metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.60D6, C23.50D12, C6.44(C4×D4), C6.32C22≀C2, C22⋊2(D6⋊C4), (C2×Dic3)⋊17D4, C23.31(C4×S3), (C22×C4).49D6, (C22×C6).68D4, C2.5(D6⋊D4), C6.86(C4⋊D4), (C23×Dic3)⋊1C2, C22.103(S3×D4), C22.44(C2×D12), C6.C42⋊16C2, C3⋊3(C23.23D4), C23.43(C3⋊D4), (C23×C6).41C22, C2.4(C23.14D6), (S3×C23).14C22, C23.295(C22×S3), (C22×C12).26C22, (C22×C6).332C23, C2.29(Dic3⋊4D4), C22.50(D4⋊2S3), C6.33(C22.D4), C2.5(C23.21D6), (C22×Dic3).186C22, (C2×D6⋊C4)⋊6C2, (C2×C3⋊D4)⋊6C4, (C6×C22⋊C4)⋊3C2, (C2×C22⋊C4)⋊5S3, C2.10(C2×D6⋊C4), (C2×C6)⋊2(C22⋊C4), (C22×S3)⋊4(C2×C4), (C2×Dic3)⋊6(C2×C4), (C2×C6).324(C2×D4), C6.37(C2×C22⋊C4), C22.129(S3×C2×C4), (C22×C6).55(C2×C4), (C22×C3⋊D4).4C2, C22.53(C2×C3⋊D4), (C2×C6).147(C4○D4), (C2×C6).111(C22×C4), SmallGroup(192,517)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.60D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=b, f2=cb=bc, ab=ba, ac=ca, eae-1=ad=da, af=fa, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce5 >
Subgroups: 808 in 286 conjugacy classes, 83 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×S3, C22×C6, C22×C6, C22×C6, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C23×C4, C22×D4, D6⋊C4, C3×C22⋊C4, C22×Dic3, C22×Dic3, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C22×C12, S3×C23, C23×C6, C23.23D4, C6.C42, C2×D6⋊C4, C6×C22⋊C4, C23×Dic3, C22×C3⋊D4, C24.60D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4○D4, C4×S3, D12, C3⋊D4, C22×S3, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, D6⋊C4, S3×C2×C4, C2×D12, S3×D4, D4⋊2S3, C2×C3⋊D4, C23.23D4, Dic3⋊4D4, D6⋊D4, C23.21D6, C2×D6⋊C4, C23.14D6, C24.60D6
►On 96 points
Generators in S96
(1 88)(2 15)(3 90)(4 17)(5 92)(6 19)(7 94)(8 21)(9 96)(10 23)(11 86)(12 13)(14 81)(16 83)(18 73)(20 75)(22 77)(24 79)(25 51)(26 64)(27 53)(28 66)(29 55)(30 68)(31 57)(32 70)(33 59)(34 72)(35 49)(36 62)(37 54)(38 67)(39 56)(40 69)(41 58)(42 71)(43 60)(44 61)(45 50)(46 63)(47 52)(48 65)(74 93)(76 95)(78 85)(80 87)(82 89)(84 91)
(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 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 57)(9 58)(10 59)(11 60)(12 49)(13 35)(14 36)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 33)(24 34)(37 92)(38 93)(39 94)(40 95)(41 96)(42 85)(43 86)(44 87)(45 88)(46 89)(47 90)(48 91)(61 80)(62 81)(63 82)(64 83)(65 84)(66 73)(67 74)(68 75)(69 76)(70 77)(71 78)(72 79)
(1 81)(2 82)(3 83)(4 84)(5 73)(6 74)(7 75)(8 76)(9 77)(10 78)(11 79)(12 80)(13 87)(14 88)(15 89)(16 90)(17 91)(18 92)(19 93)(20 94)(21 95)(22 96)(23 85)(24 86)(25 46)(26 47)(27 48)(28 37)(29 38)(30 39)(31 40)(32 41)(33 42)(34 43)(35 44)(36 45)(49 61)(50 62)(51 63)(52 64)(53 65)(54 66)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)
(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 6 56 49)(2 60 57 5)(3 4 58 59)(7 12 50 55)(8 54 51 11)(9 10 52 53)(13 45 29 94)(14 93 30 44)(15 43 31 92)(16 91 32 42)(17 41 33 90)(18 89 34 40)(19 39 35 88)(20 87 36 38)(21 37 25 86)(22 85 26 48)(23 47 27 96)(24 95 28 46)(61 81 74 68)(62 67 75 80)(63 79 76 66)(64 65 77 78)(69 73 82 72)(70 71 83 84)
