metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.27D6, C23.21D12, (C2×C12)⋊7D4, C6.34C22≀C2, (C22×C6).70D4, (C22×C4).52D6, C2.7(D6⋊3D4), C6.60(C4⋊D4), C2.8(C12⋊7D4), (C2×Dic3).57D4, (C22×S3).32D4, C22.243(S3×D4), C2.35(D6⋊D4), C6.C42⋊18C2, C6.36(C4.4D4), C22.127(C2×D12), C3⋊3(C23.10D4), (C23×C6).44C22, C2.23(C23.9D6), (S3×C23).17C22, (C22×C6).335C23, C23.383(C22×S3), (C22×C12).62C22, C2.11(C23.14D6), C22.101(C4○D12), C22.98(D4⋊2S3), C6.35(C22.D4), C2.23(C23.11D6), C2.17(C23.21D6), (C22×Dic3).47C22, (C2×D6⋊C4)⋊9C2, (C2×C4)⋊4(C3⋊D4), (C2×C22⋊C4)⋊9S3, (C6×C22⋊C4)⋊12C2, (C2×C4⋊Dic3)⋊13C2, (C2×C6).326(C2×D4), (C2×C6).81(C4○D4), (C2×C6.D4)⋊6C2, (C22×C3⋊D4).6C2, C22.129(C2×C3⋊D4), SmallGroup(192,520)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.27D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=dc=cd, f2=d, ab=ba, ac=ca, eae-1=ad=da, faf-1=abd, bc=cb, bd=db, be=eb, bf=fb, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce5 >
Subgroups: 728 in 238 conjugacy classes, 63 normal (51 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C22×C6, C22×C6, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C22×D4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C22×Dic3, C2×C3⋊D4, C22×C12, S3×C23, C23×C6, C23.10D4, C6.C42, C2×C4⋊Dic3, C2×D6⋊C4, C2×C6.D4, C6×C22⋊C4, C22×C3⋊D4, C24.27D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C3⋊D4, C22×S3, C22≀C2, C4⋊D4, C22.D4, C4.4D4, C2×D12, C4○D12, S3×D4, D4⋊2S3, C2×C3⋊D4, C23.10D4, D6⋊D4, C23.9D6, C23.11D6, C23.21D6, C12⋊7D4, D6⋊3D4, C23.14D6, C24.27D6
►On 96 points
Generators in S96
(1 68)(2 35)(3 70)(4 25)(5 72)(6 27)(7 62)(8 29)(9 64)(10 31)(11 66)(12 33)(13 37)(14 55)(15 39)(16 57)(17 41)(18 59)(19 43)(20 49)(21 45)(22 51)(23 47)(24 53)(26 95)(28 85)(30 87)(32 89)(34 91)(36 93)(38 81)(40 83)(42 73)(44 75)(46 77)(48 79)(50 76)(52 78)(54 80)(56 82)(58 84)(60 74)(61 96)(63 86)(65 88)(67 90)(69 92)(71 94)
(1 77)(2 78)(3 79)(4 80)(5 81)(6 82)(7 83)(8 84)(9 73)(10 74)(11 75)(12 76)(13 94)(14 95)(15 96)(16 85)(17 86)(18 87)(19 88)(20 89)(21 90)(22 91)(23 92)(24 93)(25 54)(26 55)(27 56)(28 57)(29 58)(30 59)(31 60)(32 49)(33 50)(34 51)(35 52)(36 53)(37 71)(38 72)(39 61)(40 62)(41 63)(42 64)(43 65)(44 66)(45 67)(46 68)(47 69)(48 70)
(1 85)(2 86)(3 87)(4 88)(5 89)(6 90)(7 91)(8 92)(9 93)(10 94)(11 95)(12 96)(13 74)(14 75)(15 76)(16 77)(17 78)(18 79)(19 80)(20 81)(21 82)(22 83)(23 84)(24 73)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 61)(34 62)(35 63)(36 64)(37 60)(38 49)(39 50)(40 51)(41 52)(42 53)(43 54)(44 55)(45 56)(46 57)(47 58)(48 59)
(1 91)(2 92)(3 93)(4 94)(5 95)(6 96)(7 85)(8 86)(9 87)(10 88)(11 89)(12 90)(13 80)(14 81)(15 82)(16 83)(17 84)(18 73)(19 74)(20 75)(21 76)(22 77)(23 78)(24 79)(25 71)(26 72)(27 61)(28 62)(29 63)(30 64)(31 65)(32 66)(33 67)(34 68)(35 69)(36 70)(37 54)(38 55)(39 56)(40 57)(41 58)(42 59)(43 60)(44 49)(45 50)(46 51)(47 52)(48 53)
(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 91 96)(2 95 92 5)(3 4 93 94)(7 12 85 90)(8 89 86 11)(9 10 87 88)(13 79 80 24)(14 23 81 78)(15 77 82 22)(16 21 83 76)(17 75 84 20)(18 19 73 74)(25 48 71 53)(26 52 72 47)(27 46 61 51)(28 50 62 45)(29 44 63 49)(30 60 64 43)(31 42 65 59)(32 58 66 41)(33 40 67 57)(34 56 68 39)(35 38 69 55)(36 54 70 37)
