metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.55D6, C23.11Dic6, C23.54(C4×S3), C22.94(S3×D4), (C22×C6).60D4, (C22×C4).38D6, (C22×C6).10Q8, C6.81(C4⋊D4), C6.C42⋊9C2, (C22×Dic3)⋊7C4, C6.15(C22⋊Q8), C3⋊2(C23.7Q8), Dic3⋊3(C22⋊C4), C22⋊2(Dic3⋊C4), (C2×Dic3).171D4, C23.42(C3⋊D4), (C23×C6).25C22, (C23×Dic3).2C2, C22.23(C2×Dic6), C6.24(C42⋊C2), C2.1(C23.14D6), (C22×C6).317C23, (C22×C12).21C22, C23.285(C22×S3), C22.41(D4⋊2S3), C2.5(Dic3.D4), C2.11(C23.16D6), (C22×Dic3).34C22, (C2×C6)⋊1(C4⋊C4), C6.29(C2×C4⋊C4), (C2×C6).30(C2×Q8), (C2×Dic3⋊C4)⋊6C2, (C2×C6).429(C2×D4), (C6×C22⋊C4).5C2, (C2×C22⋊C4).4S3, C6.27(C2×C22⋊C4), C2.27(S3×C22⋊C4), C2.5(C2×Dic3⋊C4), C22.121(S3×C2×C4), (C22×C6).45(C2×C4), C22.45(C2×C3⋊D4), (C2×C6).138(C4○D4), (C2×C6).103(C22×C4), (C2×Dic3).92(C2×C4), (C2×C6.D4).4C2, SmallGroup(192,501)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.55D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=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, 87 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C23, C23, C23, Dic3, 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×Dic3, C22×C12, C23×C6, C23.7Q8, C6.C42, C2×Dic3⋊C4, C2×C6.D4, C6×C22⋊C4, C23×Dic3, C24.55D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic6, C4×S3, C3⋊D4, C22×S3, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4, C22⋊Q8, Dic3⋊C4, C2×Dic6, S3×C2×C4, S3×D4, D4⋊2S3, C2×C3⋊D4, C23.7Q8, C23.16D6, Dic3.D4, S3×C22⋊C4, C2×Dic3⋊C4, C23.14D6, C24.55D6
►On 96 points
Generators in S96
(1 7)(2 88)(3 9)(4 90)(5 11)(6 92)(8 94)(10 96)(12 86)(13 66)(14 20)(15 68)(16 22)(17 70)(18 24)(19 72)(21 62)(23 64)(25 31)(26 56)(27 33)(28 58)(29 35)(30 60)(32 50)(34 52)(36 54)(37 82)(38 44)(39 84)(40 46)(41 74)(42 48)(43 76)(45 78)(47 80)(49 55)(51 57)(53 59)(61 67)(63 69)(65 71)(73 79)(75 81)(77 83)(85 91)(87 93)(89 95)
(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 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 49)(12 50)(13 78)(14 79)(15 80)(16 81)(17 82)(18 83)(19 84)(20 73)(21 74)(22 75)(23 76)(24 77)(25 91)(26 92)(27 93)(28 94)(29 95)(30 96)(31 85)(32 86)(33 87)(34 88)(35 89)(36 90)(37 70)(38 71)(39 72)(40 61)(41 62)(42 63)(43 64)(44 65)(45 66)(46 67)(47 68)(48 69)
(1 93)(2 94)(3 95)(4 96)(5 85)(6 86)(7 87)(8 88)(9 89)(10 90)(11 91)(12 92)(13 72)(14 61)(15 62)(16 63)(17 64)(18 65)(19 66)(20 67)(21 68)(22 69)(23 70)(24 71)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 76)(38 77)(39 78)(40 79)(41 80)(42 81)(43 82)(44 83)(45 84)(46 73)(47 74)(48 75)
(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 45 7 39)(2 71 8 65)(3 43 9 37)(4 69 10 63)(5 41 11 47)(6 67 12 61)(13 27 19 33)(14 86 20 92)(15 25 21 31)(16 96 22 90)(17 35 23 29)(18 94 24 88)(26 79 32 73)(28 77 34 83)(30 75 36 81)(38 58 44 52)(40 56 46 50)(42 54 48 60)(49 68 55 62)(51 66 57 72)(53 64 59 70)(74 85 80 91)(76 95 82 89)(78 93 84 87)
