direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Q8.14D6, C12.36C24, Dic6.31C23, C4○D4.57D6, C3⋊C8.15C23, (C2×D4).233D6, C12.428(C2×D4), (C2×C12).219D4, C6⋊5(C8.C22), C4.36(S3×C23), (C2×Q8).215D6, D4.S3⋊18C22, C3⋊Q16⋊17C22, D4.24(C22×S3), (C3×D4).24C23, (C22×C6).125D4, (C22×C4).299D6, C6.161(C22×D4), (C3×Q8).24C23, Q8.34(C22×S3), (C2×C12).558C23, (C22×Dic6)⋊21C2, (C2×Dic6)⋊70C22, (C6×D4).273C22, C23.76(C3⋊D4), C4.Dic3⋊38C22, (C6×Q8).238C22, (C22×C12).293C22, C3⋊6(C2×C8.C22), (C2×C6).77(C2×D4), C4.31(C2×C3⋊D4), (C2×D4.S3)⋊31C2, (C6×C4○D4).11C2, (C2×C4○D4).16S3, (C2×C3⋊Q16)⋊31C2, (C2×C4).96(C3⋊D4), (C2×C3⋊C8).184C22, (C2×C4.Dic3)⋊32C2, C2.34(C22×C3⋊D4), (C2×C4).247(C22×S3), (C3×C4○D4).50C22, C22.120(C2×C3⋊D4), SmallGroup(192,1382)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Q8.14D6
G = < a,b,c,d,e | a2=b4=d6=1, c2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=b2c, ece-1=bc, ede-1=d-1 >
Subgroups: 552 in 258 conjugacy classes, 111 normal (27 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, Dic3, C12, C12, C12, C2×C6, C2×C6, C2×C6, C2×C8, M4(2), SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C3⋊C8, Dic6, Dic6, C2×Dic3, C2×C12, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×C6, C22×C6, C2×M4(2), C2×SD16, C2×Q16, C8.C22, C22×Q8, C2×C4○D4, C2×C3⋊C8, C4.Dic3, D4.S3, C3⋊Q16, C2×Dic6, C2×Dic6, C22×Dic3, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4, C3×C4○D4, C2×C8.C22, C2×C4.Dic3, C2×D4.S3, C2×C3⋊Q16, Q8.14D6, C22×Dic6, C6×C4○D4, C2×Q8.14D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C3⋊D4, C22×S3, C8.C22, C22×D4, C2×C3⋊D4, S3×C23, C2×C8.C22, Q8.14D6, C22×C3⋊D4, C2×Q8.14D6
►On 96 points
Generators in S96
(1 27)(2 28)(3 29)(4 30)(5 25)(6 26)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(19 66)(20 61)(21 62)(22 63)(23 64)(24 65)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(37 53)(38 54)(39 49)(40 50)(41 51)(42 52)(55 70)(56 71)(57 72)(58 67)(59 68)(60 69)(73 84)(74 79)(75 80)(76 81)(77 82)(78 83)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)
(1 13 47 63)(2 14 48 64)(3 15 43 65)(4 16 44 66)(5 17 45 61)(6 18 46 62)(7 35 22 27)(8 36 23 28)(9 31 24 29)(10 32 19 30)(11 33 20 25)(12 34 21 26)(37 94 68 73)(38 95 69 74)(39 96 70 75)(40 91 71 76)(41 92 72 77)(42 93 67 78)(49 90 55 80)(50 85 56 81)(51 86 57 82)(52 87 58 83)(53 88 59 84)(54 89 60 79)
(1 32 47 30)(2 25 48 33)(3 34 43 26)(4 27 44 35)(5 36 45 28)(6 29 46 31)(7 16 22 66)(8 61 23 17)(9 18 24 62)(10 63 19 13)(11 14 20 64)(12 65 21 15)(37 85 68 81)(38 82 69 86)(39 87 70 83)(40 84 71 88)(41 89 72 79)(42 80 67 90)(49 93 55 78)(50 73 56 94)(51 95 57 74)(52 75 58 96)(53 91 59 76)(54 77 60 92)
(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 96 47 75)(2 95 48 74)(3 94 43 73)(4 93 44 78)(5 92 45 77)(6 91 46 76)(7 49 22 55)(8 54 23 60)(9 53 24 59)(10 52 19 58)(11 51 20 57)(12 50 21 56)(13 39 63 70)(14 38 64 69)(15 37 65 68)(16 42 66 67)(17 41 61 72)(18 40 62 71)(25 86 33 82)(26 85 34 81)(27 90 35 80)(28 89 36 79)(29 88 31 84)(30 87 32 83)
