direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Q8.13D6, C12.34C24, D12.30C23, Dic6.29C23, C4○D4⋊18D6, C6⋊5(C4○D8), C3⋊C8.30C23, D4⋊S3⋊23C22, (C2×D4).232D6, C12.263(C2×D4), (C2×C12).502D4, C4.34(S3×C23), (C2×Q8).214D6, C4○D12⋊20C22, D4.S3⋊20C22, D4.22(C22×S3), (C3×D4).22C23, C3⋊Q16⋊20C22, C6.159(C22×D4), (C22×C4).403D6, (C22×C6).123D4, (C3×Q8).22C23, Q8.32(C22×S3), (C2×C12).556C23, Q8⋊2S3⋊21C22, (C6×D4).272C22, C23.52(C3⋊D4), (C6×Q8).237C22, (C2×D12).280C22, (C22×C12).291C22, (C2×Dic6).309C22, C3⋊6(C2×C4○D8), (C6×C4○D4)⋊3C2, (C2×C4○D4)⋊7S3, (C2×D4⋊S3)⋊33C2, (C2×C3⋊C8)⋊42C22, (C22×C3⋊C8)⋊15C2, (C2×C6).76(C2×D4), C4.30(C2×C3⋊D4), (C2×C4○D12)⋊30C2, (C2×D4.S3)⋊33C2, (C2×C3⋊Q16)⋊33C2, (C2×Q8⋊2S3)⋊33C2, (C3×C4○D4)⋊17C22, C2.32(C22×C3⋊D4), (C2×C4).158(C3⋊D4), (C2×C4).636(C22×S3), C22.119(C2×C3⋊D4), SmallGroup(192,1380)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 616 in 266 conjugacy classes, 111 normal (35 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×10], S3 [×2], C6, C6 [×2], C6 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×2], D4 [×12], Q8 [×2], Q8 [×4], C23, C23 [×2], Dic3 [×2], C12 [×2], C12 [×2], C12 [×2], D6 [×4], C2×C6, C2×C6 [×2], C2×C6 [×6], C2×C8 [×6], D8 [×4], SD16 [×8], Q16 [×4], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×3], C2×Q8, C2×Q8, C4○D4 [×4], C4○D4 [×8], C3⋊C8 [×4], Dic6 [×2], Dic6, C4×S3 [×4], D12 [×2], D12, C2×Dic3, C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×4], C2×C12 [×5], C3×D4 [×2], C3×D4 [×5], C3×Q8 [×2], C3×Q8, C22×S3, C22×C6, C22×C6, C22×C8, C2×D8, C2×SD16 [×2], C2×Q16, C4○D8 [×8], C2×C4○D4, C2×C4○D4, C2×C3⋊C8 [×2], C2×C3⋊C8 [×4], D4⋊S3 [×4], D4.S3 [×4], Q8⋊2S3 [×4], C3⋊Q16 [×4], C2×Dic6, S3×C2×C4, C2×D12, C4○D12 [×4], C4○D12 [×2], C2×C3⋊D4, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4 [×4], C3×C4○D4 [×2], C2×C4○D8, C22×C3⋊C8, C2×D4⋊S3, C2×D4.S3, C2×Q8⋊2S3, C2×C3⋊Q16, Q8.13D6 [×8], C2×C4○D12, C6×C4○D4, C2×Q8.13D6
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C3⋊D4 [×4], C22×S3 [×7], C4○D8 [×2], C22×D4, C2×C3⋊D4 [×6], S3×C23, C2×C4○D8, Q8.13D6 [×2], C22×C3⋊D4, C2×Q8.13D6
Generators and relations
G = < a,b,c,d,e | a2=b4=1, c2=d6=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, cd=dc, ece-1=b-1c, ede-1=d5 >
►On 96 points
(1 77)(2 78)(3 79)(4 80)(5 81)(6 82)(7 83)(8 84)(9 73)(10 74)(11 75)(12 76)(13 70)(14 71)(15 72)(16 61)(17 62)(18 63)(19 64)(20 65)(21 66)(22 67)(23 68)(24 69)(25 38)(26 39)(27 40)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 37)(49 93)(50 94)(51 95)(52 96)(53 85)(54 86)(55 87)(56 88)(57 89)(58 90)(59 91)(60 92)
(1 19 7 13)(2 20 8 14)(3 21 9 15)(4 22 10 16)(5 23 11 17)(6 24 12 18)(25 94 31 88)(26 95 32 89)(27 96 33 90)(28 85 34 91)(29 86 35 92)(30 87 36 93)(37 49 43 55)(38 50 44 56)(39 51 45 57)(40 52 46 58)(41 53 47 59)(42 54 48 60)(61 80 67 74)(62 81 68 75)(63 82 69 76)(64 83 70 77)(65 84 71 78)(66 73 72 79)
