direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C2.D24, C22.16D24, C23.63D12, (C2×C8)⋊32D6, (C2×D12)⋊8C4, (C22×C8)⋊8S3, C2.3(C2×D24), C6.16(C2×D8), (C2×C6).23D8, D12⋊17(C2×C4), (C22×C24)⋊3C2, (C2×C4).96D12, C6⋊2(D4⋊C4), (C2×C24)⋊41C22, C4.27(D6⋊C4), C12.410(C2×D4), (C2×C12).474D4, (C2×C6).22SD16, C6.16(C2×SD16), C4⋊Dic3⋊47C22, (C22×D12).6C2, C22.53(C2×D12), (C22×C4).444D6, (C22×C6).138D4, C12.52(C22⋊C4), C12.112(C22×C4), (C2×C12).766C23, C22.49(D6⋊C4), C22.12(C24⋊C2), (C2×D12).198C22, (C22×C12).517C22, C4.70(S3×C2×C4), C3⋊3(C2×D4⋊C4), C2.4(C2×C24⋊C2), C2.24(C2×D6⋊C4), (C2×C4).116(C4×S3), (C2×C4⋊Dic3)⋊15C2, (C2×C6).156(C2×D4), C4.103(C2×C3⋊D4), C6.52(C2×C22⋊C4), (C2×C12).229(C2×C4), (C2×C4).254(C3⋊D4), (C2×C6).63(C22⋊C4), (C2×C4).714(C22×S3), SmallGroup(192,671)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C2.D24
G = < a,b,c,d | a2=b2=c24=1, d2=b, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=bc-1 >
Subgroups: 728 in 202 conjugacy classes, 79 normal (27 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×D4, C24, C24, D12, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×C6, D4⋊C4, C2×C4⋊C4, C22×C8, C22×D4, C4⋊Dic3, C4⋊Dic3, C2×C24, C2×C24, C2×D12, C2×D12, C22×Dic3, C22×C12, S3×C23, C2×D4⋊C4, C2.D24, C2×C4⋊Dic3, C22×C24, C22×D12, C2×C2.D24
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, D8, SD16, C22×C4, C2×D4, C4×S3, D12, C3⋊D4, C22×S3, D4⋊C4, C2×C22⋊C4, C2×D8, C2×SD16, C24⋊C2, D24, D6⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C2×D4⋊C4, C2.D24, C2×C24⋊C2, C2×D24, C2×D6⋊C4, C2×C2.D24
►On 96 points
(1 53)(2 54)(3 55)(4 56)(5 57)(6 58)(7 59)(8 60)(9 61)(10 62)(11 63)(12 64)(13 65)(14 66)(15 67)(16 68)(17 69)(18 70)(19 71)(20 72)(21 49)(22 50)(23 51)(24 52)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(49 77)(50 78)(51 79)(52 80)(53 81)(54 82)(55 83)(56 84)(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 73)(70 74)(71 75)(72 76)
(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 64 33 92)(2 91 34 63)(3 62 35 90)(4 89 36 61)(5 60 37 88)(6 87 38 59)(7 58 39 86)(8 85 40 57)(9 56 41 84)(10 83 42 55)(11 54 43 82)(12 81 44 53)(13 52 45 80)(14 79 46 51)(15 50 47 78)(16 77 48 49)(17 72 25 76)(18 75 26 71)(19 70 27 74)(20 73 28 69)(21 68 29 96)(22 95 30 67)(23 66 31 94)(24 93 32 65)
G:=sub<Sym(96)| (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,61)(10,62)(11,63)(12,64)(13,65)(14,66)(15,67)(16,68)(17,69)(18,70)(19,71)(20,72)(21,49)(22,50)(23,51)(24,52)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96), (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(49,77)(50,78)(51,79)(52,80)(53,81)(54,82)(55,83)(56,84)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,73)(70,74)(71,75)(72,76), (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), (1,64,33,92)(2,91,34,63)(3,62,35,90)(4,89,36,61)(5,60,37,88)(6,87,38,59)(7,58,39,86)(8,85,40,57)(9,56,41,84)(10,83,42,55)(11,54,43,82)(12,81,44,53)(13,52,45,80)(14,79,46,51)(15,50,47,78)(16,77,48,49)(17,72,25,76)(18,75,26,71)(19,70,27,74)(20,73,28,69)(21,68,29,96)(22,95,30,67)(23,66,31,94)(24,93,32,65)>;
