metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2.D8⋊7S3, D6⋊C8.8C2, C4⋊C4.53D6, (C2×C8).30D6, C6.30(C4○D8), C4.D12.9C2, C12.43(C4○D4), C4.83(C4○D12), C6.SD16⋊22C2, C2.Dic12⋊15C2, C12.Q8⋊22C2, (C22×S3).30D4, C22.234(S3×D4), C2.15(D8⋊3S3), (C2×C24).172C22, (C2×C12).304C23, C4.31(Q8⋊3S3), (C2×Dic3).169D4, C2.24(Q16⋊S3), C6.72(C8.C22), C3⋊5(C23.20D4), C2.18(D6.D4), C4⋊Dic3.127C22, (C2×Dic6).92C22, C6.48(C22.D4), (C3×C2.D8)⋊14C2, C4⋊C4⋊7S3.9C2, (C2×C6).309(C2×D4), (C2×C3⋊C8).73C22, (S3×C2×C4).42C22, (C3×C4⋊C4).97C22, (C2×C4).407(C22×S3), SmallGroup(192,447)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2.D8⋊7S3
G = < a,b,c,d,e | a2=b8=d3=e2=1, c2=a, ebe=ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, cd=dc, ece=ab4c, ede=d-1 >
Subgroups: 272 in 96 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×Q8, C3⋊C8, C24, Dic6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22⋊C8, Q8⋊C4, C4.Q8, C2.D8, C42⋊C2, C22⋊Q8, C2×C3⋊C8, C4×Dic3, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×C24, C2×Dic6, S3×C2×C4, C23.20D4, C12.Q8, C6.SD16, C2.Dic12, D6⋊C8, C3×C2.D8, C4⋊C4⋊7S3, C4.D12, C2.D8⋊7S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C22×S3, C22.D4, C4○D8, C8.C22, C4○D12, S3×D4, Q8⋊3S3, C23.20D4, D6.D4, D8⋊3S3, Q16⋊S3, C2.D8⋊7S3
►On 96 points
Generators in S96
(1 77)(2 78)(3 79)(4 80)(5 73)(6 74)(7 75)(8 76)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 81)(18 82)(19 83)(20 84)(21 85)(22 86)(23 87)(24 88)(25 94)(26 95)(27 96)(28 89)(29 90)(30 91)(31 92)(32 93)(33 69)(34 70)(35 71)(36 72)(37 65)(38 66)(39 67)(40 68)(41 49)(42 50)(43 51)(44 52)(45 53)(46 54)(47 55)(48 56)
(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 36 77 72)(2 35 78 71)(3 34 79 70)(4 33 80 69)(5 40 73 68)(6 39 74 67)(7 38 75 66)(8 37 76 65)(9 50 57 42)(10 49 58 41)(11 56 59 48)(12 55 60 47)(13 54 61 46)(14 53 62 45)(15 52 63 44)(16 51 64 43)(17 28 81 89)(18 27 82 96)(19 26 83 95)(20 25 84 94)(21 32 85 93)(22 31 86 92)(23 30 87 91)(24 29 88 90)
(1 49 18)(2 50 19)(3 51 20)(4 52 21)(5 53 22)(6 54 23)(7 55 24)(8 56 17)(9 95 71)(10 96 72)(11 89 65)(12 90 66)(13 91 67)(14 92 68)(15 93 69)(16 94 70)(25 34 64)(26 35 57)(27 36 58)(28 37 59)(29 38 60)(30 39 61)(31 40 62)(32 33 63)(41 82 77)(42 83 78)(43 84 79)(44 85 80)(45 86 73)(46 87 74)(47 88 75)(48 81 76)
(2 78)(4 80)(6 74)(8 76)(9 91)(10 31)(11 93)(12 25)(13 95)(14 27)(15 89)(16 29)(17 48)(18 49)(19 42)(20 51)(21 44)(22 53)(23 46)(24 55)(26 61)(28 63)(30 57)(32 59)(33 37)(34 66)(35 39)(36 68)(38 70)(40 72)(41 82)(43 84)(45 86)(47 88)(50 83)(52 85)(54 87)(56 81)(58 92)(60 94)(62 96)(64 90)(65 69)(67 71)
