direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22×D24, C24⋊9C23, D12⋊4C23, C12.55C24, C23.66D12, (C2×C6)⋊6D8, C6⋊1(C2×D8), (C2×C8)⋊33D6, C3⋊1(C22×D8), C8⋊8(C22×S3), (C22×C8)⋊11S3, C4.45(C2×D12), (C22×C24)⋊11C2, (C2×C24)⋊44C22, (C2×C4).100D12, C12.290(C2×D4), (C2×C12).391D4, C4.52(S3×C23), C6.22(C22×D4), (C2×D12)⋊48C22, (C22×D12)⋊11C2, (C22×C6).145D4, C22.70(C2×D12), C2.24(C22×D12), (C22×C4).459D6, (C2×C12).787C23, (C22×C12).526C22, (C2×C6).178(C2×D4), (C2×C4).736(C22×S3), SmallGroup(192,1299)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1240 in 338 conjugacy classes, 127 normal (13 characteristic)
C1, C2, C2 [×6], C2 [×8], C3, C4, C4 [×3], C22 [×7], C22 [×32], S3 [×8], C6, C6 [×6], C8 [×4], C2×C4 [×6], D4 [×20], C23, C23 [×20], C12, C12 [×3], D6 [×32], C2×C6 [×7], C2×C8 [×6], D8 [×16], C22×C4, C2×D4 [×18], C24 [×2], C24 [×4], D12 [×8], D12 [×12], C2×C12 [×6], C22×S3 [×20], C22×C6, C22×C8, C2×D8 [×12], C22×D4 [×2], D24 [×16], C2×C24 [×6], C2×D12 [×12], C2×D12 [×6], C22×C12, S3×C23 [×2], C22×D8, C2×D24 [×12], C22×C24, C22×D12 [×2], C22×D24
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], D8 [×4], C2×D4 [×6], C24, D12 [×4], C22×S3 [×7], C2×D8 [×6], C22×D4, D24 [×4], C2×D12 [×6], S3×C23, C22×D8, C2×D24 [×6], C22×D12, C22×D24
Generators and relations
G = < a,b,c,d | a2=b2=c24=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
►On 96 points
Generators in S96
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 61)(11 62)(12 63)(13 64)(14 65)(15 66)(16 67)(17 68)(18 69)(19 70)(20 71)(21 72)(22 49)(23 50)(24 51)(25 80)(26 81)(27 82)(28 83)(29 84)(30 85)(31 86)(32 87)(33 88)(34 89)(35 90)(36 91)(37 92)(38 93)(39 94)(40 95)(41 96)(42 73)(43 74)(44 75)(45 76)(46 77)(47 78)(48 79)
(1 79)(2 80)(3 81)(4 82)(5 83)(6 84)(7 85)(8 86)(9 87)(10 88)(11 89)(12 90)(13 91)(14 92)(15 93)(16 94)(17 95)(18 96)(19 73)(20 74)(21 75)(22 76)(23 77)(24 78)(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 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 87)(2 86)(3 85)(4 84)(5 83)(6 82)(7 81)(8 80)(9 79)(10 78)(11 77)(12 76)(13 75)(14 74)(15 73)(16 96)(17 95)(18 94)(19 93)(20 92)(21 91)(22 90)(23 89)(24 88)(25 59)(26 58)(27 57)(28 56)(29 55)(30 54)(31 53)(32 52)(33 51)(34 50)(35 49)(36 72)(37 71)(38 70)(39 69)(40 68)(41 67)(42 66)(43 65)(44 64)(45 63)(46 62)(47 61)(48 60)
G:=sub<Sym(96)| (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,61)(11,62)(12,63)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,70)(20,71)(21,72)(22,49)(23,50)(24,51)(25,80)(26,81)(27,82)(28,83)(29,84)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(41,96)(42,73)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79), (1,79)(2,80)(3,81)(4,82)(5,83)(6,84)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(19,73)(20,74)(21,75)(22,76)(23,77)(24,78)(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), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(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), (1,87)(2,86)(3,85)(4,84)(5,83)(6,82)(7,81)(8,80)(9,79)(10,78)(11,77)(12,76)(13,75)(14,74)(15,73)(16,96)(17,95)(18,94)(19,93)(20,92)(21,91)(22,90)(23,89)(24,88)(25,59)(26,58)(27,57)(28,56)(29,55)(30,54)(31,53)(32,52)(33,51)(34,50)(35,49)(36,72)(37,71)(38,70)(39,69)(40,68)(41,67)(42,66)(43,65)(44,64)(45,63)(46,62)(47,61)(48,60)>;
