metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.58D6, C23.48D12, C23.13Dic6, C6.86(C4×D4), C2.4(D4×Dic3), C6.31C22≀C2, C22⋊C4⋊4Dic3, C22.98(S3×D4), (C22×C6).63D4, C2.3(D6⋊D4), (C22×C6).14Q8, C22⋊2(C4⋊Dic3), C22.42(C2×D12), (C22×C4).106D6, C6.17(C22⋊Q8), C3⋊3(C23.8Q8), C6.C42⋊30C2, (C2×Dic3).174D4, (C23×C6).33C22, (C23×Dic3).5C2, C22.25(C2×Dic6), C23.18(C2×Dic3), C23.290(C22×S3), (C22×C6).325C23, (C22×C12).59C22, C22.46(D4⋊2S3), C2.7(Dic3.D4), C6.30(C22.D4), C2.4(C23.21D6), C22.39(C22×Dic3), (C22×Dic3).184C22, (C2×C6)⋊3(C4⋊C4), (C2×C12)⋊6(C2×C4), C6.31(C2×C4⋊C4), (C3×C22⋊C4)⋊7C4, (C2×C4)⋊2(C2×Dic3), C2.7(C2×C4⋊Dic3), (C2×C6).35(C2×Q8), (C2×C4⋊Dic3)⋊10C2, (C2×C6).319(C2×D4), (C2×C22⋊C4).13S3, (C6×C22⋊C4).15C2, (C22×C6).50(C2×C4), (C2×C6).142(C4○D4), (C2×C6).178(C22×C4), (C2×C6.D4).12C2, SmallGroup(192,509)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.58D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=c, f2=b, ab=ba, ac=ca, eae-1=faf-1=ad=da, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce5 >
Subgroups: 552 in 234 conjugacy classes, 91 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, C23, C23, C23, Dic3, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C22×C6, C22×C6, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C23×C4, C4⋊Dic3, C6.D4, C3×C22⋊C4, C22×Dic3, C22×Dic3, C22×Dic3, C22×C12, C23×C6, C23.8Q8, C6.C42, C2×C4⋊Dic3, C2×C6.D4, C6×C22⋊C4, C23×Dic3, C24.58D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, Dic3, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic6, D12, C2×Dic3, C22×S3, C2×C4⋊C4, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C4⋊Dic3, C2×Dic6, C2×D12, S3×D4, D4⋊2S3, C22×Dic3, C23.8Q8, Dic3.D4, D6⋊D4, C23.21D6, C2×C4⋊Dic3, D4×Dic3, C24.58D6
►On 96 points
Generators in S96
(1 84)(2 94)(3 74)(4 96)(5 76)(6 86)(7 78)(8 88)(9 80)(10 90)(11 82)(12 92)(13 52)(14 71)(15 54)(16 61)(17 56)(18 63)(19 58)(20 65)(21 60)(22 67)(23 50)(24 69)(25 85)(26 77)(27 87)(28 79)(29 89)(30 81)(31 91)(32 83)(33 93)(34 73)(35 95)(36 75)(37 59)(38 66)(39 49)(40 68)(41 51)(42 70)(43 53)(44 72)(45 55)(46 62)(47 57)(48 64)
(1 84)(2 73)(3 74)(4 75)(5 76)(6 77)(7 78)(8 79)(9 80)(10 81)(11 82)(12 83)(13 70)(14 71)(15 72)(16 61)(17 62)(18 63)(19 64)(20 65)(21 66)(22 67)(23 68)(24 69)(25 85)(26 86)(27 87)(28 88)(29 89)(30 90)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 59)(38 60)(39 49)(40 50)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)
(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 33)(2 34)(3 35)(4 36)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 42)(14 43)(15 44)(16 45)(17 46)(18 47)(19 48)(20 37)(21 38)(22 39)(23 40)(24 41)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(55 61)(56 62)(57 63)(58 64)(59 65)(60 66)(73 94)(74 95)(75 96)(76 85)(77 86)(78 87)(79 88)(80 89)(81 90)(82 91)(83 92)(84 93)
(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 42 84 52)(2 41 73 51)(3 40 74 50)(4 39 75 49)(5 38 76 60)(6 37 77 59)(7 48 78 58)(8 47 79 57)(9 46 80 56)(10 45 81 55)(11 44 82 54)(12 43 83 53)(13 93 70 33)(14 92 71 32)(15 91 72 31)(16 90 61 30)(17 89 62 29)(18 88 63 28)(19 87 64 27)(20 86 65 26)(21 85 66 25)(22 96 67 36)(23 95 68 35)(24 94 69 34)