G:=sub<Sym(96)| (1,68)(2,35)(3,70)(4,25)(5,72)(6,27)(7,62)(8,29)(9,64)(10,31)(11,66)(12,33)(13,37)(14,55)(15,39)(16,57)(17,41)(18,59)(19,43)(20,49)(21,45)(22,51)(23,47)(24,53)(26,95)(28,85)(30,87)(32,89)(34,91)(36,93)(38,81)(40,83)(42,73)(44,75)(46,77)(48,79)(50,76)(52,78)(54,80)(56,82)(58,84)(60,74)(61,96)(63,86)(65,88)(67,90)(69,92)(71,94), (1,77)(2,78)(3,79)(4,80)(5,81)(6,82)(7,83)(8,84)(9,73)(10,74)(11,75)(12,76)(13,94)(14,95)(15,96)(16,85)(17,86)(18,87)(19,88)(20,89)(21,90)(22,91)(23,92)(24,93)(25,54)(26,55)(27,56)(28,57)(29,58)(30,59)(31,60)(32,49)(33,50)(34,51)(35,52)(36,53)(37,71)(38,72)(39,61)(40,62)(41,63)(42,64)(43,65)(44,66)(45,67)(46,68)(47,69)(48,70), (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,92)(9,93)(10,94)(11,95)(12,96)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,80)(20,81)(21,82)(22,83)(23,84)(24,73)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,61)(34,62)(35,63)(36,64)(37,60)(38,49)(39,50)(40,51)(41,52)(42,53)(43,54)(44,55)(45,56)(46,57)(47,58)(48,59), (1,91)(2,92)(3,93)(4,94)(5,95)(6,96)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,80)(14,81)(15,82)(16,83)(17,84)(18,73)(19,74)(20,75)(21,76)(22,77)(23,78)(24,79)(25,71)(26,72)(27,61)(28,62)(29,63)(30,64)(31,65)(32,66)(33,67)(34,68)(35,69)(36,70)(37,54)(38,55)(39,56)(40,57)(41,58)(42,59)(43,60)(44,49)(45,50)(46,51)(47,52)(48,53), (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), (1,6,91,96)(2,95,92,5)(3,4,93,94)(7,12,85,90)(8,89,86,11)(9,10,87,88)(13,79,80,24)(14,23,81,78)(15,77,82,22)(16,21,83,76)(17,75,84,20)(18,19,73,74)(25,48,71,53)(26,52,72,47)(27,46,61,51)(28,50,62,45)(29,44,63,49)(30,60,64,43)(31,42,65,59)(32,58,66,41)(33,40,67,57)(34,56,68,39)(35,38,69,55)(36,54,70,37)>;
G:=Group( (1,68)(2,35)(3,70)(4,25)(5,72)(6,27)(7,62)(8,29)(9,64)(10,31)(11,66)(12,33)(13,37)(14,55)(15,39)(16,57)(17,41)(18,59)(19,43)(20,49)(21,45)(22,51)(23,47)(24,53)(26,95)(28,85)(30,87)(32,89)(34,91)(36,93)(38,81)(40,83)(42,73)(44,75)(46,77)(48,79)(50,76)(52,78)(54,80)(56,82)(58,84)(60,74)(61,96)(63,86)(65,88)(67,90)(69,92)(71,94), (1,77)(2,78)(3,79)(4,80)(5,81)(6,82)(7,83)(8,84)(9,73)(10,74)(11,75)(12,76)(13,94)(14,95)(15,96)(16,85)(17,86)(18,87)(19,88)(20,89)(21,90)(22,91)(23,92)(24,93)(25,54)(26,55)(27,56)(28,57)(29,58)(30,59)(31,60)(32,49)(33,50)(34,51)(35,52)(36,53)(37,71)(38,72)(39,61)(40,62)(41,63)(42,64)(43,65)(44,66)(45,67)(46,68)(47,69)(48,70), (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,92)(9,93)(10,94)(11,95)(12,96)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,80)(20,81)(21,82)(22,83)(23,84)(24,73)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,61)(34,62)(35,63)(36,64)(37,60)(38,49)(39,50)(40,51)(41,52)(42,53)(43,54)(44,55)(45,56)(46,57)(47,58)(48,59), (1,91)(2,92)(3,93)(4,94)(5,95)(6,96)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,80)(14,81)(15,82)(16,83)(17,84)(18,73)(19,74)(20,75)(21,76)(22,77)(23,78)(24,79)(25,71)(26,72)(27,61)(28,62)(29,63)(30,64)(31,65)(32,66)(33,67)(34,68)(35,69)(36,70)(37,54)(38,55)(39,56)(40,57)(41,58)(42,59)(43,60)(44,49)(45,50)(46,51)(47,52)(48,53), (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), (1,6,91,96)(2,95,92,5)(3,4,93,94)(7,12,85,90)(8,89,86,11)(9,10,87,88)(13,79,80,24)(14,23,81,78)(15,77,82,22)(16,21,83,76)(17,75,84,20)(18,19,73,74)(25,48,71,53)(26,52,72,47)(27,46,61,51)(28,50,62,45)(29,44,63,49)(30,60,64,43)(31,42,65,59)(32,58,66,41)(33,40,67,57)(34,56,68,39)(35,38,69,55)(36,54,70,37) );