G:=sub<Sym(96)| (1,77)(2,78)(3,79)(4,80)(5,73)(6,74)(7,75)(8,76)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,81)(18,82)(19,83)(20,84)(21,85)(22,86)(23,87)(24,88)(25,94)(26,95)(27,96)(28,89)(29,90)(30,91)(31,92)(32,93)(33,69)(34,70)(35,71)(36,72)(37,65)(38,66)(39,67)(40,68)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56), (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), (1,36,77,72)(2,35,78,71)(3,34,79,70)(4,33,80,69)(5,40,73,68)(6,39,74,67)(7,38,75,66)(8,37,76,65)(9,50,57,42)(10,49,58,41)(11,56,59,48)(12,55,60,47)(13,54,61,46)(14,53,62,45)(15,52,63,44)(16,51,64,43)(17,28,81,89)(18,27,82,96)(19,26,83,95)(20,25,84,94)(21,32,85,93)(22,31,86,92)(23,30,87,91)(24,29,88,90), (1,49,18)(2,50,19)(3,51,20)(4,52,21)(5,53,22)(6,54,23)(7,55,24)(8,56,17)(9,95,71)(10,96,72)(11,89,65)(12,90,66)(13,91,67)(14,92,68)(15,93,69)(16,94,70)(25,34,64)(26,35,57)(27,36,58)(28,37,59)(29,38,60)(30,39,61)(31,40,62)(32,33,63)(41,82,77)(42,83,78)(43,84,79)(44,85,80)(45,86,73)(46,87,74)(47,88,75)(48,81,76), (2,78)(4,80)(6,74)(8,76)(9,91)(10,31)(11,93)(12,25)(13,95)(14,27)(15,89)(16,29)(17,48)(18,49)(19,42)(20,51)(21,44)(22,53)(23,46)(24,55)(26,61)(28,63)(30,57)(32,59)(33,37)(34,66)(35,39)(36,68)(38,70)(40,72)(41,82)(43,84)(45,86)(47,88)(50,83)(52,85)(54,87)(56,81)(58,92)(60,94)(62,96)(64,90)(65,69)(67,71)>;
G:=Group( (1,77)(2,78)(3,79)(4,80)(5,73)(6,74)(7,75)(8,76)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,81)(18,82)(19,83)(20,84)(21,85)(22,86)(23,87)(24,88)(25,94)(26,95)(27,96)(28,89)(29,90)(30,91)(31,92)(32,93)(33,69)(34,70)(35,71)(36,72)(37,65)(38,66)(39,67)(40,68)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56), (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), (1,36,77,72)(2,35,78,71)(3,34,79,70)(4,33,80,69)(5,40,73,68)(6,39,74,67)(7,38,75,66)(8,37,76,65)(9,50,57,42)(10,49,58,41)(11,56,59,48)(12,55,60,47)(13,54,61,46)(14,53,62,45)(15,52,63,44)(16,51,64,43)(17,28,81,89)(18,27,82,96)(19,26,83,95)(20,25,84,94)(21,32,85,93)(22,31,86,92)(23,30,87,91)(24,29,88,90), (1,49,18)(2,50,19)(3,51,20)(4,52,21)(5,53,22)(6,54,23)(7,55,24)(8,56,17)(9,95,71)(10,96,72)(11,89,65)(12,90,66)(13,91,67)(14,92,68)(15,93,69)(16,94,70)(25,34,64)(26,35,57)(27,36,58)(28,37,59)(29,38,60)(30,39,61)(31,40,62)(32,33,63)(41,82,77)(42,83,78)(43,84,79)(44,85,80)(45,86,73)(46,87,74)(47,88,75)(48,81,76), (2,78)(4,80)(6,74)(8,76)(9,91)(10,31)(11,93)(12,25)(13,95)(14,27)(15,89)(16,29)(17,48)(18,49)(19,42)(20,51)(21,44)(22,53)(23,46)(24,55)(26,61)(28,63)(30,57)(32,59)(33,37)(34,66)(35,39)(36,68)(38,70)(40,72)(41,82)(43,84)(45,86)(47,88)(50,83)(52,85)(54,87)(56,81)(58,92)(60,94)(62,96)(64,90)(65,69)(67,71) );
