C24⋊C4 in GAP, Magma, Sage, TeX
C_3\times C_{24}\rtimes C_4 % in TeX
G:=Group("C3xC24:C4"); // GroupNames label
G:=SmallGroup(288,249);
// by ID
G=gap.SmallGroup(288,249);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,84,701,176,102,9414]);
// Polycyclic
G:=Group<a,b,c|a^3=b^24=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations