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, (C3×D4).22C23, D4.22(C22×S3), C3⋊Q16⋊20C22, C6.159(C22×D4), (C22×C6).123D4, (C22×C4).403D6, (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), (C2×C4○D4)⋊7S3, (C6×C4○D4)⋊3C2, (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
Generators and relations for C2×Q8.13D6
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 >
Subgroups: 616 in 266 conjugacy classes, 111 normal (35 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, S3, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, Dic3, C12, C12, C12, D6, C2×C6, C2×C6, C2×C6, C2×C8, D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C3⋊C8, Dic6, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×S3, C22×C6, C22×C6, C22×C8, C2×D8, C2×SD16, C2×Q16, C4○D8, C2×C4○D4, C2×C4○D4, C2×C3⋊C8, C2×C3⋊C8, D4⋊S3, D4.S3, Q8⋊2S3, C3⋊Q16, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C4○D12, C2×C3⋊D4, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4, C3×C4○D4, C2×C4○D8, C22×C3⋊C8, C2×D4⋊S3, C2×D4.S3, C2×Q8⋊2S3, C2×C3⋊Q16, Q8.13D6, C2×C4○D12, C6×C4○D4, C2×Q8.13D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C3⋊D4, C22×S3, C4○D8, C22×D4, C2×C3⋊D4, S3×C23, C2×C4○D8, Q8.13D6, C22×C3⋊D4, C2×Q8.13D6
►On 96 points
(1 22)(2 23)(3 24)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(10 19)(11 20)(12 21)(25 77)(26 78)(27 79)(28 80)(29 81)(30 82)(31 83)(32 84)(33 73)(34 74)(35 75)(36 76)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)(49 66)(50 67)(51 68)(52 69)(53 70)(54 71)(55 72)(56 61)(57 62)(58 63)(59 64)(60 65)
(1 33 7 27)(2 34 8 28)(3 35 9 29)(4 36 10 30)(5 25 11 31)(6 26 12 32)(13 76 19 82)(14 77 20 83)(15 78 21 84)(16 79 22 73)(17 80 23 74)(18 81 24 75)(37 66 43 72)(38 67 44 61)(39 68 45 62)(40 69 46 63)(41 70 47 64)(42 71 48 65)(49 91 55 85)(50 92 56 86)(51 93 57 87)(52 94 58 88)(53 95 59 89)(54 96 60 90)
(1 69 7 63)(2 70 8 64)(3 71 9 65)(4 72 10 66)(5 61 11 67)(6 62 12 68)(13 55 19 49)(14 56 20 50)(15 57 21 51)(16 58 22 52)(17 59 23 53)(18 60 24 54)(25 44 31 38)(26 45 32 39)(27 46 33 40)(28 47 34 41)(29 48 35 42)(30 37 36 43)(73 88 79 94)(74 89 80 95)(75 90 81 96)(76 91 82 85)(77 92 83 86)(78 93 84 87)
(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 18 19 24)(14 23 20 17)(15 16 21 22)(25 28 31 34)(26 33 32 27)(29 36 35 30)(37 65 43 71)(38 70 44 64)(39 63 45 69)(40 68 46 62)(41 61 47 67)(42 66 48 72)(49 96 55 90)(50 89 56 95)(51 94 57 88)(52 87 58 93)(53 92 59 86)(54 85 60 91)(73 84 79 78)(74 77 80 83)(75 82 81 76)
G:=sub<Sym(96)| (1,22)(2,23)(3,24)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21)(25,77)(26,78)(27,79)(28,80)(29,81)(30,82)(31,83)(32,84)(33,73)(34,74)(35,75)(36,76)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96)(49,66)(50,67)(51,68)(52,69)(53,70)(54,71)(55,72)(56,61)(57,62)(58,63)(59,64)(60,65), (1,33,7,27)(2,34,8,28)(3,35,9,29)(4,36,10,30)(5,25,11,31)(6,26,12,32)(13,76,19,82)(14,77,20,83)(15,78,21,84)(16,79,22,73)(17,80,23,74)(18,81,24,75)(37,66,43,72)(38,67,44,61)(39,68,45,62)(40,69,46,63)(41,70,47,64)(42,71,48,65)(49,91,55,85)(50,92,56,86)(51,93,57,87)(52,94,58,88)(53,95,59,89)(54,96,60,90), (1,69,7,63)(2,70,8,64)(3,71,9,65)(4,72,10,66)(5,61,11,67)(6,62,12,68)(13,55,19,49)(14,56,20,50)(15,57,21,51)(16,58,22,52)(17,59,23,53)(18,60,24,54)(25,44,31,38)(26,45,32,39)(27,46,33,40)(28,47,34,41)(29,48,35,42)(30,37,36,43)(73,88,79,94)(74,89,80,95)(75,90,81,96)(76,91,82,85)(77,92,83,86)(78,93,84,87), (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)(13,18,19,24)(14,23,20,17)(15,16,21,22)(25,28,31,34)(26,33,32,27)(29,36,35,30)(37,65,43,71)(38,70,44,64)(39,63,45,69)(40,68,46,62)(41,61,47,67)(42,66,48,72)(49,96,55,90)(50,89,56,95)(51,94,57,88)(52,87,58,93)(53,92,59,86)(54,85,60,91)(73,84,79,78)(74,77,80,83)(75,82,81,76)>;
