metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.65D6, (C2×Q8).51D6, C22⋊Q8.4S3, C6.99(C4○D8), (C2×C12).265D4, C6.Q16⋊39C2, (C22×C6).91D4, Q8⋊2Dic3⋊14C2, C12.Q8⋊38C2, (C22×C4).142D6, C12.189(C4○D4), (C6×Q8).45C22, C4.95(D4⋊2S3), (C2×C12).364C23, C12.55D4.8C2, C6.89(C8.C22), C3⋊7(C23.20D4), C23.33(C3⋊D4), C2.18(Q8.13D6), C4⋊Dic3.339C22, C2.10(Q8.11D6), (C22×C12).168C22, C23.26D6.14C2, C6.82(C22.D4), C2.16(C23.23D6), (C2×C6).495(C2×D4), (C3×C22⋊Q8).3C2, (C2×C3⋊C8).114C22, (C2×C4).172(C3⋊D4), (C3×C4⋊C4).112C22, (C2×C4).464(C22×S3), C22.170(C2×C3⋊D4), SmallGroup(192,604)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×Q8).51D6
G = < a,b,c,d,e | a2=b4=1, c2=d6=b2, e2=a, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=ab2c, ece-1=b-1c, ede-1=d5 >
Subgroups: 224 in 96 conjugacy classes, 39 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×Q8, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×C6, C22⋊C8, Q8⋊C4, C4.Q8, C2.D8, C42⋊C2, C22⋊Q8, C2×C3⋊C8, C4×Dic3, C4⋊Dic3, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C22×C12, C6×Q8, C23.20D4, C6.Q16, C12.Q8, C12.55D4, Q8⋊2Dic3, C23.26D6, C3×C22⋊Q8, (C2×Q8).51D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C22.D4, C4○D8, C8.C22, D4⋊2S3, C2×C3⋊D4, C23.20D4, C23.23D6, Q8.11D6, Q8.13D6, (C2×Q8).51D6
►On 96 points
Generators in S96
(1 33)(2 34)(3 35)(4 36)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 86)(14 87)(15 88)(16 89)(17 90)(18 91)(19 92)(20 93)(21 94)(22 95)(23 96)(24 85)(37 79)(38 80)(39 81)(40 82)(41 83)(42 84)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 72)(50 61)(51 62)(52 63)(53 64)(54 65)(55 66)(56 67)(57 68)(58 69)(59 70)(60 71)
(1 30 7 36)(2 31 8 25)(3 32 9 26)(4 33 10 27)(5 34 11 28)(6 35 12 29)(13 89 19 95)(14 90 20 96)(15 91 21 85)(16 92 22 86)(17 93 23 87)(18 94 24 88)(37 46 43 40)(38 47 44 41)(39 48 45 42)(49 52 55 58)(50 53 56 59)(51 54 57 60)(61 64 67 70)(62 65 68 71)(63 66 69 72)(73 82 79 76)(74 83 80 77)(75 84 81 78)
(1 42 7 48)(2 79 8 73)(3 44 9 38)(4 81 10 75)(5 46 11 40)(6 83 12 77)(13 61 19 67)(14 57 20 51)(15 63 21 69)(16 59 22 53)(17 65 23 71)(18 49 24 55)(25 76 31 82)(26 41 32 47)(27 78 33 84)(28 43 34 37)(29 80 35 74)(30 45 36 39)(50 92 56 86)(52 94 58 88)(54 96 60 90)(62 87 68 93)(64 89 70 95)(66 91 72 85)
(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 22 33 95)(2 15 34 88)(3 20 35 93)(4 13 36 86)(5 18 25 91)(6 23 26 96)(7 16 27 89)(8 21 28 94)(9 14 29 87)(10 19 30 92)(11 24 31 85)(12 17 32 90)(37 55 79 66)(38 60 80 71)(39 53 81 64)(40 58 82 69)(41 51 83 62)(42 56 84 67)(43 49 73 72)(44 54 74 65)(45 59 75 70)(46 52 76 63)(47 57 77 68)(48 50 78 61)