G=PermutationGroup([[(1,68),(2,35),(3,70),(4,25),(5,72),(6,27),(7,62),(8,29),(9,64),(10,31),(11,66),(12,33),(13,37),(14,55),(15,39),(16,57),(17,41),(18,59),(19,43),(20,49),(21,45),(22,51),(23,47),(24,53),(26,95),(28,85),(30,87),(32,89),(34,91),(36,93),(38,81),(40,83),(42,73),(44,75),(46,77),(48,79),(50,76),(52,78),(54,80),(56,82),(58,84),(60,74),(61,96),(63,86),(65,88),(67,90),(69,92),(71,94)], [(1,77),(2,78),(3,79),(4,80),(5,81),(6,82),(7,83),(8,84),(9,73),(10,74),(11,75),(12,76),(13,94),(14,95),(15,96),(16,85),(17,86),(18,87),(19,88),(20,89),(21,90),(22,91),(23,92),(24,93),(25,54),(26,55),(27,56),(28,57),(29,58),(30,59),(31,60),(32,49),(33,50),(34,51),(35,52),(36,53),(37,71),(38,72),(39,61),(40,62),(41,63),(42,64),(43,65),(44,66),(45,67),(46,68),(47,69),(48,70)], [(1,85),(2,86),(3,87),(4,88),(5,89),(6,90),(7,91),(8,92),(9,93),(10,94),(11,95),(12,96),(13,74),(14,75),(15,76),(16,77),(17,78),(18,79),(19,80),(20,81),(21,82),(22,83),(23,84),(24,73),(25,65),(26,66),(27,67),(28,68),(29,69),(30,70),(31,71),(32,72),(33,61),(34,62),(35,63),(36,64),(37,60),(38,49),(39,50),(40,51),(41,52),(42,53),(43,54),(44,55),(45,56),(46,57),(47,58),(48,59)], [(1,91),(2,92),(3,93),(4,94),(5,95),(6,96),(7,85),(8,86),(9,87),(10,88),(11,89),(12,90),(13,80),(14,81),(15,82),(16,83),(17,84),(18,73),(19,74),(20,75),(21,76),(22,77),(23,78),(24,79),(25,71),(26,72),(27,61),(28,62),(29,63),(30,64),(31,65),(32,66),(33,67),(34,68),(35,69),(36,70),(37,54),(38,55),(39,56),(40,57),(41,58),(42,59),(43,60),(44,49),(45,50),(46,51),(47,52),(48,53)], [(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)], [(1,6,91,96),(2,95,92,5),(3,4,93,94),(7,12,85,90),(8,89,86,11),(9,10,87,88),(13,79,80,24),(14,23,81,78),(15,77,82,22),(16,21,83,76),(17,75,84,20),(18,19,73,74),(25,48,71,53),(26,52,72,47),(27,46,61,51),(28,50,62,45),(29,44,63,49),(30,60,64,43),(31,42,65,59),(32,58,66,41),(33,40,67,57),(34,56,68,39),(35,38,69,55),(36,54,70,37)]])
42 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 6A | ··· | 6G | 6H | 6I | 6J | 6K | 12A | ··· | 12H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 12 | 12 | 2 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D4 | D6 | D6 | C4○D4 | C3⋊D4 | D12 | C4○D12 | S3×D4 | D4⋊2S3 |
| kernel | C24.27D6 | C6.C42 | C2×C4⋊Dic3 | C2×D6⋊C4 | C2×C6.D4 | C6×C22⋊C4 | C22×C3⋊D4 | C2×C22⋊C4 | C2×Dic3 | C2×C12 | C22×S3 | C22×C6 | C22×C4 | C24 | C2×C6 | C2×C4 | C23 | C22 | C22 | C22 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 6 | 4 | 4 | 4 | 2 | 2 |
Matrix representation of C24.27D6 ►in GL6(𝔽13)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 8 | 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 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 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 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 2 | 0 | 0 |
| 0 | 0 | 1 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 10 |
| 0 | 0 | 0 | 0 | 3 | 6 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 11 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 3 |
| 0 | 0 | 0 | 0 | 6 | 3 |
G:=sub<GL(6,GF(13))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,8,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[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],[8,0,0,0,0,0,0,5,0,0,0,0,0,0,5,1,0,0,0,0,2,8,0,0,0,0,0,0,3,3,0,0,0,0,10,6],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,11,5,0,0,0,0,0,0,10,6,0,0,0,0,3,3] >;
C24.27D6 in GAP, Magma, Sage, TeX
C_2^4._{27}D_6 % in TeX
G:=Group("C2^4.27D6"); // GroupNames label
G:=SmallGroup(192,520);
// by ID
G=gap.SmallGroup(192,520);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,254,387,6278]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=d*c=c*d,f^2=d,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,f*a*f^-1=a*b*d,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,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