metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22.4D24, C23.44D12, C6.6(C2×D8), (C2×C6).5D8, (C2×C8).4D6, C22⋊C8⋊4S3, C24⋊1C4⋊4C2, C2.8(C2×D24), C2.D24⋊6C2, (C2×C4).34D12, (C2×C12).45D4, C12⋊7D4.3C2, (C2×C24).4C22, (C22×C6).56D4, (C22×C4).102D6, C3⋊1(C22.D8), C12.283(C4○D4), (C2×C12).746C23, C2.13(C8.D6), (C2×D12).11C22, C22.109(C2×D12), C6.10(C8.C22), C4.107(D4⋊2S3), C4⋊Dic3.271C22, (C22×C12).53C22, C6.18(C22.D4), C2.14(C23.21D6), (C3×C22⋊C8)⋊6C2, (C2×C4⋊Dic3)⋊6C2, (C2×C6).129(C2×D4), (C2×C4).691(C22×S3), SmallGroup(192,295)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22.D24
G = < a,b,c,d | a2=b2=c24=1, d2=b, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=bc-1 >
Subgroups: 384 in 114 conjugacy classes, 43 normal (25 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C24, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C22⋊C8, D4⋊C4, C2.D8, C2×C4⋊C4, C4⋊D4, C4⋊Dic3, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C2×C24, C2×D12, C22×Dic3, C2×C3⋊D4, C22×C12, C22.D8, C24⋊1C4, C2.D24, C3×C22⋊C8, C2×C4⋊Dic3, C12⋊7D4, C22.D24
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C4○D4, D12, C22×S3, C22.D4, C2×D8, C8.C22, D24, C2×D12, D4⋊2S3, C22.D8, C23.21D6, C2×D24, C8.D6, C22.D24
►On 96 points
Generators in S96
(1 96)(2 26)(3 74)(4 28)(5 76)(6 30)(7 78)(8 32)(9 80)(10 34)(11 82)(12 36)(13 84)(14 38)(15 86)(16 40)(17 88)(18 42)(19 90)(20 44)(21 92)(22 46)(23 94)(24 48)(25 71)(27 49)(29 51)(31 53)(33 55)(35 57)(37 59)(39 61)(41 63)(43 65)(45 67)(47 69)(50 75)(52 77)(54 79)(56 81)(58 83)(60 85)(62 87)(64 89)(66 91)(68 93)(70 95)(72 73)
(1 71)(2 72)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 55)(10 56)(11 57)(12 58)(13 59)(14 60)(15 61)(16 62)(17 63)(18 64)(19 65)(20 66)(21 67)(22 68)(23 69)(24 70)(25 96)(26 73)(27 74)(28 75)(29 76)(30 77)(31 78)(32 79)(33 80)(34 81)(35 82)(36 83)(37 84)(38 85)(39 86)(40 87)(41 88)(42 89)(43 90)(44 91)(45 92)(46 93)(47 94)(48 95)
(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 70 71 24)(2 23 72 69)(3 68 49 22)(4 21 50 67)(5 66 51 20)(6 19 52 65)(7 64 53 18)(8 17 54 63)(9 62 55 16)(10 15 56 61)(11 60 57 14)(12 13 58 59)(25 48 96 95)(26 94 73 47)(27 46 74 93)(28 92 75 45)(29 44 76 91)(30 90 77 43)(31 42 78 89)(32 88 79 41)(33 40 80 87)(34 86 81 39)(35 38 82 85)(36 84 83 37)
G:=sub<Sym(96)| (1,96)(2,26)(3,74)(4,28)(5,76)(6,30)(7,78)(8,32)(9,80)(10,34)(11,82)(12,36)(13,84)(14,38)(15,86)(16,40)(17,88)(18,42)(19,90)(20,44)(21,92)(22,46)(23,94)(24,48)(25,71)(27,49)(29,51)(31,53)(33,55)(35,57)(37,59)(39,61)(41,63)(43,65)(45,67)(47,69)(50,75)(52,77)(54,79)(56,81)(58,83)(60,85)(62,87)(64,89)(66,91)(68,93)(70,95)(72,73), (1,71)(2,72)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,63)(18,64)(19,65)(20,66)(21,67)(22,68)(23,69)(24,70)(25,96)(26,73)(27,74)(28,75)(29,76)(30,77)(31,78)(32,79)(33,80)(34,81)(35,82)(36,83)(37,84)(38,85)(39,86)(40,87)(41,88)(42,89)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95), (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), (1,70,71,24)(2,23,72,69)(3,68,49,22)(4,21,50,67)(5,66,51,20)(6,19,52,65)(7,64,53,18)(8,17,54,63)(9,62,55,16)(10,15,56,61)(11,60,57,14)(12,13,58,59)(25,48,96,95)(26,94,73,47)(27,46,74,93)(28,92,75,45)(29,44,76,91)(30,90,77,43)(31,42,78,89)(32,88,79,41)(33,40,80,87)(34,86,81,39)(35,38,82,85)(36,84,83,37)>;