G:=sub<Sym(96)| (1,88)(2,15)(3,90)(4,17)(5,92)(6,19)(7,94)(8,21)(9,96)(10,23)(11,86)(12,13)(14,81)(16,83)(18,73)(20,75)(22,77)(24,79)(25,51)(26,64)(27,53)(28,66)(29,55)(30,68)(31,57)(32,70)(33,59)(34,72)(35,49)(36,62)(37,54)(38,67)(39,56)(40,69)(41,58)(42,71)(43,60)(44,61)(45,50)(46,63)(47,52)(48,65)(74,93)(76,95)(78,85)(80,87)(82,89)(84,91), (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), (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,57)(9,58)(10,59)(11,60)(12,49)(13,35)(14,36)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(23,33)(24,34)(37,92)(38,93)(39,94)(40,95)(41,96)(42,85)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91)(61,80)(62,81)(63,82)(64,83)(65,84)(66,73)(67,74)(68,75)(69,76)(70,77)(71,78)(72,79), (1,81)(2,82)(3,83)(4,84)(5,73)(6,74)(7,75)(8,76)(9,77)(10,78)(11,79)(12,80)(13,87)(14,88)(15,89)(16,90)(17,91)(18,92)(19,93)(20,94)(21,95)(22,96)(23,85)(24,86)(25,46)(26,47)(27,48)(28,37)(29,38)(30,39)(31,40)(32,41)(33,42)(34,43)(35,44)(36,45)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72), (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), (1,6,56,49)(2,60,57,5)(3,4,58,59)(7,12,50,55)(8,54,51,11)(9,10,52,53)(13,45,29,94)(14,93,30,44)(15,43,31,92)(16,91,32,42)(17,41,33,90)(18,89,34,40)(19,39,35,88)(20,87,36,38)(21,37,25,86)(22,85,26,48)(23,47,27,96)(24,95,28,46)(61,81,74,68)(62,67,75,80)(63,79,76,66)(64,65,77,78)(69,73,82,72)(70,71,83,84)>;
G:=Group( (1,88)(2,15)(3,90)(4,17)(5,92)(6,19)(7,94)(8,21)(9,96)(10,23)(11,86)(12,13)(14,81)(16,83)(18,73)(20,75)(22,77)(24,79)(25,51)(26,64)(27,53)(28,66)(29,55)(30,68)(31,57)(32,70)(33,59)(34,72)(35,49)(36,62)(37,54)(38,67)(39,56)(40,69)(41,58)(42,71)(43,60)(44,61)(45,50)(46,63)(47,52)(48,65)(74,93)(76,95)(78,85)(80,87)(82,89)(84,91), (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), (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,57)(9,58)(10,59)(11,60)(12,49)(13,35)(14,36)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(23,33)(24,34)(37,92)(38,93)(39,94)(40,95)(41,96)(42,85)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91)(61,80)(62,81)(63,82)(64,83)(65,84)(66,73)(67,74)(68,75)(69,76)(70,77)(71,78)(72,79), (1,81)(2,82)(3,83)(4,84)(5,73)(6,74)(7,75)(8,76)(9,77)(10,78)(11,79)(12,80)(13,87)(14,88)(15,89)(16,90)(17,91)(18,92)(19,93)(20,94)(21,95)(22,96)(23,85)(24,86)(25,46)(26,47)(27,48)(28,37)(29,38)(30,39)(31,40)(32,41)(33,42)(34,43)(35,44)(36,45)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72), (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), (1,6,56,49)(2,60,57,5)(3,4,58,59)(7,12,50,55)(8,54,51,11)(9,10,52,53)(13,45,29,94)(14,93,30,44)(15,43,31,92)(16,91,32,42)(17,41,33,90)(18,89,34,40)(19,39,35,88)(20,87,36,38)(21,37,25,86)(22,85,26,48)(23,47,27,96)(24,95,28,46)(61,81,74,68)(62,67,75,80)(63,79,76,66)(64,65,77,78)(69,73,82,72)(70,71,83,84) );
