metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.27D12, (C2×C24)⋊10C4, (C2×C8)⋊6Dic3, C24.73(C2×C4), C8⋊Dic3⋊29C2, C24⋊1C4⋊29C2, (C2×C8).307D6, C12.47(C4⋊C4), (C2×C12).61Q8, C12.75(C2×Q8), C6.15(C4○D8), C2.4(C4○D24), (C2×C12).401D4, (C2×C4).169D12, (C22×C8).13S3, (C2×C4).50Dic6, C4.41(C2×Dic6), C8.17(C2×Dic3), (C22×C24).17C2, C4.23(C4⋊Dic3), C22.52(C2×D12), (C22×C4).441D6, (C22×C6).137D4, (C2×C12).765C23, (C2×C24).392C22, C12.171(C22×C4), C3⋊4(C23.25D4), C4.25(C22×Dic3), C4⋊Dic3.281C22, C22.13(C4⋊Dic3), C23.26D6.5C2, (C22×C12).539C22, C6.46(C2×C4⋊C4), (C2×C6).51(C4⋊C4), (C2×C6).155(C2×D4), C2.12(C2×C4⋊Dic3), (C2×C12).306(C2×C4), (C2×C4).83(C2×Dic3), (C2×C4).712(C22×S3), SmallGroup(192,665)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.27D12
G = < a,b,c,d,e | a2=b2=c2=1, d12=c, e2=cb=bc, ab=ba, eae-1=ac=ca, ad=da, bd=db, be=eb, cd=dc, ce=ec, ede-1=d11 >
Subgroups: 248 in 114 conjugacy classes, 71 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C24, C2×Dic3, C2×C12, C2×C12, C22×C6, C4.Q8, C2.D8, C42⋊C2, C22×C8, C4×Dic3, C4⋊Dic3, C6.D4, C2×C24, C2×C24, C22×C12, C23.25D4, C8⋊Dic3, C24⋊1C4, C23.26D6, C22×C24, C23.27D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, Dic3, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic6, D12, C2×Dic3, C22×S3, C2×C4⋊C4, C4○D8, C4⋊Dic3, C2×Dic6, C2×D12, C22×Dic3, C23.25D4, C4○D24, C2×C4⋊Dic3, C23.27D12
►On 96 points
Generators in S96
(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(73 85)(74 86)(75 87)(76 88)(77 89)(78 90)(79 91)(80 92)(81 93)(82 94)(83 95)(84 96)
(1 64)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 71)(9 72)(10 49)(11 50)(12 51)(13 52)(14 53)(15 54)(16 55)(17 56)(18 57)(19 58)(20 59)(21 60)(22 61)(23 62)(24 63)(25 78)(26 79)(27 80)(28 81)(29 82)(30 83)(31 84)(32 85)(33 86)(34 87)(35 88)(36 89)(37 90)(38 91)(39 92)(40 93)(41 94)(42 95)(43 96)(44 73)(45 74)(46 75)(47 76)(48 77)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(49 61)(50 62)(51 63)(52 64)(53 65)(54 66)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)(73 85)(74 86)(75 87)(76 88)(77 89)(78 90)(79 91)(80 92)(81 93)(82 94)(83 95)(84 96)
(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 32 52 73)(2 43 53 84)(3 30 54 95)(4 41 55 82)(5 28 56 93)(6 39 57 80)(7 26 58 91)(8 37 59 78)(9 48 60 89)(10 35 61 76)(11 46 62 87)(12 33 63 74)(13 44 64 85)(14 31 65 96)(15 42 66 83)(16 29 67 94)(17 40 68 81)(18 27 69 92)(19 38 70 79)(20 25 71 90)(21 36 72 77)(22 47 49 88)(23 34 50 75)(24 45 51 86)
G:=sub<Sym(96)| (25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96), (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,49)(11,50)(12,51)(13,52)(14,53)(15,54)(16,55)(17,56)(18,57)(19,58)(20,59)(21,60)(22,61)(23,62)(24,63)(25,78)(26,79)(27,80)(28,81)(29,82)(30,83)(31,84)(32,85)(33,86)(34,87)(35,88)(36,89)(37,90)(38,91)(39,92)(40,93)(41,94)(42,95)(43,96)(44,73)(45,74)(46,75)(47,76)(48,77), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96), (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), (1,32,52,73)(2,43,53,84)(3,30,54,95)(4,41,55,82)(5,28,56,93)(6,39,57,80)(7,26,58,91)(8,37,59,78)(9,48,60,89)(10,35,61,76)(11,46,62,87)(12,33,63,74)(13,44,64,85)(14,31,65,96)(15,42,66,83)(16,29,67,94)(17,40,68,81)(18,27,69,92)(19,38,70,79)(20,25,71,90)(21,36,72,77)(22,47,49,88)(23,34,50,75)(24,45,51,86)>;