G:=Group( (1,96)(2,26)(3,74)(4,28)(5,76)(6,30)(7,78)(8,32)(9,80)(10,34)(11,82)(12,36)(13,84)(14,38)(15,86)(16,40)(17,88)(18,42)(19,90)(20,44)(21,92)(22,46)(23,94)(24,48)(25,71)(27,49)(29,51)(31,53)(33,55)(35,57)(37,59)(39,61)(41,63)(43,65)(45,67)(47,69)(50,75)(52,77)(54,79)(56,81)(58,83)(60,85)(62,87)(64,89)(66,91)(68,93)(70,95)(72,73), (1,71)(2,72)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,63)(18,64)(19,65)(20,66)(21,67)(22,68)(23,69)(24,70)(25,96)(26,73)(27,74)(28,75)(29,76)(30,77)(31,78)(32,79)(33,80)(34,81)(35,82)(36,83)(37,84)(38,85)(39,86)(40,87)(41,88)(42,89)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95), (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), (1,70,71,24)(2,23,72,69)(3,68,49,22)(4,21,50,67)(5,66,51,20)(6,19,52,65)(7,64,53,18)(8,17,54,63)(9,62,55,16)(10,15,56,61)(11,60,57,14)(12,13,58,59)(25,48,96,95)(26,94,73,47)(27,46,74,93)(28,92,75,45)(29,44,76,91)(30,90,77,43)(31,42,78,89)(32,88,79,41)(33,40,80,87)(34,86,81,39)(35,38,82,85)(36,84,83,37) );
G=PermutationGroup([[(1,96),(2,26),(3,74),(4,28),(5,76),(6,30),(7,78),(8,32),(9,80),(10,34),(11,82),(12,36),(13,84),(14,38),(15,86),(16,40),(17,88),(18,42),(19,90),(20,44),(21,92),(22,46),(23,94),(24,48),(25,71),(27,49),(29,51),(31,53),(33,55),(35,57),(37,59),(39,61),(41,63),(43,65),(45,67),(47,69),(50,75),(52,77),(54,79),(56,81),(58,83),(60,85),(62,87),(64,89),(66,91),(68,93),(70,95),(72,73)], [(1,71),(2,72),(3,49),(4,50),(5,51),(6,52),(7,53),(8,54),(9,55),(10,56),(11,57),(12,58),(13,59),(14,60),(15,61),(16,62),(17,63),(18,64),(19,65),(20,66),(21,67),(22,68),(23,69),(24,70),(25,96),(26,73),(27,74),(28,75),(29,76),(30,77),(31,78),(32,79),(33,80),(34,81),(35,82),(36,83),(37,84),(38,85),(39,86),(40,87),(41,88),(42,89),(43,90),(44,91),(45,92),(46,93),(47,94),(48,95)], [(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)], [(1,70,71,24),(2,23,72,69),(3,68,49,22),(4,21,50,67),(5,66,51,20),(6,19,52,65),(7,64,53,18),(8,17,54,63),(9,62,55,16),(10,15,56,61),(11,60,57,14),(12,13,58,59),(25,48,96,95),(26,94,73,47),(27,46,74,93),(28,92,75,45),(29,44,76,91),(30,90,77,43),(31,42,78,89),(32,88,79,41),(33,40,80,87),(34,86,81,39),(35,38,82,85),(36,84,83,37)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 24 | 2 | 2 | 2 | 4 | 12 | 12 | 12 | 12 | 24 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | C4○D4 | D8 | D12 | D12 | D24 | C8.C22 | D4⋊2S3 | C8.D6 |
| kernel | C22.D24 | C24⋊1C4 | C2.D24 | C3×C22⋊C8 | C2×C4⋊Dic3 | C12⋊7D4 | C22⋊C8 | C2×C12 | C22×C6 | C2×C8 | C22×C4 | C12 | C2×C6 | C2×C4 | C23 | C22 | C6 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 4 | 4 | 2 | 2 | 8 | 1 | 2 | 2 |
Matrix representation of C22.D24 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 17 | 8 |
| 0 | 0 | 37 | 56 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 18 | 50 | 0 | 0 |
| 23 | 68 | 0 | 0 |
| 0 | 0 | 52 | 3 |
| 0 | 0 | 48 | 21 |
| 23 | 68 | 0 | 0 |
| 18 | 50 | 0 | 0 |
| 0 | 0 | 52 | 3 |
| 0 | 0 | 23 | 21 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,17,37,0,0,8,56],[1,0,0,0,0,1,0,0,0,0,72,0,0,0,0,72],[18,23,0,0,50,68,0,0,0,0,52,48,0,0,3,21],[23,18,0,0,68,50,0,0,0,0,52,23,0,0,3,21] >;
C22.D24 in GAP, Magma, Sage, TeX
C_2^2.D_{24} % in TeX
G:=Group("C2^2.D24"); // GroupNames label
G:=SmallGroup(192,295);
// by ID
G=gap.SmallGroup(192,295);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,254,219,310,1123,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^24=1,d^2=b,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=b*c^-1>;
// generators/relations