G=PermutationGroup([[(1,77),(2,78),(3,79),(4,80),(5,73),(6,74),(7,75),(8,76),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,81),(18,82),(19,83),(20,84),(21,85),(22,86),(23,87),(24,88),(25,94),(26,95),(27,96),(28,89),(29,90),(30,91),(31,92),(32,93),(33,69),(34,70),(35,71),(36,72),(37,65),(38,66),(39,67),(40,68),(41,49),(42,50),(43,51),(44,52),(45,53),(46,54),(47,55),(48,56)], [(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)], [(1,36,77,72),(2,35,78,71),(3,34,79,70),(4,33,80,69),(5,40,73,68),(6,39,74,67),(7,38,75,66),(8,37,76,65),(9,50,57,42),(10,49,58,41),(11,56,59,48),(12,55,60,47),(13,54,61,46),(14,53,62,45),(15,52,63,44),(16,51,64,43),(17,28,81,89),(18,27,82,96),(19,26,83,95),(20,25,84,94),(21,32,85,93),(22,31,86,92),(23,30,87,91),(24,29,88,90)], [(1,49,18),(2,50,19),(3,51,20),(4,52,21),(5,53,22),(6,54,23),(7,55,24),(8,56,17),(9,95,71),(10,96,72),(11,89,65),(12,90,66),(13,91,67),(14,92,68),(15,93,69),(16,94,70),(25,34,64),(26,35,57),(27,36,58),(28,37,59),(29,38,60),(30,39,61),(31,40,62),(32,33,63),(41,82,77),(42,83,78),(43,84,79),(44,85,80),(45,86,73),(46,87,74),(47,88,75),(48,81,76)], [(2,78),(4,80),(6,74),(8,76),(9,91),(10,31),(11,93),(12,25),(13,95),(14,27),(15,89),(16,29),(17,48),(18,49),(19,42),(20,51),(21,44),(22,53),(23,46),(24,55),(26,61),(28,63),(30,57),(32,59),(33,37),(34,66),(35,39),(36,68),(38,70),(40,72),(41,82),(43,84),(45,86),(47,88),(50,83),(52,85),(54,87),(56,81),(58,92),(60,94),(62,96),(64,90),(65,69),(67,71)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 12 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 8 | 12 | 12 | 24 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | C4○D4 | C4○D8 | C4○D12 | C8.C22 | Q8⋊3S3 | S3×D4 | D8⋊3S3 | Q16⋊S3 |
| kernel | C2.D8⋊7S3 | C12.Q8 | C6.SD16 | C2.Dic12 | D6⋊C8 | C3×C2.D8 | C4⋊C4⋊7S3 | C4.D12 | C2.D8 | C2×Dic3 | C22×S3 | C4⋊C4 | C2×C8 | C12 | C6 | C4 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 4 | 4 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of C2.D8⋊7S3 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 0 |
| 0 | 0 | 0 | 0 | 2 | 10 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 27 | 0 | 0 |
| 0 | 0 | 27 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 52 | 53 |
| 0 | 0 | 0 | 0 | 22 | 21 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 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 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 49 | 72 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,22,2,0,0,0,0,0,10],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,27,0,0,0,0,27,0,0,0,0,0,0,0,52,22,0,0,0,0,53,21],[72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,49,0,0,0,0,0,72] >;
C2.D8⋊7S3 in GAP, Magma, Sage, TeX
C_2.D_8\rtimes_7S_3
% in TeX
G:=Group("C2.D8:7S3"); // GroupNames label
G:=SmallGroup(192,447);
// by ID
G=gap.SmallGroup(192,447);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,64,254,219,268,851,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^8=d^3=e^2=1,c^2=a,e*b*e=a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e=a*b^4*c,e*d*e=d^-1>;
// generators/relations