direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C4○D24, D24⋊23C22, C24.62C23, C12.56C24, C23.34D12, D12.21C23, Dic12⋊20C22, Dic6.20C23, (C2×C8)⋊34D6, C6⋊1(C4○D8), (C2×D24)⋊27C2, (C22×C8)⋊12S3, C4.46(C2×D12), (C22×C24)⋊12C2, (C2×C24)⋊45C22, (C2×C12).404D4, C12.291(C2×D4), (C2×C4).101D12, C8.51(C22×S3), C4.53(S3×C23), C6.23(C22×D4), C24⋊C2⋊22C22, (C2×Dic12)⋊27C2, C4○D12⋊16C22, (C22×C4).460D6, (C22×C6).146D4, C2.25(C22×D12), C22.71(C2×D12), (C2×C12).797C23, (C2×D12).229C22, (C22×C12).545C22, (C2×Dic6).257C22, C3⋊1(C2×C4○D8), (C2×C24⋊C2)⋊33C2, (C2×C4○D12)⋊13C2, (C2×C6).179(C2×D4), (C2×C4).737(C22×S3), SmallGroup(192,1300)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 728 in 266 conjugacy classes, 111 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×10], S3 [×4], C6, C6 [×2], C6 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×14], Q8 [×6], C23, C23 [×2], Dic3 [×4], C12 [×2], C12 [×2], D6 [×8], C2×C6, C2×C6 [×2], C2×C6 [×2], C2×C8 [×2], C2×C8 [×4], D8 [×4], SD16 [×8], Q16 [×4], C22×C4, C22×C4 [×2], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×12], C24 [×4], Dic6 [×4], Dic6 [×2], C4×S3 [×8], D12 [×4], D12 [×2], C2×Dic3 [×2], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×4], C22×S3 [×2], C22×C6, C22×C8, C2×D8, C2×SD16 [×2], C2×Q16, C4○D8 [×8], C2×C4○D4 [×2], C24⋊C2 [×8], D24 [×4], Dic12 [×4], C2×C24 [×2], C2×C24 [×4], C2×Dic6 [×2], S3×C2×C4 [×2], C2×D12 [×2], C4○D12 [×8], C4○D12 [×4], C2×C3⋊D4 [×2], C22×C12, C2×C4○D8, C2×C24⋊C2 [×2], C2×D24, C4○D24 [×8], C2×Dic12, C22×C24, C2×C4○D12 [×2], C2×C4○D24
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, D12 [×4], C22×S3 [×7], C4○D8 [×2], C22×D4, C2×D12 [×6], S3×C23, C2×C4○D8, C4○D24 [×2], C22×D12, C2×C4○D24
Generators and relations
G = < a,b,c,d | a2=b4=d2=1, c12=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c11 >
►On 96 points
(1 77)(2 78)(3 79)(4 80)(5 81)(6 82)(7 83)(8 84)(9 85)(10 86)(11 87)(12 88)(13 89)(14 90)(15 91)(16 92)(17 93)(18 94)(19 95)(20 96)(21 73)(22 74)(23 75)(24 76)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 61)(34 62)(35 63)(36 64)(37 65)(38 66)(39 67)(40 68)(41 69)(42 70)(43 71)(44 72)(45 49)(46 50)(47 51)(48 52)
(1 67 13 55)(2 68 14 56)(3 69 15 57)(4 70 16 58)(5 71 17 59)(6 72 18 60)(7 49 19 61)(8 50 20 62)(9 51 21 63)(10 52 22 64)(11 53 23 65)(12 54 24 66)(25 75 37 87)(26 76 38 88)(27 77 39 89)(28 78 40 90)(29 79 41 91)(30 80 42 92)(31 81 43 93)(32 82 44 94)(33 83 45 95)(34 84 46 96)(35 85 47 73)(36 86 48 74)
(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 76)(2 75)(3 74)(4 73)(5 96)(6 95)(7 94)(8 93)(9 92)(10 91)(11 90)(12 89)(13 88)(14 87)(15 86)(16 85)(17 84)(18 83)(19 82)(20 81)(21 80)(22 79)(23 78)(24 77)(25 56)(26 55)(27 54)(28 53)(29 52)(30 51)(31 50)(32 49)(33 72)(34 71)(35 70)(36 69)(37 68)(38 67)(39 66)(40 65)(41 64)(42 63)(43 62)(44 61)(45 60)(46 59)(47 58)(48 57)
G:=sub<Sym(96)| (1,77)(2,78)(3,79)(4,80)(5,81)(6,82)(7,83)(8,84)(9,85)(10,86)(11,87)(12,88)(13,89)(14,90)(15,91)(16,92)(17,93)(18,94)(19,95)(20,96)(21,73)(22,74)(23,75)(24,76)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,61)(34,62)(35,63)(36,64)(37,65)(38,66)(39,67)(40,68)(41,69)(42,70)(43,71)(44,72)(45,49)(46,50)(47,51)(48,52), (1,67,13,55)(2,68,14,56)(3,69,15,57)(4,70,16,58)(5,71,17,59)(6,72,18,60)(7,49,19,61)(8,50,20,62)(9,51,21,63)(10,52,22,64)(11,53,23,65)(12,54,24,66)(25,75,37,87)(26,76,38,88)(27,77,39,89)(28,78,40,90)(29,79,41,91)(30,80,42,92)(31,81,43,93)(32,82,44,94)(33,83,45,95)(34,84,46,96)(35,85,47,73)(36,86,48,74), (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), (1,76)(2,75)(3,74)(4,73)(5,96)(6,95)(7,94)(8,93)(9,92)(10,91)(11,90)(12,89)(13,88)(14,87)(15,86)(16,85)(17,84)(18,83)(19,82)(20,81)(21,80)(22,79)(23,78)(24,77)(25,56)(26,55)(27,54)(28,53)(29,52)(30,51)(31,50)(32,49)(33,72)(34,71)(35,70)(36,69)(37,68)(38,67)(39,66)(40,65)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)>;