G:=Group( (25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96), (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,49)(11,50)(12,51)(13,52)(14,53)(15,54)(16,55)(17,56)(18,57)(19,58)(20,59)(21,60)(22,61)(23,62)(24,63)(25,78)(26,79)(27,80)(28,81)(29,82)(30,83)(31,84)(32,85)(33,86)(34,87)(35,88)(36,89)(37,90)(38,91)(39,92)(40,93)(41,94)(42,95)(43,96)(44,73)(45,74)(46,75)(47,76)(48,77), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96), (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), (1,32,52,73)(2,43,53,84)(3,30,54,95)(4,41,55,82)(5,28,56,93)(6,39,57,80)(7,26,58,91)(8,37,59,78)(9,48,60,89)(10,35,61,76)(11,46,62,87)(12,33,63,74)(13,44,64,85)(14,31,65,96)(15,42,66,83)(16,29,67,94)(17,40,68,81)(18,27,69,92)(19,38,70,79)(20,25,71,90)(21,36,72,77)(22,47,49,88)(23,34,50,75)(24,45,51,86) );
G=PermutationGroup([[(25,37),(26,38),(27,39),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48),(73,85),(74,86),(75,87),(76,88),(77,89),(78,90),(79,91),(80,92),(81,93),(82,94),(83,95),(84,96)], [(1,64),(2,65),(3,66),(4,67),(5,68),(6,69),(7,70),(8,71),(9,72),(10,49),(11,50),(12,51),(13,52),(14,53),(15,54),(16,55),(17,56),(18,57),(19,58),(20,59),(21,60),(22,61),(23,62),(24,63),(25,78),(26,79),(27,80),(28,81),(29,82),(30,83),(31,84),(32,85),(33,86),(34,87),(35,88),(36,89),(37,90),(38,91),(39,92),(40,93),(41,94),(42,95),(43,96),(44,73),(45,74),(46,75),(47,76),(48,77)], [(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(25,37),(26,38),(27,39),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48),(49,61),(50,62),(51,63),(52,64),(53,65),(54,66),(55,67),(56,68),(57,69),(58,70),(59,71),(60,72),(73,85),(74,86),(75,87),(76,88),(77,89),(78,90),(79,91),(80,92),(81,93),(82,94),(83,95),(84,96)], [(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)], [(1,32,52,73),(2,43,53,84),(3,30,54,95),(4,41,55,82),(5,28,56,93),(6,39,57,80),(7,26,58,91),(8,37,59,78),(9,48,60,89),(10,35,61,76),(11,46,62,87),(12,33,63,74),(13,44,64,85),(14,31,65,96),(15,42,66,83),(16,29,67,94),(17,40,68,81),(18,27,69,92),(19,38,70,79),(20,25,71,90),(21,36,72,77),(22,47,49,88),(23,34,50,75),(24,45,51,86)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 6A | ··· | 6G | 8A | ··· | 8H | 12A | ··· | 12H | 24A | ··· | 24P |
| order | 1 | 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 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 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 |
| type | + | + | + | + | + | + | + | - | + | - | + | + | - | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | Q8 | D4 | Dic3 | D6 | D6 | Dic6 | D12 | D12 | C4○D8 | C4○D24 |
| kernel | C23.27D12 | C8⋊Dic3 | C24⋊1C4 | C23.26D6 | C22×C24 | C2×C24 | C22×C8 | C2×C12 | C2×C12 | C22×C6 | C2×C8 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 8 | 1 | 1 | 2 | 1 | 4 | 2 | 1 | 4 | 2 | 2 | 8 | 16 |
Matrix representation of C23.27D12 ►in GL3(𝔽73) generated by
| 72 | 0 | 0 |
| 0 | 1 | 2 |
| 0 | 0 | 72 |
| 72 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 72 | 0 |
| 0 | 0 | 72 |
| 1 | 0 | 0 |
| 0 | 56 | 13 |
| 0 | 0 | 43 |
| 46 | 0 | 0 |
| 0 | 39 | 54 |
| 0 | 34 | 34 |
G:=sub<GL(3,GF(73))| [72,0,0,0,1,0,0,2,72],[72,0,0,0,1,0,0,0,1],[1,0,0,0,72,0,0,0,72],[1,0,0,0,56,0,0,13,43],[46,0,0,0,39,34,0,54,34] >;
C23.27D12 in GAP, Magma, Sage, TeX
C_2^3._{27}D_{12} % in TeX
G:=Group("C2^3.27D12"); // GroupNames label
G:=SmallGroup(192,665);
// by ID
G=gap.SmallGroup(192,665);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,232,422,100,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^12=c,e^2=c*b=b*c,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^11>;
// generators/relations