G:=sub<Sym(96)| (1,84)(2,94)(3,74)(4,96)(5,76)(6,86)(7,78)(8,88)(9,80)(10,90)(11,82)(12,92)(13,52)(14,71)(15,54)(16,61)(17,56)(18,63)(19,58)(20,65)(21,60)(22,67)(23,50)(24,69)(25,85)(26,77)(27,87)(28,79)(29,89)(30,81)(31,91)(32,83)(33,93)(34,73)(35,95)(36,75)(37,59)(38,66)(39,49)(40,68)(41,51)(42,70)(43,53)(44,72)(45,55)(46,62)(47,57)(48,64), (1,84)(2,73)(3,74)(4,75)(5,76)(6,77)(7,78)(8,79)(9,80)(10,81)(11,82)(12,83)(13,70)(14,71)(15,72)(16,61)(17,62)(18,63)(19,64)(20,65)(21,66)(22,67)(23,68)(24,69)(25,85)(26,86)(27,87)(28,88)(29,89)(30,90)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,59)(38,60)(39,49)(40,50)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58), (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), (1,33)(2,34)(3,35)(4,36)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,42)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,37)(21,38)(22,39)(23,40)(24,41)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,61)(56,62)(57,63)(58,64)(59,65)(60,66)(73,94)(74,95)(75,96)(76,85)(77,86)(78,87)(79,88)(80,89)(81,90)(82,91)(83,92)(84,93), (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), (1,42,84,52)(2,41,73,51)(3,40,74,50)(4,39,75,49)(5,38,76,60)(6,37,77,59)(7,48,78,58)(8,47,79,57)(9,46,80,56)(10,45,81,55)(11,44,82,54)(12,43,83,53)(13,93,70,33)(14,92,71,32)(15,91,72,31)(16,90,61,30)(17,89,62,29)(18,88,63,28)(19,87,64,27)(20,86,65,26)(21,85,66,25)(22,96,67,36)(23,95,68,35)(24,94,69,34)>;
G:=Group( (1,84)(2,94)(3,74)(4,96)(5,76)(6,86)(7,78)(8,88)(9,80)(10,90)(11,82)(12,92)(13,52)(14,71)(15,54)(16,61)(17,56)(18,63)(19,58)(20,65)(21,60)(22,67)(23,50)(24,69)(25,85)(26,77)(27,87)(28,79)(29,89)(30,81)(31,91)(32,83)(33,93)(34,73)(35,95)(36,75)(37,59)(38,66)(39,49)(40,68)(41,51)(42,70)(43,53)(44,72)(45,55)(46,62)(47,57)(48,64), (1,84)(2,73)(3,74)(4,75)(5,76)(6,77)(7,78)(8,79)(9,80)(10,81)(11,82)(12,83)(13,70)(14,71)(15,72)(16,61)(17,62)(18,63)(19,64)(20,65)(21,66)(22,67)(23,68)(24,69)(25,85)(26,86)(27,87)(28,88)(29,89)(30,90)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,59)(38,60)(39,49)(40,50)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58), (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), (1,33)(2,34)(3,35)(4,36)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,42)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,37)(21,38)(22,39)(23,40)(24,41)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,61)(56,62)(57,63)(58,64)(59,65)(60,66)(73,94)(74,95)(75,96)(76,85)(77,86)(78,87)(79,88)(80,89)(81,90)(82,91)(83,92)(84,93), (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), (1,42,84,52)(2,41,73,51)(3,40,74,50)(4,39,75,49)(5,38,76,60)(6,37,77,59)(7,48,78,58)(8,47,79,57)(9,46,80,56)(10,45,81,55)(11,44,82,54)(12,43,83,53)(13,93,70,33)(14,92,71,32)(15,91,72,31)(16,90,61,30)(17,89,62,29)(18,88,63,28)(19,87,64,27)(20,86,65,26)(21,85,66,25)(22,96,67,36)(23,95,68,35)(24,94,69,34) );