G:=sub<Sym(96)| (1,7)(2,88)(3,9)(4,90)(5,11)(6,92)(8,94)(10,96)(12,86)(13,66)(14,20)(15,68)(16,22)(17,70)(18,24)(19,72)(21,62)(23,64)(25,31)(26,56)(27,33)(28,58)(29,35)(30,60)(32,50)(34,52)(36,54)(37,82)(38,44)(39,84)(40,46)(41,74)(42,48)(43,76)(45,78)(47,80)(49,55)(51,57)(53,59)(61,67)(63,69)(65,71)(73,79)(75,81)(77,83)(85,91)(87,93)(89,95), (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), (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,49)(12,50)(13,78)(14,79)(15,80)(16,81)(17,82)(18,83)(19,84)(20,73)(21,74)(22,75)(23,76)(24,77)(25,91)(26,92)(27,93)(28,94)(29,95)(30,96)(31,85)(32,86)(33,87)(34,88)(35,89)(36,90)(37,70)(38,71)(39,72)(40,61)(41,62)(42,63)(43,64)(44,65)(45,66)(46,67)(47,68)(48,69), (1,93)(2,94)(3,95)(4,96)(5,85)(6,86)(7,87)(8,88)(9,89)(10,90)(11,91)(12,92)(13,72)(14,61)(15,62)(16,63)(17,64)(18,65)(19,66)(20,67)(21,68)(22,69)(23,70)(24,71)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,76)(38,77)(39,78)(40,79)(41,80)(42,81)(43,82)(44,83)(45,84)(46,73)(47,74)(48,75), (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), (1,45,7,39)(2,71,8,65)(3,43,9,37)(4,69,10,63)(5,41,11,47)(6,67,12,61)(13,27,19,33)(14,86,20,92)(15,25,21,31)(16,96,22,90)(17,35,23,29)(18,94,24,88)(26,79,32,73)(28,77,34,83)(30,75,36,81)(38,58,44,52)(40,56,46,50)(42,54,48,60)(49,68,55,62)(51,66,57,72)(53,64,59,70)(74,85,80,91)(76,95,82,89)(78,93,84,87)>;
G:=Group( (1,7)(2,88)(3,9)(4,90)(5,11)(6,92)(8,94)(10,96)(12,86)(13,66)(14,20)(15,68)(16,22)(17,70)(18,24)(19,72)(21,62)(23,64)(25,31)(26,56)(27,33)(28,58)(29,35)(30,60)(32,50)(34,52)(36,54)(37,82)(38,44)(39,84)(40,46)(41,74)(42,48)(43,76)(45,78)(47,80)(49,55)(51,57)(53,59)(61,67)(63,69)(65,71)(73,79)(75,81)(77,83)(85,91)(87,93)(89,95), (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), (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,49)(12,50)(13,78)(14,79)(15,80)(16,81)(17,82)(18,83)(19,84)(20,73)(21,74)(22,75)(23,76)(24,77)(25,91)(26,92)(27,93)(28,94)(29,95)(30,96)(31,85)(32,86)(33,87)(34,88)(35,89)(36,90)(37,70)(38,71)(39,72)(40,61)(41,62)(42,63)(43,64)(44,65)(45,66)(46,67)(47,68)(48,69), (1,93)(2,94)(3,95)(4,96)(5,85)(6,86)(7,87)(8,88)(9,89)(10,90)(11,91)(12,92)(13,72)(14,61)(15,62)(16,63)(17,64)(18,65)(19,66)(20,67)(21,68)(22,69)(23,70)(24,71)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,76)(38,77)(39,78)(40,79)(41,80)(42,81)(43,82)(44,83)(45,84)(46,73)(47,74)(48,75), (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), (1,45,7,39)(2,71,8,65)(3,43,9,37)(4,69,10,63)(5,41,11,47)(6,67,12,61)(13,27,19,33)(14,86,20,92)(15,25,21,31)(16,96,22,90)(17,35,23,29)(18,94,24,88)(26,79,32,73)(28,77,34,83)(30,75,36,81)(38,58,44,52)(40,56,46,50)(42,54,48,60)(49,68,55,62)(51,66,57,72)(53,64,59,70)(74,85,80,91)(76,95,82,89)(78,93,84,87) );
G=PermutationGroup([[(1,7),(2,88),(3,9),(4,90),(5,11),(6,92),(8,94),(10,96),(12,86),(13,66),(14,20),(15,68),(16,22),(17,70),(18,24),(19,72),(21,62),(23,64),(25,31),(26,56),(27,33),(28,58),(29,35),(30,60),(32,50),(34,52),(36,54),(37,82),(38,44),(39,84),(40,46),(41,74),(42,48),(43,76),(45,78),(47,80),(49,55),(51,57),(53,59),(61,67),(63,69),(65,71),(73,79),(75,81),(77,83),(85,91),(87,93),(89,95)], [(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)], [(1,51),(2,52),(3,53),(4,54),(5,55),(6,56),(7,57),(8,58),(9,59),(10,60),(11,49),(12,50),(13,78),(14,79),(15,80),(16,81),(17,82),(18,83),(19,84),(20,73),(21,74),(22,75),(23,76),(24,77),(25,91),(26,92),(27,93),(28,94),(29,95),(30,96),(31,85),(32,86),(33,87),(34,88),(35,89),(36,90),(37,70),(38,71),(39,72),(40,61),(41,62),(42,63),(43,64),(44,65),(45,66),(46,67),(47,68),(48,69)], [(1,93),(2,94),(3,95),(4,96),(5,85),(6,86),(7,87),(8,88),(9,89),(10,90),(11,91),(12,92),(13,72),(14,61),(15,62),(16,63),(17,64),(18,65),(19,66),(20,67),(21,68),(22,69),(23,70),(24,71),(25,49),(26,50),(27,51),(28,52),(29,53),(30,54),(31,55),(32,56),(33,57),(34,58),(35,59),(36,60),(37,76),(38,77),(39,78),(40,79),(41,80),(42,81),(43,82),(44,83),(45,84),(46,73),(47,74),(48,75)], [(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)], [(1,45,7,39),(2,71,8,65),(3,43,9,37),(4,69,10,63),(5,41,11,47),(6,67,12,61),(13,27,19,33),(14,86,20,92),(15,25,21,31),(16,96,22,90),(17,35,23,29),(18,94,24,88),(26,79,32,73),(28,77,34,83),(30,75,36,81),(38,58,44,52),(40,56,46,50),(42,54,48,60),(49,68,55,62),(51,66,57,72),(53,64,59,70),(74,85,80,91),(76,95,82,89),(78,93,84,87)]])
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 | C24.55D6 | C6.C42 | C2×Dic3⋊C4 | C2×C6.D4 | C6×C22⋊C4 | C23×Dic3 | C22×Dic3 | 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 C24.55D6 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 5 | 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 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 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 |
| 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 |
| 8 | 11 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 |
| 12 | 10 | 0 | 0 | 0 | 0 |
| 5 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 0 | 0 | 0 | 0 | 6 | 0 |
G:=sub<GL(6,GF(13))| [12,5,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[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],[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],[8,0,0,0,0,0,11,5,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,6,0,0,0,0,0,0,11],[12,5,0,0,0,0,10,1,0,0,0,0,0,0,0,7,0,0,0,0,11,0,0,0,0,0,0,0,0,6,0,0,0,0,2,0] >;
C24.55D6 in GAP, Magma, Sage, TeX
C_2^4._{55}D_6 % in TeX
G:=Group("C2^4.55D6"); // GroupNames label
G:=SmallGroup(192,501);
// by ID
G=gap.SmallGroup(192,501);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,422,387,58,6278]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=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