(1 92 7 86)(2 93 8 87)(3 94 9 88)(4 95 10 89)(5 96 11 90)(6 85 12 91)(13 29 19 35)(14 30 20 36)(15 31 21 25)(16 32 22 26)(17 33 23 27)(18 34 24 28)(37 71 43 65)(38 72 44 66)(39 61 45 67)(40 62 46 68)(41 63 47 69)(42 64 48 70)(49 84 55 78)(50 73 56 79)(51 74 57 80)(52 75 58 81)(53 76 59 82)(54 77 60 83)
(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 6 7 12)(2 11 8 5)(3 4 9 10)(13 24 19 18)(14 17 20 23)(15 22 21 16)(25 89 31 95)(26 94 32 88)(27 87 33 93)(28 92 34 86)(29 85 35 91)(30 90 36 96)(37 52 43 58)(38 57 44 51)(39 50 45 56)(40 55 46 49)(41 60 47 54)(42 53 48 59)(61 72 67 66)(62 65 68 71)(63 70 69 64)(73 74 79 80)(75 84 81 78)(76 77 82 83)
G:=sub<Sym(96)| (1,77)(2,78)(3,79)(4,80)(5,81)(6,82)(7,83)(8,84)(9,73)(10,74)(11,75)(12,76)(13,70)(14,71)(15,72)(16,61)(17,62)(18,63)(19,64)(20,65)(21,66)(22,67)(23,68)(24,69)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,37)(49,93)(50,94)(51,95)(52,96)(53,85)(54,86)(55,87)(56,88)(57,89)(58,90)(59,91)(60,92), (1,19,7,13)(2,20,8,14)(3,21,9,15)(4,22,10,16)(5,23,11,17)(6,24,12,18)(25,94,31,88)(26,95,32,89)(27,96,33,90)(28,85,34,91)(29,86,35,92)(30,87,36,93)(37,49,43,55)(38,50,44,56)(39,51,45,57)(40,52,46,58)(41,53,47,59)(42,54,48,60)(61,80,67,74)(62,81,68,75)(63,82,69,76)(64,83,70,77)(65,84,71,78)(66,73,72,79), (1,92,7,86)(2,93,8,87)(3,94,9,88)(4,95,10,89)(5,96,11,90)(6,85,12,91)(13,29,19,35)(14,30,20,36)(15,31,21,25)(16,32,22,26)(17,33,23,27)(18,34,24,28)(37,71,43,65)(38,72,44,66)(39,61,45,67)(40,62,46,68)(41,63,47,69)(42,64,48,70)(49,84,55,78)(50,73,56,79)(51,74,57,80)(52,75,58,81)(53,76,59,82)(54,77,60,83), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,24,19,18)(14,17,20,23)(15,22,21,16)(25,89,31,95)(26,94,32,88)(27,87,33,93)(28,92,34,86)(29,85,35,91)(30,90,36,96)(37,52,43,58)(38,57,44,51)(39,50,45,56)(40,55,46,49)(41,60,47,54)(42,53,48,59)(61,72,67,66)(62,65,68,71)(63,70,69,64)(73,74,79,80)(75,84,81,78)(76,77,82,83)>;
G:=Group( (1,77)(2,78)(3,79)(4,80)(5,81)(6,82)(7,83)(8,84)(9,73)(10,74)(11,75)(12,76)(13,70)(14,71)(15,72)(16,61)(17,62)(18,63)(19,64)(20,65)(21,66)(22,67)(23,68)(24,69)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,37)(49,93)(50,94)(51,95)(52,96)(53,85)(54,86)(55,87)(56,88)(57,89)(58,90)(59,91)(60,92), (1,19,7,13)(2,20,8,14)(3,21,9,15)(4,22,10,16)(5,23,11,17)(6,24,12,18)(25,94,31,88)(26,95,32,89)(27,96,33,90)(28,85,34,91)(29,86,35,92)(30,87,36,93)(37,49,43,55)(38,50,44,56)(39,51,45,57)(40,52,46,58)(41,53,47,59)(42,54,48,60)(61,80,67,74)(62,81,68,75)(63,82,69,76)(64,83,70,77)(65,84,71,78)(66,73,72,79), (1,92,7,86)(2,93,8,87)(3,94,9,88)(4,95,10,89)(5,96,11,90)(6,85,12,91)(13,29,19,35)(14,30,20,36)(15,31,21,25)(16,32,22,26)(17,33,23,27)(18,34,24,28)(37,71,43,65)(38,72,44,66)(39,61,45,67)(40,62,46,68)(41,63,47,69)(42,64,48,70)(49,84,55,78)(50,73,56,79)(51,74,57,80)(52,75,58,81)(53,76,59,82)(54,77,60,83), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,24,19,18)(14,17,20,23)(15,22,21,16)(25,89,31,95)(26,94,32,88)(27,87,33,93)(28,92,34,86)(29,85,35,91)(30,90,36,96)(37,52,43,58)(38,57,44,51)(39,50,45,56)(40,55,46,49)(41,60,47,54)(42,53,48,59)(61,72,67,66)(62,65,68,71)(63,70,69,64)(73,74,79,80)(75,84,81,78)(76,77,82,83) );