G:=sub<Sym(96)| (1,33)(2,34)(3,35)(4,36)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,86)(14,87)(15,88)(16,89)(17,90)(18,91)(19,92)(20,93)(21,94)(22,95)(23,96)(24,85)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,72)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(56,67)(57,68)(58,69)(59,70)(60,71), (1,30,7,36)(2,31,8,25)(3,32,9,26)(4,33,10,27)(5,34,11,28)(6,35,12,29)(13,89,19,95)(14,90,20,96)(15,91,21,85)(16,92,22,86)(17,93,23,87)(18,94,24,88)(37,46,43,40)(38,47,44,41)(39,48,45,42)(49,52,55,58)(50,53,56,59)(51,54,57,60)(61,64,67,70)(62,65,68,71)(63,66,69,72)(73,82,79,76)(74,83,80,77)(75,84,81,78), (1,42,7,48)(2,79,8,73)(3,44,9,38)(4,81,10,75)(5,46,11,40)(6,83,12,77)(13,61,19,67)(14,57,20,51)(15,63,21,69)(16,59,22,53)(17,65,23,71)(18,49,24,55)(25,76,31,82)(26,41,32,47)(27,78,33,84)(28,43,34,37)(29,80,35,74)(30,45,36,39)(50,92,56,86)(52,94,58,88)(54,96,60,90)(62,87,68,93)(64,89,70,95)(66,91,72,85), (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), (1,22,33,95)(2,15,34,88)(3,20,35,93)(4,13,36,86)(5,18,25,91)(6,23,26,96)(7,16,27,89)(8,21,28,94)(9,14,29,87)(10,19,30,92)(11,24,31,85)(12,17,32,90)(37,55,79,66)(38,60,80,71)(39,53,81,64)(40,58,82,69)(41,51,83,62)(42,56,84,67)(43,49,73,72)(44,54,74,65)(45,59,75,70)(46,52,76,63)(47,57,77,68)(48,50,78,61)>;
G:=Group( (1,33)(2,34)(3,35)(4,36)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,86)(14,87)(15,88)(16,89)(17,90)(18,91)(19,92)(20,93)(21,94)(22,95)(23,96)(24,85)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,72)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(56,67)(57,68)(58,69)(59,70)(60,71), (1,30,7,36)(2,31,8,25)(3,32,9,26)(4,33,10,27)(5,34,11,28)(6,35,12,29)(13,89,19,95)(14,90,20,96)(15,91,21,85)(16,92,22,86)(17,93,23,87)(18,94,24,88)(37,46,43,40)(38,47,44,41)(39,48,45,42)(49,52,55,58)(50,53,56,59)(51,54,57,60)(61,64,67,70)(62,65,68,71)(63,66,69,72)(73,82,79,76)(74,83,80,77)(75,84,81,78), (1,42,7,48)(2,79,8,73)(3,44,9,38)(4,81,10,75)(5,46,11,40)(6,83,12,77)(13,61,19,67)(14,57,20,51)(15,63,21,69)(16,59,22,53)(17,65,23,71)(18,49,24,55)(25,76,31,82)(26,41,32,47)(27,78,33,84)(28,43,34,37)(29,80,35,74)(30,45,36,39)(50,92,56,86)(52,94,58,88)(54,96,60,90)(62,87,68,93)(64,89,70,95)(66,91,72,85), (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), (1,22,33,95)(2,15,34,88)(3,20,35,93)(4,13,36,86)(5,18,25,91)(6,23,26,96)(7,16,27,89)(8,21,28,94)(9,14,29,87)(10,19,30,92)(11,24,31,85)(12,17,32,90)(37,55,79,66)(38,60,80,71)(39,53,81,64)(40,58,82,69)(41,51,83,62)(42,56,84,67)(43,49,73,72)(44,54,74,65)(45,59,75,70)(46,52,76,63)(47,57,77,68)(48,50,78,61) );
G=PermutationGroup([[(1,33),(2,34),(3,35),(4,36),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,86),(14,87),(15,88),(16,89),(17,90),(18,91),(19,92),(20,93),(21,94),(22,95),(23,96),(24,85),(37,79),(38,80),(39,81),(40,82),(41,83),(42,84),(43,73),(44,74),(45,75),(46,76),(47,77),(48,78),(49,72),(50,61),(51,62),(52,63),(53,64),(54,65),(55,66),(56,67),(57,68),(58,69),(59,70),(60,71)], [(1,30,7,36),(2,31,8,25),(3,32,9,26),(4,33,10,27),(5,34,11,28),(6,35,12,29),(13,89,19,95),(14,90,20,96),(15,91,21,85),(16,92,22,86),(17,93,23,87),(18,94,24,88),(37,46,43,40),(38,47,44,41),(39,48,45,42),(49,52,55,58),(50,53,56,59),(51,54,57,60),