G:=Group( (1,22)(2,23)(3,24)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21)(25,77)(26,78)(27,79)(28,80)(29,81)(30,82)(31,83)(32,84)(33,73)(34,74)(35,75)(36,76)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96)(49,66)(50,67)(51,68)(52,69)(53,70)(54,71)(55,72)(56,61)(57,62)(58,63)(59,64)(60,65), (1,33,7,27)(2,34,8,28)(3,35,9,29)(4,36,10,30)(5,25,11,31)(6,26,12,32)(13,76,19,82)(14,77,20,83)(15,78,21,84)(16,79,22,73)(17,80,23,74)(18,81,24,75)(37,66,43,72)(38,67,44,61)(39,68,45,62)(40,69,46,63)(41,70,47,64)(42,71,48,65)(49,91,55,85)(50,92,56,86)(51,93,57,87)(52,94,58,88)(53,95,59,89)(54,96,60,90), (1,69,7,63)(2,70,8,64)(3,71,9,65)(4,72,10,66)(5,61,11,67)(6,62,12,68)(13,55,19,49)(14,56,20,50)(15,57,21,51)(16,58,22,52)(17,59,23,53)(18,60,24,54)(25,44,31,38)(26,45,32,39)(27,46,33,40)(28,47,34,41)(29,48,35,42)(30,37,36,43)(73,88,79,94)(74,89,80,95)(75,90,81,96)(76,91,82,85)(77,92,83,86)(78,93,84,87), (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)(13,18,19,24)(14,23,20,17)(15,16,21,22)(25,28,31,34)(26,33,32,27)(29,36,35,30)(37,65,43,71)(38,70,44,64)(39,63,45,69)(40,68,46,62)(41,61,47,67)(42,66,48,72)(49,96,55,90)(50,89,56,95)(51,94,57,88)(52,87,58,93)(53,92,59,86)(54,85,60,91)(73,84,79,78)(74,77,80,83)(75,82,81,76) );
G=PermutationGroup([[(1,22),(2,23),(3,24),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18),(10,19),(11,20),(12,21),(25,77),(26,78),(27,79),(28,80),(29,81),(30,82),(31,83),(32,84),(33,73),(34,74),(35,75),(36,76),(37,85),(38,86),(39,87),(40,88),(41,89),(42,90),(43,91),(44,92),(45,93),(46,94),(47,95),(48,96),(49,66),(50,67),(51,68),(52,69),(53,70),(54,71),(55,72),(56,61),(57,62),(58,63),(59,64),(60,65)], [(1,33,7,27),(2,34,8,28),(3,35,9,29),(4,36,10,30),(5,25,11,31),(6,26,12,32),(13,76,19,82),(14,77,20,83),(15,78,21,84),(16,79,22,73),(17,80,23,74),(18,81,24,75),(37,66,43,72),(38,67,44,61),(39,68,45,62),(40,69,46,63),(41,70,47,64),(42,71,48,65),(49,91,55,85),(50,92,56,86),(51,93,57,87),(52,94,58,88),(53,95,59,89),(54,96,60,90)], [(1,69,7,63),(2,70,8,64),(3,71,9,65),(4,72,10,66),(5,61,11,67),(6,62,12,68),(13,55,19,49),(14,56,20,50),(15,57,21,51),(16,58,22,52),(17,59,23,53),(18,60,24,54),(25,44,31,38),(26,45,32,39),(27,46,33,40),(28,47,34,41),(29,48,35,42),(30,37,36,43),(73,88,79,94),(74,89,80,95),(75,90,81,96),(76,91,82,85),(77,92,83,86),(78,93,84,87)], [(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),(13,18,19,24),(14,23,20,17),(15,16,21,22),(25,28,31,34),(26,33,32,27),(29,36,35,30),(37,65,43,71),(38,70,44,64),(39,63,45,69),(40,68,46,62),(41,61,47,67),(42,66,48,72),(49,96,55,90),(50,89,56,95),(51,94,57,88),(52,87,58,93),(53,92,59,86),(54,85,60,91),(73,84,79,78),(74,77,80,83),(75,82,81,76)]])
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 |
Matrix representation of C2×Q8.13D6 ►in 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] >;
C2×Q8.13D6 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