direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D6⋊3Q8, D6⋊4(C2×Q8), (C2×Q8)⋊31D6, C6⋊5(C22⋊Q8), (C22×S3)⋊7Q8, C12.258(C2×D4), (C2×C12).214D4, (C22×Q8)⋊12S3, (C6×Q8)⋊35C22, C22.37(S3×Q8), C6.53(C22×Q8), (C2×C6).305C24, C4⋊Dic3⋊79C22, C6.153(C22×D4), (C22×C4).400D6, (C2×C12).552C23, Dic3⋊C4⋊75C22, D6⋊C4.157C22, (C22×C6).423C23, C23.351(C22×S3), C22.316(S3×C23), (C22×S3).241C23, (S3×C23).114C22, (C22×C12).285C22, C22.39(Q8⋊3S3), (C2×Dic3).157C23, (C22×Dic3).164C22, (Q8×C2×C6)⋊4C2, C3⋊6(C2×C22⋊Q8), C2.35(C2×S3×Q8), C4.98(C2×C3⋊D4), (S3×C22×C4).9C2, (C2×C6).98(C2×Q8), (C2×D6⋊C4).27C2, (C2×C4⋊Dic3)⋊46C2, C6.127(C2×C4○D4), (C2×C6).588(C2×D4), (C2×Dic3⋊C4)⋊49C2, (S3×C2×C4).262C22, C2.34(C2×Q8⋊3S3), C2.26(C22×C3⋊D4), (C2×C6).200(C4○D4), (C2×C4).202(C3⋊D4), (C2×C4).242(C22×S3), C22.116(C2×C3⋊D4), SmallGroup(192,1372)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 744 in 322 conjugacy classes, 135 normal (21 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C3, C4 [×4], C4 [×10], C22, C22 [×6], C22 [×16], S3 [×4], C6 [×3], C6 [×4], C2×C4 [×10], C2×C4 [×24], Q8 [×8], C23, C23 [×10], Dic3 [×6], C12 [×4], C12 [×4], D6 [×4], D6 [×12], C2×C6, C2×C6 [×6], C22⋊C4 [×8], C4⋊C4 [×12], C22×C4, C22×C4 [×2], C22×C4 [×11], C2×Q8 [×4], C2×Q8 [×4], C24, C4×S3 [×8], C2×Dic3 [×6], C2×Dic3 [×6], C2×C12 [×10], C2×C12 [×4], C3×Q8 [×8], C22×S3 [×6], C22×S3 [×4], C22×C6, C2×C22⋊C4 [×2], C2×C4⋊C4 [×3], C22⋊Q8 [×8], C23×C4, C22×Q8, Dic3⋊C4 [×8], C4⋊Dic3 [×4], D6⋊C4 [×8], S3×C2×C4 [×4], S3×C2×C4 [×4], C22×Dic3, C22×Dic3 [×2], C22×C12, C22×C12 [×2], C6×Q8 [×4], C6×Q8 [×4], S3×C23, C2×C22⋊Q8, C2×Dic3⋊C4 [×2], C2×C4⋊Dic3, C2×D6⋊C4 [×2], D6⋊3Q8 [×8], S3×C22×C4, Q8×C2×C6, C2×D6⋊3Q8
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], Q8 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, C3⋊D4 [×4], C22×S3 [×7], C22⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4, S3×Q8 [×2], Q8⋊3S3 [×2], C2×C3⋊D4 [×6], S3×C23, C2×C22⋊Q8, D6⋊3Q8 [×4], C2×S3×Q8, C2×Q8⋊3S3, C22×C3⋊D4, C2×D6⋊3Q8
Generators and relations
G = < a,b,c,d,e | a2=b6=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=b3c, ce=ec, ede-1=d-1 >
►On 96 points
(1 64)(2 65)(3 66)(4 61)(5 62)(6 63)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 59)(14 60)(15 55)(16 56)(17 57)(18 58)(19 75)(20 76)(21 77)(22 78)(23 73)(24 74)(25 71)(26 72)(27 67)(28 68)(29 69)(30 70)(31 87)(32 88)(33 89)(34 90)(35 85)(36 86)(37 83)(38 84)(39 79)(40 80)(41 81)(42 82)(49 95)(50 96)(51 91)(52 92)(53 93)(54 94)
(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 63)(2 62)(3 61)(4 66)(5 65)(6 64)(7 47)(8 46)(9 45)(10 44)(11 43)(12 48)(13 58)(14 57)(15 56)(16 55)(17 60)(18 59)(19 73)(20 78)(21 77)(22 76)(23 75)(24 74)(25 67)(26 72)(27 71)(28 70)(29 69)(30 68)(31 88)(32 87)(33 86)(34 85)(35 90)(36 89)(37 82)(38 81)(39 80)(40 79)(41 84)(42 83)(49 91)(50 96)(51 95)(52 94)(53 93)(54 92)
(1 28 16 20)(2 29 17 21)(3 30 18 22)(4 25 13 23)(5 26 14 24)(6 27 15 19)(7 79 91 87)(8 80 92 88)(9 81 93 89)(10 82 94 90)(11 83 95 85)(12 84 96 86)(31 43 39 51)(32 44 40 52)(33 45 41 53)(34 46 42 54)(35 47 37 49)(36 48 38 50)(55 75 63 67)(56 76 64 68)(57 77 65 69)(58 78 66 70)(59 73 61 71)(60 74 62 72)
(1 40 16 32)(2 41 17 33)(3 42 18 34)(4 37 13 35)(5 38 14 36)(6 39 15 31)(7 75 91 67)(8 76 92 68)(9 77 93 69)(10 78 94 70)(11 73 95 71)(12 74 96 72)(19 51 27 43)(20 52 28 44)(21 53 29 45)(22 54 30 46)(23 49 25 47)(24 50 26 48)(55 87 63 79)(56 88 64 80)(57 89 65 81)(58 90 66 82)(59 85 61 83)(60 86 62 84)
G:=sub<Sym(96)| (1,64)(2,65)(3,66)(4,61)(5,62)(6,63)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,59)(14,60)(15,55)(16,56)(17,57)(18,58)(19,75)(20,76)(21,77)(22,78)(23,73)(24,74)(25,71)(26,72)(27,67)(28,68)(29,69)(30,70)(31,87)(32,88)(33,89)(34,90)(35,85)(36,86)(37,83)(38,84)(39,79)(40,80)(41,81)(42,82)(49,95)(50,96)(51,91)(52,92)(53,93)(54,94), (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), (1,63)(2,62)(3,61)(4,66)(5,65)(6,64)(7,47)(8,46)(9,45)(10,44)(11,43)(12,48)(13,58)(14,57)(15,56)(16,55)(17,60)(18,59)(19,73)(20,78)(21,77)(22,76)(23,75)(24,74)(25,67)(26,72)(27,71)(28,70)(29,69)(30,68)(31,88)(32,87)(33,86)(34,85)(35,90)(36,89)(37,82)(38,81)(39,80)(40,79)(41,84)(42,83)(49,91)(50,96)(51,95)(52,94)(53,93)(54,92), (1,28,16,20)(2,29,17,21)(3,30,18,22)(4,25,13,23)(5,26,14,24)(6,27,15,19)(7,79,91,87)(8,80,92,88)(9,81,93,89)(10,82,94,90)(11,83,95,85)(12,84,96,86)(31,43,39,51)(32,44,40,52)(33,45,41,53)(34,46,42,54)(35,47,37,49)(36,48,38,50)(55,75,63,67)(56,76,64,68)(57,77,65,69)(58,78,66,70)(59,73,61,71)(60,74,62,72), (1,40,16,32)(2,41,17,33)(3,42,18,34)(4,37,13,35)(5,38,14,36)(6,39,15,31)(7,75,91,67)(8,76,92,68)(9,77,93,69)(10,78,94,70)(11,73,95,71)(12,74,96,72)(19,51,27,43)(20,52,28,44)(21,53,29,45)(22,54,30,46)(23,49,25,47)(24,50,26,48)(55,87,63,79)(56,88,64,80)(57,89,65,81)(58,90,66,82)(59,85,61,83)(60,86,62,84)>;
G:=Group( (1,64)(2,65)(3,66)(4,61)(5,62)(6,63)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,59)(14,60)(15,55)(16,56)(17,57)(18,58)(19,75)(20,76)(21,77)(22,78)(23,73)(24,74)(25,71)(26,72)(27,67)(28,68)(29,69)(30,70)(31,87)(32,88)(33,89)(34,90)(35,85)(36,86)(37,83)(38,84)(39,79)(40,80)(41,81)(42,82)(49,95)(50,96)(51,91)(52,92)(53,93)(54,94), (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), (1,63)(2,62)(3,61)(4,66)(5,65)(6,64)(7,47)(8,46)(9,45)(10,44)(11,43)(12,48)(13,58)(14,57)(15,56)(16,55)(17,60)(18,59)(19,73)(20,78)(21,77)(22,76)(23,75)(24,74)(25,67)(26,72)(27,71)(28,70)(29,69)(30,68)(31,88)(32,87)(33,86)(34,85)(35,90)(36,89)(37,82)(38,81)(39,80)(40,79)(41,84)(42,83)(49,91)(50,96)(51,95)(52,94)(53,93)(54,92), (1,28,16,20)(2,29,17,21)(3,30,18,22)(4,25,13,23)(5,26,14,24)(6,27,15,19)(7,79,91,87)(8,80,92,88)(9,81,93,89)(10,82,94,90)(11,83,95,85)(12,84,96,86)(31,43,39,51)(32,44,40,52)(33,45,41,53)(34,46,42,54)(35,47,37,49)(36,48,38,50)(55,75,63,67)(56,76,64,68)(57,77,65,69)(58,78,66,70)(59,73,61,71)(60,74,62,72), (1,40,16,32)(2,41,17,33)(3,42,18,34)(4,37,13,35)(5,38,14,36)(6,39,15,31)(7,75,91,67)(8,76,92,68)(9,77,93,69)(10,78,94,70)(11,73,95,71)(12,74,96,72)(19,51,27,43)(20,52,28,44)(21,53,29,45)(22,54,30,46)(23,49,25,47)(24,50,26,48)(55,87,63,79)(56,88,64,80)(57,89,65,81)(58,90,66,82)(59,85,61,83)(60,86,62,84) );
