metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.17D6, C23.6Dic6, (C2×C12).49D4, (C22×C4).44D6, C2.6(D6⋊3D4), (C22×C6).12Q8, C6.29(C4⋊D4), (C2×Dic3).54D4, C22.238(S3×D4), C6.56(C22⋊Q8), C2.32(Dic3⋊D4), C3⋊2(C23.Q8), C6.C42⋊13C2, (C23×C6).31C22, C22.45(C2×Dic6), C6.13(C42⋊2C2), C2.8(C23.14D6), C2.7(C12.48D4), C22.95(C4○D12), (C22×C12).57C22, (C22×C6).323C23, C23.377(C22×S3), C22.93(D4⋊2S3), C2.12(C23.8D6), C2.21(Dic3.D4), (C22×Dic3).39C22, (C2×C4⋊Dic3)⋊8C2, (C2×C6).33(C2×Q8), (C2×C6).317(C2×D4), (C2×Dic3⋊C4)⋊20C2, (C2×C6).77(C4○D4), (C2×C4).28(C3⋊D4), (C2×C22⋊C4).11S3, (C6×C22⋊C4).13C2, C22.123(C2×C3⋊D4), (C2×C6.D4).10C2, SmallGroup(192,507)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.17D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=f2=dc=cd, 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: 472 in 186 conjugacy classes, 63 normal (51 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, C23, C23, Dic3, C12, C2×C6, C2×C6, 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, Dic3⋊C4, C4⋊Dic3, C6.D4, C3×C22⋊C4, C22×Dic3, C22×C12, C23×C6, C23.Q8, C6.C42, C2×Dic3⋊C4, C2×C4⋊Dic3, C2×C6.D4, C6×C22⋊C4, C24.17D6
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, Dic6, C3⋊D4, C22×S3, C4⋊D4, C22⋊Q8, C42⋊2C2, C2×Dic6, C4○D12, S3×D4, D4⋊2S3, C2×C3⋊D4, C23.Q8, Dic3.D4, C23.8D6, Dic3⋊D4, C12.48D4, D6⋊3D4, C23.14D6, C24.17D6
►On 96 points
Generators in S96
(2 55)(4 57)(6 59)(8 49)(10 51)(12 53)(13 73)(14 66)(15 75)(16 68)(17 77)(18 70)(19 79)(20 72)(21 81)(22 62)(23 83)(24 64)(26 85)(28 87)(30 89)(32 91)(34 93)(36 95)(37 67)(38 76)(39 69)(40 78)(41 71)(42 80)(43 61)(44 82)(45 63)(46 84)(47 65)(48 74)
(1 92)(2 93)(3 94)(4 95)(5 96)(6 85)(7 86)(8 87)(9 88)(10 89)(11 90)(12 91)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 79)(20 80)(21 81)(22 82)(23 83)(24 84)(25 58)(26 59)(27 60)(28 49)(29 50)(30 51)(31 52)(32 53)(33 54)(34 55)(35 56)(36 57)(37 67)(38 68)(39 69)(40 70)(41 71)(42 72)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)
(1 60)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 55)(9 56)(10 57)(11 58)(12 59)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 37)(22 38)(23 39)(24 40)(25 90)(26 91)(27 92)(28 93)(29 94)(30 95)(31 96)(32 85)(33 86)(34 87)(35 88)(36 89)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(71 73)(72 74)
(1 54)(2 55)(3 56)(4 57)(5 58)(6 59)(7 60)(8 49)(9 50)(10 51)(11 52)(12 53)(13 47)(14 48)(15 37)(16 38)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)(23 45)(24 46)(25 96)(26 85)(27 86)(28 87)(29 88)(30 89)(31 90)(32 91)(33 92)(34 93)(35 94)(36 95)(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 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 44 7 38)(2 21 8 15)(3 42 9 48)(4 19 10 13)(5 40 11 46)(6 17 12 23)(14 56 20 50)(16 54 22 60)(18 52 24 58)(25 78 31 84)(26 69 32 63)(27 76 33 82)(28 67 34 61)(29 74 35 80)(30 65 36 71)(37 55 43 49)(39 53 45 59)(41 51 47 57)(62 86 68 92)(64 96 70 90)(66 94 72 88)(73 95 79 89)(75 93 81 87)(77 91 83 85)
G:=sub<Sym(96)| (2,55)(4,57)(6,59)(8,49)(10,51)(12,53)(13,73)(14,66)(15,75)(16,68)(17,77)(18,70)(19,79)(20,72)(21,81)(22,62)(23,83)(24,64)(26,85)(28,87)(30,89)(32,91)(34,93)(36,95)(37,67)(38,76)(39,69)(40,78)(41,71)(42,80)(43,61)(44,82)(45,63)(46,84)(47,65)(48,74), (1,92)(2,93)(3,94)(4,95)(5,96)(6,85)(7,86)(8,87)(9,88)(10,89)(11,90)(12,91)(13,73)(14,74)(15,75)(16,76)(17,77)(18,78)(19,79)(20,80)(21,81)(22,82)(23,83)(24,84)(25,58)(26,59)(27,60)(28,49)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,57)(37,67)(38,68)(39,69)(40,70)(41,71)(42,72)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66), (1,60)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,57)(11,58)(12,59)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,37)(22,38)(23,39)(24,40)(25,90)(26,91)(27,92)(28,93)(29,94)(30,95)(31,96)(32,85)(33,86)(34,87)(35,88)(36,89)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(71,73)(72,74), (1,54)(2,55)(3,56)(4,57)(5,58)(6,59)(7,60)(8,49)(9,50)(10,51)(11,52)(12,53)(13,47)(14,48)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(23,45)(24,46)(25,96)(26,85)(27,86)(28,87)(29,88)(30,89)(31,90)(32,91)(33,92)(34,93)(35,94)(36,95)(61,81)(62,82)(63,83)(64,84)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(71,79)(72,80), (1,2,3,4,5,6,7,8,9,10,11,12)(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), (1,44,7,38)(2,21,8,15)(3,42,9,48)(4,19,10,13)(5,40,11,46)(6,17,12,23)(14,56,20,50)(16,54,22,60)(18,52,24,58)(25,78,31,84)(26,69,32,63)(27,76,33,82)(28,67,34,61)(29,74,35,80)(30,65,36,71)(37,55,43,49)(39,53,45,59)(41,51,47,57)(62,86,68,92)(64,96,70,90)(66,94,72,88)(73,95,79,89)(75,93,81,87)(77,91,83,85)>;
G:=Group( (2,55)(4,57)(6,59)(8,49)(10,51)(12,53)(13,73)(14,66)(15,75)(16,68)(17,77)(18,70)(19,79)(20,72)(21,81)(22,62)(23,83)(24,64)(26,85)(28,87)(30,89)(32,91)(34,93)(36,95)(37,67)(38,76)(39,69)(40,78)(41,71)(42,80)(43,61)(44,82)(45,63)(46,84)(47,65)(48,74), (1,92)(2,93)(3,94)(4,95)(5,96)(6,85)(7,86)(8,87)(9,88)(10,89)(11,90)(12,91)(13,73)(14,74)(15,75)(16,76)(17,77)(18,78)(19,79)(20,80)(21,81)(22,82)(23,83)(24,84)(25,58)(26,59)(27,60)(28,49)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,57)(37,67)(38,68)(39,69)(40,70)(41,71)(42,72)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66), (1,60)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,57)(11,58)(12,59)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,37)(22,38)(23,39)(24,40)(25,90)(26,91)(27,92)(28,93)(29,94)(30,95)(31,96)(32,85)(33,86)(34,87)(35,88)(36,89)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(71,73)(72,74), (1,54)(2,55)(3,56)(4,57)(5,58)(6,59)(7,60)(8,49)(9,50)(10,51)(11,52)(12,53)(13,47)(14,48)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(23,45)(24,46)(25,96)(26,85)(27,86)(28,87)(29,88)(30,89)(31,90)(32,91)(33,92)(34,93)(35,94)(36,95)(61,81)(62,82)(63,83)(64,84)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(71,79)(72,80), (1,2,3,4,5,6,7,8,9,10,11,12)(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), (1,44,7,38)(2,21,8,15)(3,42,9,48)(4,19,10,13)(5,40,11,46)(6,17,12,23)(14,56,20,50)(16,54,22,60)(18,52,24,58)(25,78,31,84)(26,69,32,63)(27,76,33,82)(28,67,34,61)(29,74,35,80)(30,65,36,71)(37,55,43,49)(39,53,45,59)(41,51,47,57)(62,86,68,92)(64,96,70,90)(66,94,72,88)(73,95,79,89)(75,93,81,87)(77,91,83,85) );
