metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.57D6, C23.12Dic6, C6.43(C4×D4), C6.35C22≀C2, C23.30(C4×S3), (C22×C4).42D6, (C22×C6).62D4, C22.97(S3×D4), C6.D4⋊10C4, C2.1(C23⋊2D6), (C22×C6).11Q8, C6.16(C22⋊Q8), C3⋊2(C23.8Q8), C6.C42⋊12C2, C22⋊3(Dic3⋊C4), (C2×Dic3).173D4, C23.58(C3⋊D4), (C23×C6).29C22, (C23×Dic3).4C2, C22.24(C2×Dic6), C23.289(C22×S3), (C22×C6).321C23, (C22×C12).23C22, C2.26(Dic3⋊4D4), C22.45(D4⋊2S3), C2.6(Dic3.D4), C6.72(C22.D4), C2.3(C23.23D6), (C22×Dic3).37C22, (C2×C6)⋊2(C4⋊C4), C6.30(C2×C4⋊C4), (C2×C6).31(C2×Q8), (C2×Dic3⋊C4)⋊8C2, (C2×Dic3)⋊4(C2×C4), (C2×C6).315(C2×D4), (C2×C22⋊C4).8S3, (C6×C22⋊C4).7C2, C2.6(C2×Dic3⋊C4), C22.125(S3×C2×C4), (C22×C6).49(C2×C4), C22.49(C2×C3⋊D4), (C2×C6).141(C4○D4), (C2×C6).107(C22×C4), (C2×C6.D4).8C2, SmallGroup(192,505)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.57D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=d, f2=b, ab=ba, eae-1=faf-1=ac=ca, ad=da, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=be5 >
Subgroups: 552 in 234 conjugacy classes, 83 normal (27 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C23, C23, C23, Dic3, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, C2×Dic3, C2×Dic3, C2×C12, C22×C6, C22×C6, C22×C6, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C23×C4, Dic3⋊C4, C6.D4, C3×C22⋊C4, C22×Dic3, C22×Dic3, C22×C12, C23×C6, C23.8Q8, C6.C42, C2×Dic3⋊C4, C2×C6.D4, C6×C22⋊C4, C23×Dic3, C24.57D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic6, C4×S3, C3⋊D4, C22×S3, C2×C4⋊C4, C4×D4, C22≀C2, C22⋊Q8, C22.D4, Dic3⋊C4, C2×Dic6, S3×C2×C4, S3×D4, D4⋊2S3, C2×C3⋊D4, C23.8Q8, Dic3.D4, Dic3⋊4D4, C2×Dic3⋊C4, C23.23D6, C23⋊2D6, C24.57D6
►On 96 points
Generators in S96
(1 90)(2 46)(3 92)(4 48)(5 94)(6 38)(7 96)(8 40)(9 86)(10 42)(11 88)(12 44)(13 84)(14 65)(15 74)(16 67)(17 76)(18 69)(19 78)(20 71)(21 80)(22 61)(23 82)(24 63)(25 64)(26 73)(27 66)(28 75)(29 68)(30 77)(31 70)(32 79)(33 72)(34 81)(35 62)(36 83)(37 51)(39 53)(41 55)(43 57)(45 59)(47 49)(50 93)(52 95)(54 85)(56 87)(58 89)(60 91)
(1 90)(2 91)(3 92)(4 93)(5 94)(6 95)(7 96)(8 85)(9 86)(10 87)(11 88)(12 89)(13 84)(14 73)(15 74)(16 75)(17 76)(18 77)(19 78)(20 79)(21 80)(22 81)(23 82)(24 83)(25 64)(26 65)(27 66)(28 67)(29 68)(30 69)(31 70)(32 71)(33 72)(34 61)(35 62)(36 63)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(43 57)(44 58)(45 59)(46 60)(47 49)(48 50)
(1 59)(2 60)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 55)(10 56)(11 57)(12 58)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)(37 94)(38 95)(39 96)(40 85)(41 86)(42 87)(43 88)(44 89)(45 90)(46 91)(47 92)(48 93)(61 81)(62 82)(63 83)(64 84)(65 73)(66 74)(67 75)(68 76)(69 77)(70 78)(71 79)(72 80)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 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 79 90 20)(2 13 91 84)(3 77 92 18)(4 23 93 82)(5 75 94 16)(6 21 95 80)(7 73 96 14)(8 19 85 78)(9 83 86 24)(10 17 87 76)(11 81 88 22)(12 15 89 74)(25 46 64 60)(26 53 65 39)(27 44 66 58)(28 51 67 37)(29 42 68 56)(30 49 69 47)(31 40 70 54)(32 59 71 45)(33 38 72 52)(34 57 61 43)(35 48 62 50)(36 55 63 41)