G:=sub<Sym(96)| (1,27)(2,28)(3,29)(4,30)(5,25)(6,26)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(19,66)(20,61)(21,62)(22,63)(23,64)(24,65)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(37,53)(38,54)(39,49)(40,50)(41,51)(42,52)(55,70)(56,71)(57,72)(58,67)(59,68)(60,69)(73,84)(74,79)(75,80)(76,81)(77,82)(78,83)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96), (1,13,47,63)(2,14,48,64)(3,15,43,65)(4,16,44,66)(5,17,45,61)(6,18,46,62)(7,35,22,27)(8,36,23,28)(9,31,24,29)(10,32,19,30)(11,33,20,25)(12,34,21,26)(37,94,68,73)(38,95,69,74)(39,96,70,75)(40,91,71,76)(41,92,72,77)(42,93,67,78)(49,90,55,80)(50,85,56,81)(51,86,57,82)(52,87,58,83)(53,88,59,84)(54,89,60,79), (1,32,47,30)(2,25,48,33)(3,34,43,26)(4,27,44,35)(5,36,45,28)(6,29,46,31)(7,16,22,66)(8,61,23,17)(9,18,24,62)(10,63,19,13)(11,14,20,64)(12,65,21,15)(37,85,68,81)(38,82,69,86)(39,87,70,83)(40,84,71,88)(41,89,72,79)(42,80,67,90)(49,93,55,78)(50,73,56,94)(51,95,57,74)(52,75,58,96)(53,91,59,76)(54,77,60,92), (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), (1,96,47,75)(2,95,48,74)(3,94,43,73)(4,93,44,78)(5,92,45,77)(6,91,46,76)(7,49,22,55)(8,54,23,60)(9,53,24,59)(10,52,19,58)(11,51,20,57)(12,50,21,56)(13,39,63,70)(14,38,64,69)(15,37,65,68)(16,42,66,67)(17,41,61,72)(18,40,62,71)(25,86,33,82)(26,85,34,81)(27,90,35,80)(28,89,36,79)(29,88,31,84)(30,87,32,83)>;
G:=Group( (1,27)(2,28)(3,29)(4,30)(5,25)(6,26)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(19,66)(20,61)(21,62)(22,63)(23,64)(24,65)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(37,53)(38,54)(39,49)(40,50)(41,51)(42,52)(55,70)(56,71)(57,72)(58,67)(59,68)(60,69)(73,84)(74,79)(75,80)(76,81)(77,82)(78,83)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96), (1,13,47,63)(2,14,48,64)(3,15,43,65)(4,16,44,66)(5,17,45,61)(6,18,46,62)(7,35,22,27)(8,36,23,28)(9,31,24,29)(10,32,19,30)(11,33,20,25)(12,34,21,26)(37,94,68,73)(38,95,69,74)(39,96,70,75)(40,91,71,76)(41,92,72,77)(42,93,67,78)(49,90,55,80)(50,85,56,81)(51,86,57,82)(52,87,58,83)(53,88,59,84)(54,89,60,79), (1,32,47,30)(2,25,48,33)(3,34,43,26)(4,27,44,35)(5,36,45,28)(6,29,46,31)(7,16,22,66)(8,61,23,17)(9,18,24,62)(10,63,19,13)(11,14,20,64)(12,65,21,15)(37,85,68,81)(38,82,69,86)(39,87,70,83)(40,84,71,88)(41,89,72,79)(42,80,67,90)(49,93,55,78)(50,73,56,94)(51,95,57,74)(52,75,58,96)(53,91,59,76)(54,77,60,92), (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), (1,96,47,75)(2,95,48,74)(3,94,43,73)(4,93,44,78)(5,92,45,77)(6,91,46,76)(7,49,22,55)(8,54,23,60)(9,53,24,59)(10,52,19,58)(11,51,20,57)(12,50,21,56)(13,39,63,70)(14,38,64,69)(15,37,65,68)(16,42,66,67)(17,41,61,72)(18,40,62,71)(25,86,33,82)(26,85,34,81)(27,90,35,80)(28,89,36,79)(29,88,31,84)(30,87,32,83) );