G=PermutationGroup([[(1,88),(2,15),(3,90),(4,17),(5,92),(6,19),(7,94),(8,21),(9,96),(10,23),(11,86),(12,13),(14,81),(16,83),(18,73),(20,75),(22,77),(24,79),(25,51),(26,64),(27,53),(28,66),(29,55),(30,68),(31,57),(32,70),(33,59),(34,72),(35,49),(36,62),(37,54),(38,67),(39,56),(40,69),(41,58),(42,71),(43,60),(44,61),(45,50),(46,63),(47,52),(48,65),(74,93),(76,95),(78,85),(80,87),(82,89),(84,91)], [(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)], [(1,50),(2,51),(3,52),(4,53),(5,54),(6,55),(7,56),(8,57),(9,58),(10,59),(11,60),(12,49),(13,35),(14,36),(15,25),(16,26),(17,27),(18,28),(19,29),(20,30),(21,31),(22,32),(23,33),(24,34),(37,92),(38,93),(39,94),(40,95),(41,96),(42,85),(43,86),(44,87),(45,88),(46,89),(47,90),(48,91),(61,80),(62,81),(63,82),(64,83),(65,84),(66,73),(67,74),(68,75),(69,76),(70,77),(71,78),(72,79)], [(1,81),(2,82),(3,83),(4,84),(5,73),(6,74),(7,75),(8,76),(9,77),(10,78),(11,79),(12,80),(13,87),(14,88),(15,89),(16,90),(17,91),(18,92),(19,93),(20,94),(21,95),(22,96),(23,85),(24,86),(25,46),(26,47),(27,48),(28,37),(29,38),(30,39),(31,40),(32,41),(33,42),(34,43),(35,44),(36,45),(49,61),(50,62),(51,63),(52,64),(53,65),(54,66),(55,67),(56,68),(57,69),(58,70),(59,71),(60,72)], [(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)], [(1,6,56,49),(2,60,57,5),(3,4,58,59),(7,12,50,55),(8,54,51,11),(9,10,52,53),(13,45,29,94),(14,93,30,44),(15,43,31,92),(16,91,32,42),(17,41,33,90),(18,89,34,40),(19,39,35,88),(20,87,36,38),(21,37,25,86),(22,85,26,48),(23,47,27,96),(24,95,28,46),(61,81,74,68),(62,67,75,80),(63,79,76,66),(64,65,77,78),(69,73,82,72),(70,71,83,84)]])
48 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | 4N | 6A | ··· | 6G | 6H | 6I | 6J | 6K | 12A | ··· | 12H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 12 | 12 | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 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 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D6 | D6 | C4○D4 | C4×S3 | D12 | C3⋊D4 | S3×D4 | D4⋊2S3 |
| kernel | C24.60D6 | C6.C42 | C2×D6⋊C4 | C6×C22⋊C4 | C23×Dic3 | C22×C3⋊D4 | C2×C3⋊D4 | C2×C22⋊C4 | C2×Dic3 | C22×C6 | C22×C4 | C24 | C2×C6 | C23 | C23 | C23 | C22 | C22 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 1 | 4 | 4 | 2 | 1 | 4 | 4 | 4 | 4 | 2 | 2 |
Matrix representation of C24.60D6 ►in GL6(𝔽13)
| 0 | 5 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 2 |
| 0 | 0 | 0 | 0 | 1 | 5 |
| 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 |
| 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 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 5 | 12 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [0,8,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,1,0,0,0,0,2,5],[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],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,5,8,0,0,0,0,5,0,0,0,0,0,0,0,1,5,0,0,0,0,0,12],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,8,5,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C24.60D6 in GAP, Magma, Sage, TeX
C_2^4._{60}D_6 % in TeX
G:=Group("C2^4.60D6"); // GroupNames label
G:=SmallGroup(192,517);
// by ID
G=gap.SmallGroup(192,517);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,422,387,58,6278]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=b,f^2=c*b=b*c,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,a*f=f*a,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