G:=sub<Sym(96)| (1,90)(2,46)(3,92)(4,48)(5,94)(6,38)(7,96)(8,40)(9,86)(10,42)(11,88)(12,44)(13,84)(14,65)(15,74)(16,67)(17,76)(18,69)(19,78)(20,71)(21,80)(22,61)(23,82)(24,63)(25,64)(26,73)(27,66)(28,75)(29,68)(30,77)(31,70)(32,79)(33,72)(34,81)(35,62)(36,83)(37,51)(39,53)(41,55)(43,57)(45,59)(47,49)(50,93)(52,95)(54,85)(56,87)(58,89)(60,91), (1,90)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,85)(9,86)(10,87)(11,88)(12,89)(13,84)(14,73)(15,74)(16,75)(17,76)(18,77)(19,78)(20,79)(21,80)(22,81)(23,82)(24,83)(25,64)(26,65)(27,66)(28,67)(29,68)(30,69)(31,70)(32,71)(33,72)(34,61)(35,62)(36,63)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(43,57)(44,58)(45,59)(46,60)(47,49)(48,50), (1,59)(2,60)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,57)(12,58)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)(37,94)(38,95)(39,96)(40,85)(41,86)(42,87)(43,88)(44,89)(45,90)(46,91)(47,92)(48,93)(61,81)(62,82)(63,83)(64,84)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(71,79)(72,80), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96), (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), (1,79,90,20)(2,13,91,84)(3,77,92,18)(4,23,93,82)(5,75,94,16)(6,21,95,80)(7,73,96,14)(8,19,85,78)(9,83,86,24)(10,17,87,76)(11,81,88,22)(12,15,89,74)(25,46,64,60)(26,53,65,39)(27,44,66,58)(28,51,67,37)(29,42,68,56)(30,49,69,47)(31,40,70,54)(32,59,71,45)(33,38,72,52)(34,57,61,43)(35,48,62,50)(36,55,63,41)>;
G:=Group( (1,90)(2,46)(3,92)(4,48)(5,94)(6,38)(7,96)(8,40)(9,86)(10,42)(11,88)(12,44)(13,84)(14,65)(15,74)(16,67)(17,76)(18,69)(19,78)(20,71)(21,80)(22,61)(23,82)(24,63)(25,64)(26,73)(27,66)(28,75)(29,68)(30,77)(31,70)(32,79)(33,72)(34,81)(35,62)(36,83)(37,51)(39,53)(41,55)(43,57)(45,59)(47,49)(50,93)(52,95)(54,85)(56,87)(58,89)(60,91), (1,90)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,85)(9,86)(10,87)(11,88)(12,89)(13,84)(14,73)(15,74)(16,75)(17,76)(18,77)(19,78)(20,79)(21,80)(22,81)(23,82)(24,83)(25,64)(26,65)(27,66)(28,67)(29,68)(30,69)(31,70)(32,71)(33,72)(34,61)(35,62)(36,63)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(43,57)(44,58)(45,59)(46,60)(47,49)(48,50), (1,59)(2,60)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,57)(12,58)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)(37,94)(38,95)(39,96)(40,85)(41,86)(42,87)(43,88)(44,89)(45,90)(46,91)(47,92)(48,93)(61,81)(62,82)(63,83)(64,84)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(71,79)(72,80), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96), (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), (1,79,90,20)(2,13,91,84)(3,77,92,18)(4,23,93,82)(5,75,94,16)(6,21,95,80)(7,73,96,14)(8,19,85,78)(9,83,86,24)(10,17,87,76)(11,81,88,22)(12,15,89,74)(25,46,64,60)(26,53,65,39)(27,44,66,58)(28,51,67,37)(29,42,68,56)(30,49,69,47)(31,40,70,54)(32,59,71,45)(33,38,72,52)(34,57,61,43)(35,48,62,50)(36,55,63,41) );