G=PermutationGroup([[(1,27),(2,28),(3,29),(4,30),(5,25),(6,26),(7,13),(8,14),(9,15),(10,16),(11,17),(12,18),(19,66),(20,61),(21,62),(22,63),(23,64),(24,65),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48),(37,53),(38,54),(39,49),(40,50),(41,51),(42,52),(55,70),(56,71),(57,72),(58,67),(59,68),(60,69),(73,84),(74,79),(75,80),(76,81),(77,82),(78,83),(85,91),(86,92),(87,93),(88,94),(89,95),(90,96)], [(1,13,47,63),(2,14,48,64),(3,15,43,65),(4,16,44,66),(5,17,45,61),(6,18,46,62),(7,35,22,27),(8,36,23,28),(9,31,24,29),(10,32,19,30),(11,33,20,25),(12,34,21,26),(37,94,68,73),(38,95,69,74),(39,96,70,75),(40,91,71,76),(41,92,72,77),(42,93,67,78),(49,90,55,80),(50,85,56,81),(51,86,57,82),(52,87,58,83),(53,88,59,84),(54,89,60,79)], [(1,32,47,30),(2,25,48,33),(3,34,43,26),(4,27,44,35),(5,36,45,28),(6,29,46,31),(7,16,22,66),(8,61,23,17),(9,18,24,62),(10,63,19,13),(11,14,20,64),(12,65,21,15),(37,85,68,81),(38,82,69,86),(39,87,70,83),(40,84,71,88),(41,89,72,79),(42,80,67,90),(49,93,55,78),(50,73,56,94),(51,95,57,74),(52,75,58,96),(53,91,59,76),(54,77,60,92)], [(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)], [(1,96,47,75),(2,95,48,74),(3,94,43,73),(4,93,44,78),(5,92,45,77),(6,91,46,76),(7,49,22,55),(8,54,23,60),(9,53,24,59),(10,52,19,58),(11,51,20,57),(12,50,21,56),(13,39,63,70),(14,38,64,69),(15,37,65,68),(16,42,66,67),(17,41,61,72),(18,40,62,71),(25,86,33,82),(26,85,34,81),(27,90,35,80),(28,89,36,79),(29,88,31,84),(30,87,32,83)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 6D | ··· | 6I | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 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 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | D6 | C3⋊D4 | C3⋊D4 | C8.C22 | Q8.14D6 |
| kernel | C2×Q8.14D6 | C2×C4.Dic3 | C2×D4.S3 | C2×C3⋊Q16 | Q8.14D6 | C22×Dic6 | C6×C4○D4 | C2×C4○D4 | C2×C12 | C22×C6 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 3 | 1 | 1 | 1 | 1 | 4 | 6 | 2 | 2 | 4 |
Matrix representation of C2×Q8.14D6 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 45 | 3 | 0 | 0 | 0 | 0 |
| 31 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 55 | 71 | 71 |
| 0 | 0 | 55 | 55 | 71 | 2 |
| 0 | 0 | 71 | 71 | 18 | 55 |
| 0 | 0 | 71 | 2 | 55 | 55 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,72,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,72,0,0],[0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,72,0,0,0,0,0,0,72,0,0],[45,31,0,0,0,0,3,28,0,0,0,0,0,0,18,55,71,71,0,0,55,55,71,2,0,0,71,71,18,55,0,0,71,2,55,55] >;
C2×Q8.14D6 in GAP, Magma, Sage, TeX
C_2\times Q_8._{14}D_6 % in TeX
G:=Group("C2xQ8.14D6"); // GroupNames label
G:=SmallGroup(192,1382);
// by ID
G=gap.SmallGroup(192,1382);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,675,297,1684,235,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=d^6=1,c^2=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=b^2*c,e*c*e^-1=b*c,e*d*e^-1=d^-1>;
// generators/relations