G:=Group( (1,77)(2,78)(3,79)(4,80)(5,81)(6,82)(7,83)(8,84)(9,85)(10,86)(11,87)(12,88)(13,89)(14,90)(15,91)(16,92)(17,93)(18,94)(19,95)(20,96)(21,73)(22,74)(23,75)(24,76)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,61)(34,62)(35,63)(36,64)(37,65)(38,66)(39,67)(40,68)(41,69)(42,70)(43,71)(44,72)(45,49)(46,50)(47,51)(48,52), (1,67,13,55)(2,68,14,56)(3,69,15,57)(4,70,16,58)(5,71,17,59)(6,72,18,60)(7,49,19,61)(8,50,20,62)(9,51,21,63)(10,52,22,64)(11,53,23,65)(12,54,24,66)(25,75,37,87)(26,76,38,88)(27,77,39,89)(28,78,40,90)(29,79,41,91)(30,80,42,92)(31,81,43,93)(32,82,44,94)(33,83,45,95)(34,84,46,96)(35,85,47,73)(36,86,48,74), (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), (1,76)(2,75)(3,74)(4,73)(5,96)(6,95)(7,94)(8,93)(9,92)(10,91)(11,90)(12,89)(13,88)(14,87)(15,86)(16,85)(17,84)(18,83)(19,82)(20,81)(21,80)(22,79)(23,78)(24,77)(25,56)(26,55)(27,54)(28,53)(29,52)(30,51)(31,50)(32,49)(33,72)(34,71)(35,70)(36,69)(37,68)(38,67)(39,66)(40,65)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57) );
G=PermutationGroup([(1,77),(2,78),(3,79),(4,80),(5,81),(6,82),(7,83),(8,84),(9,85),(10,86),(11,87),(12,88),(13,89),(14,90),(15,91),(16,92),(17,93),(18,94),(19,95),(20,96),(21,73),(22,74),(23,75),(24,76),(25,53),(26,54),(27,55),(28,56),(29,57),(30,58),(31,59),(32,60),(33,61),(34,62),(35,63),(36,64),(37,65),(38,66),(39,67),(40,68),(41,69),(42,70),(43,71),(44,72),(45,49),(46,50),(47,51),(48,52)], [(1,67,13,55),(2,68,14,56),(3,69,15,57),(4,70,16,58),(5,71,17,59),(6,72,18,60),(7,49,19,61),(8,50,20,62),(9,51,21,63),(10,52,22,64),(11,53,23,65),(12,54,24,66),(25,75,37,87),(26,76,38,88),(27,77,39,89),(28,78,40,90),(29,79,41,91),(30,80,42,92),(31,81,43,93),(32,82,44,94),(33,83,45,95),(34,84,46,96),(35,85,47,73),(36,86,48,74)], [(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)], [(1,76),(2,75),(3,74),(4,73),(5,96),(6,95),(7,94),(8,93),(9,92),(10,91),(11,90),(12,89),(13,88),(14,87),(15,86),(16,85),(17,84),(18,83),(19,82),(20,81),(21,80),(22,79),(23,78),(24,77),(25,56),(26,55),(27,54),(28,53),(29,52),(30,51),(31,50),(32,49),(33,72),(34,71),(35,70),(36,69),(37,68),(38,67),(39,66),(40,65),(41,64),(42,63),(43,62),(44,61),(45,60),(46,59),(47,58),(48,57)])
Matrix representation ►G ⊆ GL3(𝔽73) generated by
| 72 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 72 | 0 | 0 |
| 0 | 46 | 0 |
| 0 | 0 | 46 |
| 72 | 0 | 0 |
| 0 | 50 | 18 |
| 0 | 55 | 68 |
| 72 | 0 | 0 |
| 0 | 55 | 68 |
| 0 | 50 | 18 |
G:=sub<GL(3,GF(73))| [72,0,0,0,1,0,0,0,1],[72,0,0,0,46,0,0,0,46],[72,0,0,0,50,55,0,18,68],[72,0,0,0,55,50,0,68,18] >;
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | ··· | 6G | 8A | ··· | 8H | 12A | ··· | 12H | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D12 | D12 | C4○D8 | C4○D24 |
| kernel | C2×C4○D24 | C2×C24⋊C2 | C2×D24 | C4○D24 | C2×Dic12 | C22×C24 | C2×C4○D12 | C22×C8 | C2×C12 | C22×C6 | C2×C8 | C22×C4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 2 | 1 | 8 | 1 | 1 | 2 | 1 | 3 | 1 | 6 | 1 | 6 | 2 | 8 | 16 |
In GAP, Magma, Sage, TeX
C_2\times C_4\circ D_{24} % in TeX
G:=Group("C2xC4oD24"); // GroupNames label
G:=SmallGroup(192,1300);
// by ID
G=gap.SmallGroup(192,1300);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,675,80,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=d^2=1,c^12=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^11>;
// generators/relations