G:=Group( (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,61)(10,62)(11,63)(12,64)(13,65)(14,66)(15,67)(16,68)(17,69)(18,70)(19,71)(20,72)(21,49)(22,50)(23,51)(24,52)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96), (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(49,77)(50,78)(51,79)(52,80)(53,81)(54,82)(55,83)(56,84)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,73)(70,74)(71,75)(72,76), (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), (1,64,33,92)(2,91,34,63)(3,62,35,90)(4,89,36,61)(5,60,37,88)(6,87,38,59)(7,58,39,86)(8,85,40,57)(9,56,41,84)(10,83,42,55)(11,54,43,82)(12,81,44,53)(13,52,45,80)(14,79,46,51)(15,50,47,78)(16,77,48,49)(17,72,25,76)(18,75,26,71)(19,70,27,74)(20,73,28,69)(21,68,29,96)(22,95,30,67)(23,66,31,94)(24,93,32,65) );
G=PermutationGroup([[(1,53),(2,54),(3,55),(4,56),(5,57),(6,58),(7,59),(8,60),(9,61),(10,62),(11,63),(12,64),(13,65),(14,66),(15,67),(16,68),(17,69),(18,70),(19,71),(20,72),(21,49),(22,50),(23,51),(24,52),(25,73),(26,74),(27,75),(28,76),(29,77),(30,78),(31,79),(32,80),(33,81),(34,82),(35,83),(36,84),(37,85),(38,86),(39,87),(40,88),(41,89),(42,90),(43,91),(44,92),(45,93),(46,94),(47,95),(48,96)], [(1,33),(2,34),(3,35),(4,36),(5,37),(6,38),(7,39),(8,40),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32),(49,77),(50,78),(51,79),(52,80),(53,81),(54,82),(55,83),(56,84),(57,85),(58,86),(59,87),(60,88),(61,89),(62,90),(63,91),(64,92),(65,93),(66,94),(67,95),(68,96),(69,73),(70,74),(71,75),(72,76)], [(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)], [(1,64,33,92),(2,91,34,63),(3,62,35,90),(4,89,36,61),(5,60,37,88),(6,87,38,59),(7,58,39,86),(8,85,40,57),(9,56,41,84),(10,83,42,55),(11,54,43,82),(12,81,44,53),(13,52,45,80),(14,79,46,51),(15,50,47,78),(16,77,48,49),(17,72,25,76),(18,75,26,71),(19,70,27,74),(20,73,28,69),(21,68,29,96),(22,95,30,67),(23,66,31,94),(24,93,32,65)]])
60 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6G | 8A | ··· | 8H | 12A | ··· | 12H | 24A | ··· | 24P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | ··· | 1 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D6 | D6 | D8 | SD16 | C4×S3 | D12 | C3⋊D4 | D12 | C24⋊C2 | D24 |
| kernel | C2×C2.D24 | C2.D24 | C2×C4⋊Dic3 | C22×C24 | C22×D12 | C2×D12 | C22×C8 | C2×C12 | C22×C6 | C2×C8 | C22×C4 | C2×C6 | C2×C6 | C2×C4 | C2×C4 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 1 | 3 | 1 | 2 | 1 | 4 | 4 | 4 | 2 | 4 | 2 | 8 | 8 |
Matrix representation of C2×C2.D24 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 27 | 27 | 0 | 0 | 0 | 0 |
| 46 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 25 |
| 0 | 0 | 0 | 0 | 48 | 11 |
| 46 | 46 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 25 |
| 0 | 0 | 0 | 0 | 62 | 37 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[27,46,0,0,0,0,27,0,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,36,48,0,0,0,0,25,11],[46,0,0,0,0,0,46,27,0,0,0,0,0,0,1,0,0,0,0,0,1,72,0,0,0,0,0,0,36,62,0,0,0,0,25,37] >;
C2×C2.D24 in GAP, Magma, Sage, TeX
C_2\times C_2.D_{24} % in TeX
G:=Group("C2xC2.D24"); // GroupNames label
G:=SmallGroup(192,671);
// by ID
G=gap.SmallGroup(192,671);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,142,1123,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^24=1,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=b*c^-1>;
// generators/relations