(61,64,67,70),(62,65,68,71),(63,66,69,72),(73,82,79,76),(74,83,80,77),(75,84,81,78)], [(1,42,7,48),(2,79,8,73),(3,44,9,38),(4,81,10,75),(5,46,11,40),(6,83,12,77),(13,61,19,67),(14,57,20,51),(15,63,21,69),(16,59,22,53),(17,65,23,71),(18,49,24,55),(25,76,31,82),(26,41,32,47),(27,78,33,84),(28,43,34,37),(29,80,35,74),(30,45,36,39),(50,92,56,86),(52,94,58,88),(54,96,60,90),(62,87,68,93),(64,89,70,95),(66,91,72,85)], [(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)], [(1,22,33,95),(2,15,34,88),(3,20,35,93),(4,13,36,86),(5,18,25,91),(6,23,26,96),(7,16,27,89),(8,21,28,94),(9,14,29,87),(10,19,30,92),(11,24,31,85),(12,17,32,90),(37,55,79,66),(38,60,80,71),(39,53,81,64),(40,58,82,69),(41,51,83,62),(42,56,84,67),(43,49,73,72),(44,54,74,65),(45,59,75,70),(46,52,76,63),(47,57,77,68),(48,50,78,61)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H |
| order | 1 | 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 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | C4○D4 | C3⋊D4 | C3⋊D4 | C4○D8 | C8.C22 | D4⋊2S3 | Q8.11D6 | Q8.13D6 |
| kernel | (C2×Q8).51D6 | C6.Q16 | C12.Q8 | C12.55D4 | Q8⋊2Dic3 | C23.26D6 | C3×C22⋊Q8 | C22⋊Q8 | C2×C12 | C22×C6 | C4⋊C4 | C22×C4 | C2×Q8 | C12 | C2×C4 | C23 | C6 | C6 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 4 | 1 | 2 | 2 | 2 |
Matrix representation of (C2×Q8).51D6 ►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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 32 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 17 | 71 | 0 | 0 | 0 | 0 |
| 71 | 56 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 2 | 0 | 0 |
| 0 | 0 | 37 | 61 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 13 |
| 0 | 0 | 0 | 0 | 60 | 43 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 56 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 0 | 0 | 0 |
| 0 | 0 | 0 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 46 | 0 | 0 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 45 | 44 | 0 | 0 |
| 0 | 0 | 17 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 2 |
| 0 | 0 | 0 | 0 | 62 | 13 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,32,0,0,0,0,0,27,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[17,71,0,0,0,0,71,56,0,0,0,0,0,0,12,37,0,0,0,0,2,61,0,0,0,0,0,0,30,60,0,0,0,0,13,43],[72,56,0,0,0,0,0,1,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,0,72,0,0,0,0,1,1],[46,0,0,0,0,0,0,46,0,0,0,0,0,0,45,17,0,0,0,0,44,28,0,0,0,0,0,0,60,62,0,0,0,0,2,13] >;
(C2×Q8).51D6 in GAP, Magma, Sage, TeX
(C_2\times Q_8)._{51}D_6 % in TeX
G:=Group("(C2xQ8).51D6"); // GroupNames label
G:=SmallGroup(192,604);
// by ID
G=gap.SmallGroup(192,604);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,232,254,219,184,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=1,c^2=d^6=b^2,e^2=a,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=a*b^2*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^5>;
// generators/relations