G=PermutationGroup([(1,77),(2,78),(3,79),(4,80),(5,81),(6,82),(7,83),(8,84),(9,73),(10,74),(11,75),(12,76),(13,70),(14,71),(15,72),(16,61),(17,62),(18,63),(19,64),(20,65),(21,66),(22,67),(23,68),(24,69),(25,38),(26,39),(27,40),(28,41),(29,42),(30,43),(31,44),(32,45),(33,46),(34,47),(35,48),(36,37),(49,93),(50,94),(51,95),(52,96),(53,85),(54,86),(55,87),(56,88),(57,89),(58,90),(59,91),(60,92)], [(1,19,7,13),(2,20,8,14),(3,21,9,15),(4,22,10,16),(5,23,11,17),(6,24,12,18),(25,94,31,88),(26,95,32,89),(27,96,33,90),(28,85,34,91),(29,86,35,92),(30,87,36,93),(37,49,43,55),(38,50,44,56),(39,51,45,57),(40,52,46,58),(41,53,47,59),(42,54,48,60),(61,80,67,74),(62,81,68,75),(63,82,69,76),(64,83,70,77),(65,84,71,78),(66,73,72,79)], [(1,92,7,86),(2,93,8,87),(3,94,9,88),(4,95,10,89),(5,96,11,90),(6,85,12,91),(13,29,19,35),(14,30,20,36),(15,31,21,25),(16,32,22,26),(17,33,23,27),(18,34,24,28),(37,71,43,65),(38,72,44,66),(39,61,45,67),(40,62,46,68),(41,63,47,69),(42,64,48,70),(49,84,55,78),(50,73,56,79),(51,74,57,80),(52,75,58,81),(53,76,59,82),(54,77,60,83)], [(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)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,24,19,18),(14,17,20,23),(15,22,21,16),(25,89,31,95),(26,94,32,88),(27,87,33,93),(28,92,34,86),(29,85,35,91),(30,90,36,96),(37,52,43,58),(38,57,44,51),(39,50,45,56),(40,55,46,49),(41,60,47,54),(42,53,48,59),(61,72,67,66),(62,65,68,71),(63,70,69,64),(73,74,79,80),(75,84,81,78),(76,77,82,83)])
Matrix representation ►G ⊆ GL5(𝔽73)
| 72 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 72 |
| 72 | 0 | 0 | 0 | 0 |
| 0 | 67 | 67 | 0 | 0 |
| 0 | 67 | 6 | 0 | 0 |
| 0 | 0 | 0 | 30 | 60 |
| 0 | 0 | 0 | 13 | 43 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 1 | 1 |
| 72 | 0 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,72,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,72],[72,0,0,0,0,0,67,67,0,0,0,67,6,0,0,0,0,0,30,13,0,0,0,60,43],[1,0,0,0,0,0,46,0,0,0,0,0,46,0,0,0,0,0,0,1,0,0,0,72,1],[72,0,0,0,0,0,46,0,0,0,0,0,27,0,0,0,0,0,0,1,0,0,0,1,0] >;
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 6D | ··· | 6I | 8A | ··· | 8H | 12A | 12B | 12C | 12D | 12E | ··· | 12J |
| order | 1 | 2 | 2 | 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 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 12 | 12 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | D6 | C3⋊D4 | C3⋊D4 | C4○D8 | Q8.13D6 |
| kernel | C2×Q8.13D6 | C22×C3⋊C8 | C2×D4⋊S3 | C2×D4.S3 | C2×Q8⋊2S3 | C2×C3⋊Q16 | Q8.13D6 | C2×C4○D12 | 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 | 1 | 1 | 1 | 1 | 8 | 1 | 1 | 1 | 3 | 1 | 1 | 1 | 1 | 4 | 6 | 2 | 8 | 4 |
In GAP, Magma, Sage, TeX
C_2\times Q_8._{13}D_6 % in TeX
G:=Group("C2xQ8.13D6"); // GroupNames label
G:=SmallGroup(192,1380);
// by ID
G=gap.SmallGroup(192,1380);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,675,1684,235,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=1,c^2=d^6=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,c*d=d*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^5>;
// generators/relations