G=PermutationGroup([[(2,55),(4,57),(6,59),(8,49),(10,51),(12,53),(13,73),(14,66),(15,75),(16,68),(17,77),(18,70),(19,79),(20,72),(21,81),(22,62),(23,83),(24,64),(26,85),(28,87),(30,89),(32,91),(34,93),(36,95),(37,67),(38,76),(39,69),(40,78),(41,71),(42,80),(43,61),(44,82),(45,63),(46,84),(47,65),(48,74)], [(1,92),(2,93),(3,94),(4,95),(5,96),(6,85),(7,86),(8,87),(9,88),(10,89),(11,90),(12,91),(13,73),(14,74),(15,75),(16,76),(17,77),(18,78),(19,79),(20,80),(21,81),(22,82),(23,83),(24,84),(25,58),(26,59),(27,60),(28,49),(29,50),(30,51),(31,52),(32,53),(33,54),(34,55),(35,56),(36,57),(37,67),(38,68),(39,69),(40,70),(41,71),(42,72),(43,61),(44,62),(45,63),(46,64),(47,65),(48,66)], [(1,60),(2,49),(3,50),(4,51),(5,52),(6,53),(7,54),(8,55),(9,56),(10,57),(11,58),(12,59),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,37),(22,38),(23,39),(24,40),(25,90),(26,91),(27,92),(28,93),(29,94),(30,95),(31,96),(32,85),(33,86),(34,87),(35,88),(36,89),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,81),(68,82),(69,83),(70,84),(71,73),(72,74)], [(1,54),(2,55),(3,56),(4,57),(5,58),(6,59),(7,60),(8,49),(9,50),(10,51),(11,52),(12,53),(13,47),(14,48),(15,37),(16,38),(17,39),(18,40),(19,41),(20,42),(21,43),(22,44),(23,45),(24,46),(25,96),(26,85),(27,86),(28,87),(29,88),(30,89),(31,90),(32,91),(33,92),(34,93),(35,94),(36,95),(61,81),(62,82),(63,83),(64,84),(65,73),(66,74),(67,75),(68,76),(69,77),(70,78),(71,79),(72,80)], [(1,2,3,4,5,6,7,8,9,10,11,12),(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)], [(1,44,7,38),(2,21,8,15),(3,42,9,48),(4,19,10,13),(5,40,11,46),(6,17,12,23),(14,56,20,50),(16,54,22,60),(18,52,24,58),(25,78,31,84),(26,69,32,63),(27,76,33,82),(28,67,34,61),(29,74,35,80),(30,65,36,71),(37,55,43,49),(39,53,45,59),(41,51,47,57),(62,86,68,92),(64,96,70,90),(66,94,72,88),(73,95,79,89),(75,93,81,87),(77,91,83,85)]])
42 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 6A | ··· | 6G | 6H | 6I | 6J | 6K | 12A | ··· | 12H |
| order | 1 | 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 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | Q8 | D6 | D6 | C4○D4 | C3⋊D4 | Dic6 | C4○D12 | S3×D4 | D4⋊2S3 |
| kernel | C24.17D6 | C6.C42 | C2×Dic3⋊C4 | C2×C4⋊Dic3 | C2×C6.D4 | C6×C22⋊C4 | C2×C22⋊C4 | C2×Dic3 | C2×C12 | C22×C6 | C22×C4 | C24 | C2×C6 | C2×C4 | C23 | C22 | C22 | C22 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 2 | 2 | 2 | 1 | 6 | 4 | 4 | 4 | 2 | 2 |
Matrix representation of C24.17D6 ►in GL6(𝔽13)
| 1 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 9 | 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 |
| 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 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 0 | 1 | 4 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 8 |
| 0 | 0 | 0 | 0 | 12 | 10 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,9,0,0,0,0,0,12],[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],[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],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,10,1,0,0,0,0,0,4],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,3,12,0,0,0,0,8,10] >;
C24.17D6 in GAP, Magma, Sage, TeX
C_2^4._{17}D_6 % in TeX
G:=Group("C2^4.17D6"); // GroupNames label
G:=SmallGroup(192,507);
// by ID
G=gap.SmallGroup(192,507);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,344,254,387,6278]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=f^2=d*c=c*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