G=PermutationGroup([(1,64),(2,65),(3,66),(4,61),(5,62),(6,63),(7,43),(8,44),(9,45),(10,46),(11,47),(12,48),(13,59),(14,60),(15,55),(16,56),(17,57),(18,58),(19,75),(20,76),(21,77),(22,78),(23,73),(24,74),(25,71),(26,72),(27,67),(28,68),(29,69),(30,70),(31,87),(32,88),(33,89),(34,90),(35,85),(36,86),(37,83),(38,84),(39,79),(40,80),(41,81),(42,82),(49,95),(50,96),(51,91),(52,92),(53,93),(54,94)], [(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)], [(1,63),(2,62),(3,61),(4,66),(5,65),(6,64),(7,47),(8,46),(9,45),(10,44),(11,43),(12,48),(13,58),(14,57),(15,56),(16,55),(17,60),(18,59),(19,73),(20,78),(21,77),(22,76),(23,75),(24,74),(25,67),(26,72),(27,71),(28,70),(29,69),(30,68),(31,88),(32,87),(33,86),(34,85),(35,90),(36,89),(37,82),(38,81),(39,80),(40,79),(41,84),(42,83),(49,91),(50,96),(51,95),(52,94),(53,93),(54,92)], [(1,28,16,20),(2,29,17,21),(3,30,18,22),(4,25,13,23),(5,26,14,24),(6,27,15,19),(7,79,91,87),(8,80,92,88),(9,81,93,89),(10,82,94,90),(11,83,95,85),(12,84,96,86),(31,43,39,51),(32,44,40,52),(33,45,41,53),(34,46,42,54),(35,47,37,49),(36,48,38,50),(55,75,63,67),(56,76,64,68),(57,77,65,69),(58,78,66,70),(59,73,61,71),(60,74,62,72)], [(1,40,16,32),(2,41,17,33),(3,42,18,34),(4,37,13,35),(5,38,14,36),(6,39,15,31),(7,75,91,67),(8,76,92,68),(9,77,93,69),(10,78,94,70),(11,73,95,71),(12,74,96,72),(19,51,27,43),(20,52,28,44),(21,53,29,45),(22,54,30,46),(23,49,25,47),(24,50,26,48),(55,87,63,79),(56,88,64,80),(57,89,65,81),(58,90,66,82),(59,85,61,83),(60,86,62,84)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 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 |
| 1 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 2 | 9 | 0 | 0 | 0 | 0 |
| 4 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 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 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
G:=sub<GL(6,GF(13))| [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],[1,1,0,0,0,0,12,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,1,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1],[2,4,0,0,0,0,9,11,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,0,0,0,0,0,0,5] >;
48 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 6A | ··· | 6G | 12A | ··· | 12L |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | D6 | D6 | C4○D4 | C3⋊D4 | S3×Q8 | Q8⋊3S3 |
| kernel | C2×D6⋊3Q8 | C2×Dic3⋊C4 | C2×C4⋊Dic3 | C2×D6⋊C4 | D6⋊3Q8 | S3×C22×C4 | Q8×C2×C6 | C22×Q8 | C2×C12 | C22×S3 | C22×C4 | C2×Q8 | C2×C6 | C2×C4 | C22 | C22 |
| # reps | 1 | 2 | 1 | 2 | 8 | 1 | 1 | 1 | 4 | 4 | 3 | 4 | 4 | 8 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_2\times D_6\rtimes_3Q_8
% in TeX
G:=Group("C2xD6:3Q8"); // GroupNames label
G:=SmallGroup(192,1372);
// by ID
G=gap.SmallGroup(192,1372);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,675,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^6=c^2=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b^3*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations