direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×C24⋊C2, C12⋊8SD16, C42.258D6, C8⋊11(C4×S3), (C4×C8)⋊11S3, C6.6(C4×D4), C3⋊1(C4×SD16), (C4×C24)⋊16C2, C24⋊23(C2×C4), C2.9(C4×D12), Dic6⋊8(C2×C4), (C4×Dic6)⋊1C2, (C4×D12).3C2, C8⋊Dic3⋊28C2, C6.2(C4○D8), (C2×C8).285D6, (C2×C4).60D12, C6.3(C2×SD16), D12.12(C2×C4), C2.1(C4○D24), (C2×C12).350D4, C2.Dic12⋊43C2, C2.D24.17C2, C22.27(C2×D12), C12.216(C4○D4), C4.100(C4○D12), (C2×C24).345C22, (C2×C12).720C23, C12.101(C22×C4), (C4×C12).325C22, (C2×D12).187C22, C4⋊Dic3.262C22, (C2×Dic6).206C22, C4.59(S3×C2×C4), C2.1(C2×C24⋊C2), (C2×C6).103(C2×D4), (C2×C24⋊C2).10C2, (C2×C4).663(C22×S3), SmallGroup(192,250)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×C24⋊C2
G = < a,b,c | a4=b24=c2=1, ab=ba, ac=ca, cbc=b11 >
Subgroups: 360 in 122 conjugacy classes, 55 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C8, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, C12, D6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C24, C24, Dic6, Dic6, C4×S3, D12, D12, C2×Dic3, C2×C12, C22×S3, C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C4×D4, C4×Q8, C2×SD16, C24⋊C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C4×C12, C2×C24, C2×Dic6, S3×C2×C4, C2×D12, C4×SD16, C2.Dic12, C8⋊Dic3, C2.D24, C4×C24, C4×Dic6, C4×D12, C2×C24⋊C2, C4×C24⋊C2
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, SD16, C22×C4, C2×D4, C4○D4, C4×S3, D12, C22×S3, C4×D4, C2×SD16, C4○D8, C24⋊C2, S3×C2×C4, C2×D12, C4○D12, C4×SD16, C4×D12, C2×C24⋊C2, C4○D24, C4×C24⋊C2
►On 96 points
(1 30 51 86)(2 31 52 87)(3 32 53 88)(4 33 54 89)(5 34 55 90)(6 35 56 91)(7 36 57 92)(8 37 58 93)(9 38 59 94)(10 39 60 95)(11 40 61 96)(12 41 62 73)(13 42 63 74)(14 43 64 75)(15 44 65 76)(16 45 66 77)(17 46 67 78)(18 47 68 79)(19 48 69 80)(20 25 70 81)(21 26 71 82)(22 27 72 83)(23 28 49 84)(24 29 50 85)
(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)
(2 12)(3 23)(4 10)(5 21)(6 8)(7 19)(9 17)(11 15)(14 24)(16 22)(18 20)(25 47)(26 34)(27 45)(28 32)(29 43)(31 41)(33 39)(35 37)(36 48)(38 46)(40 44)(49 53)(50 64)(52 62)(54 60)(55 71)(56 58)(57 69)(59 67)(61 65)(66 72)(68 70)(73 87)(75 85)(76 96)(77 83)(78 94)(79 81)(80 92)(82 90)(84 88)(89 95)(91 93)
G:=sub<Sym(96)| (1,30,51,86)(2,31,52,87)(3,32,53,88)(4,33,54,89)(5,34,55,90)(6,35,56,91)(7,36,57,92)(8,37,58,93)(9,38,59,94)(10,39,60,95)(11,40,61,96)(12,41,62,73)(13,42,63,74)(14,43,64,75)(15,44,65,76)(16,45,66,77)(17,46,67,78)(18,47,68,79)(19,48,69,80)(20,25,70,81)(21,26,71,82)(22,27,72,83)(23,28,49,84)(24,29,50,85), (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), (2,12)(3,23)(4,10)(5,21)(6,8)(7,19)(9,17)(11,15)(14,24)(16,22)(18,20)(25,47)(26,34)(27,45)(28,32)(29,43)(31,41)(33,39)(35,37)(36,48)(38,46)(40,44)(49,53)(50,64)(52,62)(54,60)(55,71)(56,58)(57,69)(59,67)(61,65)(66,72)(68,70)(73,87)(75,85)(76,96)(77,83)(78,94)(79,81)(80,92)(82,90)(84,88)(89,95)(91,93)>;