G:=Group( (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,61)(11,62)(12,63)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,70)(20,71)(21,72)(22,49)(23,50)(24,51)(25,80)(26,81)(27,82)(28,83)(29,84)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(41,96)(42,73)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79), (1,79)(2,80)(3,81)(4,82)(5,83)(6,84)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(19,73)(20,74)(21,75)(22,76)(23,77)(24,78)(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), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(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), (1,87)(2,86)(3,85)(4,84)(5,83)(6,82)(7,81)(8,80)(9,79)(10,78)(11,77)(12,76)(13,75)(14,74)(15,73)(16,96)(17,95)(18,94)(19,93)(20,92)(21,91)(22,90)(23,89)(24,88)(25,59)(26,58)(27,57)(28,56)(29,55)(30,54)(31,53)(32,52)(33,51)(34,50)(35,49)(36,72)(37,71)(38,70)(39,69)(40,68)(41,67)(42,66)(43,65)(44,64)(45,63)(46,62)(47,61)(48,60) );
G=PermutationGroup([(1,52),(2,53),(3,54),(4,55),(5,56),(6,57),(7,58),(8,59),(9,60),(10,61),(11,62),(12,63),(13,64),(14,65),(15,66),(16,67),(17,68),(18,69),(19,70),(20,71),(21,72),(22,49),(23,50),(24,51),(25,80),(26,81),(27,82),(28,83),(29,84),(30,85),(31,86),(32,87),(33,88),(34,89),(35,90),(36,91),(37,92),(38,93),(39,94),(40,95),(41,96),(42,73),(43,74),(44,75),(45,76),(46,77),(47,78),(48,79)], [(1,79),(2,80),(3,81),(4,82),(5,83),(6,84),(7,85),(8,86),(9,87),(10,88),(11,89),(12,90),(13,91),(14,92),(15,93),(16,94),(17,95),(18,96),(19,73),(20,74),(21,75),(22,76),(23,77),(24,78),(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)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24),(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)], [(1,87),(2,86),(3,85),(4,84),(5,83),(6,82),(7,81),(8,80),(9,79),(10,78),(11,77),(12,76),(13,75),(14,74),(15,73),(16,96),(17,95),(18,94),(19,93),(20,92),(21,91),(22,90),(23,89),(24,88),(25,59),(26,58),(27,57),(28,56),(29,55),(30,54),(31,53),(32,52),(33,51),(34,50),(35,49),(36,72),(37,71),(38,70),(39,69),(40,68),(41,67),(42,66),(43,65),(44,64),(45,63),(46,62),(47,61),(48,60)])
Matrix representation ►G ⊆ GL7(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 57 | 16 | 0 | 0 | 0 | 0 |
| 0 | 57 | 57 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 33 | 4 | 0 | 0 |
| 0 | 0 | 0 | 56 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 72 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 53 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 | 1 |
G:=sub<GL(7,GF(73))| [72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,0,57,57,0,0,0,0,0,16,57,0,0,0,0,0,0,0,33,56,0,0,0,0,0,4,40,0,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0],[72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,53,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,1] >;
60 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 3 | 4A | 4B | 4C | 4D | 6A | ··· | 6G | 8A | ··· | 8H | 12A | ··· | 12H | 24A | ··· | 24P |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | ··· | 1 | 12 | ··· | 12 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D8 | D12 | D12 | D24 |
| kernel | C22×D24 | C2×D24 | C22×C24 | C22×D12 | C22×C8 | C2×C12 | C22×C6 | C2×C8 | C22×C4 | C2×C6 | C2×C4 | C23 | C22 |
| # reps | 1 | 12 | 1 | 2 | 1 | 3 | 1 | 6 | 1 | 8 | 6 | 2 | 16 |
In GAP, Magma, Sage, TeX
C_2^2\times D_{24} % in TeX
G:=Group("C2^2xD24"); // GroupNames label
G:=SmallGroup(192,1299);
// by ID
G=gap.SmallGroup(192,1299);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,675,192,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^24=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