G=PermutationGroup([[(1,90),(2,46),(3,92),(4,48),(5,94),(6,38),(7,96),(8,40),(9,86),(10,42),(11,88),(12,44),(13,84),(14,65),(15,74),(16,67),(17,76),(18,69),(19,78),(20,71),(21,80),(22,61),(23,82),(24,63),(25,64),(26,73),(27,66),(28,75),(29,68),(30,77),(31,70),(32,79),(33,72),(34,81),(35,62),(36,83),(37,51),(39,53),(41,55),(43,57),(45,59),(47,49),(50,93),(52,95),(54,85),(56,87),(58,89),(60,91)], [(1,90),(2,91),(3,92),(4,93),(5,94),(6,95),(7,96),(8,85),(9,86),(10,87),(11,88),(12,89),(13,84),(14,73),(15,74),(16,75),(17,76),(18,77),(19,78),(20,79),(21,80),(22,81),(23,82),(24,83),(25,64),(26,65),(27,66),(28,67),(29,68),(30,69),(31,70),(32,71),(33,72),(34,61),(35,62),(36,63),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56),(43,57),(44,58),(45,59),(46,60),(47,49),(48,50)], [(1,59),(2,60),(3,49),(4,50),(5,51),(6,52),(7,53),(8,54),(9,55),(10,56),(11,57),(12,58),(13,25),(14,26),(15,27),(16,28),(17,29),(18,30),(19,31),(20,32),(21,33),(22,34),(23,35),(24,36),(37,94),(38,95),(39,96),(40,85),(41,86),(42,87),(43,88),(44,89),(45,90),(46,91),(47,92),(48,93),(61,81),(62,82),(63,83),(64,84),(65,73),(66,74),(67,75),(68,76),(69,77),(70,78),(71,79),(72,80)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48),(49,55),(50,56),(51,57),(52,58),(53,59),(54,60),(61,67),(62,68),(63,69),(64,70),(65,71),(66,72),(73,79),(74,80),(75,81),(76,82),(77,83),(78,84),(85,91),(86,92),(87,93),(88,94),(89,95),(90,96)], [(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)], [(1,79,90,20),(2,13,91,84),(3,77,92,18),(4,23,93,82),(5,75,94,16),(6,21,95,80),(7,73,96,14),(8,19,85,78),(9,83,86,24),(10,17,87,76),(11,81,88,22),(12,15,89,74),(25,46,64,60),(26,53,65,39),(27,44,66,58),(28,51,67,37),(29,42,68,56),(30,49,69,47),(31,40,70,54),(32,59,71,45),(33,38,72,52),(34,57,61,43),(35,48,62,50),(36,55,63,41)]])
48 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | 4N | 4O | 4P | 6A | ··· | 6G | 6H | 6I | 6J | 6K | 12A | ··· | 12H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | Q8 | D6 | D6 | C4○D4 | Dic6 | C4×S3 | C3⋊D4 | S3×D4 | D4⋊2S3 |
| kernel | C24.57D6 | C6.C42 | C2×Dic3⋊C4 | C2×C6.D4 | C6×C22⋊C4 | C23×Dic3 | C6.D4 | C2×C22⋊C4 | C2×Dic3 | C22×C6 | C22×C6 | C22×C4 | C24 | C2×C6 | C23 | C23 | C23 | C22 | C22 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 1 | 4 | 2 | 2 | 2 | 1 | 4 | 4 | 4 | 4 | 2 | 2 |
Matrix representation of C24.57D6 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 5 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 11 | 0 | 0 |
| 0 | 0 | 12 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 7 |
| 0 | 0 | 0 | 0 | 10 | 3 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,12,1,0,0,0,0,0,0,5,8,0,0,0,0,5,0],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,7,12,0,0,0,0,11,6,0,0,0,0,0,0,10,10,0,0,0,0,7,3] >;
C24.57D6 in GAP, Magma, Sage, TeX
C_2^4._{57}D_6 % in TeX
G:=Group("C2^4.57D6"); // GroupNames label
G:=SmallGroup(192,505);
// by ID
G=gap.SmallGroup(192,505);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,254,219,184,6278]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=d,f^2=b,a*b=b*a,e*a*e^-1=f*a*f^-1=a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=b*e^5>;
// generators/relations