G:=Group( (1,30,51,86)(2,31,52,87)(3,32,53,88)(4,33,54,89)(5,34,55,90)(6,35,56,91)(7,36,57,92)(8,37,58,93)(9,38,59,94)(10,39,60,95)(11,40,61,96)(12,41,62,73)(13,42,63,74)(14,43,64,75)(15,44,65,76)(16,45,66,77)(17,46,67,78)(18,47,68,79)(19,48,69,80)(20,25,70,81)(21,26,71,82)(22,27,72,83)(23,28,49,84)(24,29,50,85), (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), (2,12)(3,23)(4,10)(5,21)(6,8)(7,19)(9,17)(11,15)(14,24)(16,22)(18,20)(25,47)(26,34)(27,45)(28,32)(29,43)(31,41)(33,39)(35,37)(36,48)(38,46)(40,44)(49,53)(50,64)(52,62)(54,60)(55,71)(56,58)(57,69)(59,67)(61,65)(66,72)(68,70)(73,87)(75,85)(76,96)(77,83)(78,94)(79,81)(80,92)(82,90)(84,88)(89,95)(91,93) );
G=PermutationGroup([[(1,30,51,86),(2,31,52,87),(3,32,53,88),(4,33,54,89),(5,34,55,90),(6,35,56,91),(7,36,57,92),(8,37,58,93),(9,38,59,94),(10,39,60,95),(11,40,61,96),(12,41,62,73),(13,42,63,74),(14,43,64,75),(15,44,65,76),(16,45,66,77),(17,46,67,78),(18,47,68,79),(19,48,69,80),(20,25,70,81),(21,26,71,82),(22,27,72,83),(23,28,49,84),(24,29,50,85)], [(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)], [(2,12),(3,23),(4,10),(5,21),(6,8),(7,19),(9,17),(11,15),(14,24),(16,22),(18,20),(25,47),(26,34),(27,45),(28,32),(29,43),(31,41),(33,39),(35,37),(36,48),(38,46),(40,44),(49,53),(50,64),(52,62),(54,60),(55,71),(56,58),(57,69),(59,67),(61,65),(66,72),(68,70),(73,87),(75,85),(76,96),(77,83),(78,94),(79,81),(80,92),(82,90),(84,88),(89,95),(91,93)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 6A | 6B | 6C | 8A | ··· | 8H | 12A | ··· | 12L | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 12 | ··· | 12 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 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 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D6 | SD16 | C4○D4 | C4×S3 | D12 | C4○D8 | C24⋊C2 | C4○D12 | C4○D24 |
| kernel | C4×C24⋊C2 | C2.Dic12 | C8⋊Dic3 | C2.D24 | C4×C24 | C4×Dic6 | C4×D12 | C2×C24⋊C2 | C24⋊C2 | C4×C8 | C2×C12 | C42 | C2×C8 | C12 | C12 | C8 | C2×C4 | C6 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 1 | 2 | 4 | 2 | 4 | 4 | 4 | 8 | 4 | 8 |
Matrix representation of C4×C24⋊C2 ►in GL3(𝔽73) generated by
| 46 | 0 | 0 |
| 0 | 27 | 0 |
| 0 | 0 | 27 |
| 1 | 0 | 0 |
| 0 | 62 | 37 |
| 0 | 36 | 25 |
| 72 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 72 | 72 |
G:=sub<GL(3,GF(73))| [46,0,0,0,27,0,0,0,27],[1,0,0,0,62,36,0,37,25],[72,0,0,0,1,72,0,0,72] >;
C4×C24⋊C2 in GAP, Magma, Sage, TeX
C_4\times C_{24}\rtimes C_2 % in TeX
G:=Group("C4xC24:C2"); // GroupNames label
G:=SmallGroup(192,250);
// by ID
G=gap.SmallGroup(192,250);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,58,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^4=b^24=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^11>;
// generators/relations