G=PermutationGroup([[(1,84),(2,94),(3,74),(4,96),(5,76),(6,86),(7,78),(8,88),(9,80),(10,90),(11,82),(12,92),(13,52),(14,71),(15,54),(16,61),(17,56),(18,63),(19,58),(20,65),(21,60),(22,67),(23,50),(24,69),(25,85),(26,77),(27,87),(28,79),(29,89),(30,81),(31,91),(32,83),(33,93),(34,73),(35,95),(36,75),(37,59),(38,66),(39,49),(40,68),(41,51),(42,70),(43,53),(44,72),(45,55),(46,62),(47,57),(48,64)], [(1,84),(2,73),(3,74),(4,75),(5,76),(6,77),(7,78),(8,79),(9,80),(10,81),(11,82),(12,83),(13,70),(14,71),(15,72),(16,61),(17,62),(18,63),(19,64),(20,65),(21,66),(22,67),(23,68),(24,69),(25,85),(26,86),(27,87),(28,88),(29,89),(30,90),(31,91),(32,92),(33,93),(34,94),(35,95),(36,96),(37,59),(38,60),(39,49),(40,50),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58)], [(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)], [(1,33),(2,34),(3,35),(4,36),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,42),(14,43),(15,44),(16,45),(17,46),(18,47),(19,48),(20,37),(21,38),(22,39),(23,40),(24,41),(49,67),(50,68),(51,69),(52,70),(53,71),(54,72),(55,61),(56,62),(57,63),(58,64),(59,65),(60,66),(73,94),(74,95),(75,96),(76,85),(77,86),(78,87),(79,88),(80,89),(81,90),(82,91),(83,92),(84,93)], [(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)], [(1,42,84,52),(2,41,73,51),(3,40,74,50),(4,39,75,49),(5,38,76,60),(6,37,77,59),(7,48,78,58),(8,47,79,57),(9,46,80,56),(10,45,81,55),(11,44,82,54),(12,43,83,53),(13,93,70,33),(14,92,71,32),(15,91,72,31),(16,90,61,30),(17,89,62,29),(18,88,63,28),(19,87,64,27),(20,86,65,26),(21,85,66,25),(22,96,67,36),(23,95,68,35),(24,94,69,34)]])
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 | Dic3 | D6 | D6 | C4○D4 | Dic6 | D12 | S3×D4 | D4⋊2S3 |
| kernel | C24.58D6 | C6.C42 | C2×C4⋊Dic3 | C2×C6.D4 | C6×C22⋊C4 | C23×Dic3 | C3×C22⋊C4 | C2×C22⋊C4 | C2×Dic3 | C22×C6 | C22×C6 | C22⋊C4 | C22×C4 | C24 | C2×C6 | C23 | C23 | C22 | C22 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 1 | 4 | 2 | 2 | 4 | 2 | 1 | 4 | 4 | 4 | 2 | 2 |
Matrix representation of C24.58D6 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 6 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 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 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 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 |
| 12 | 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 |
| 4 | 3 | 0 | 0 | 0 | 0 |
| 8 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 3 |
| 0 | 0 | 0 | 0 | 10 | 10 |
| 4 | 3 | 0 | 0 | 0 | 0 |
| 8 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 5 |
| 0 | 0 | 0 | 0 | 0 | 5 |
G:=sub<GL(6,GF(13))| [1,6,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[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],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,8,0,0,0,0,3,9,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,7,10,0,0,0,0,3,10],[4,8,0,0,0,0,3,9,0,0,0,0,0,0,0,8,0,0,0,0,8,0,0,0,0,0,0,0,8,0,0,0,0,0,5,5] >;
C24.58D6 in GAP, Magma, Sage, TeX
C_2^4._{58}D_6 % in TeX
G:=Group("C2^4.58D6"); // GroupNames label
G:=SmallGroup(192,509);
// by ID
G=gap.SmallGroup(192,509);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,422,387,100,6278]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=c,f^2=b,a*b=b*a,a*c=c*a,e*a*e^-1=f*a*f^-1=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=c*e^5>;
// generators/relations