direct product, metabelian, supersoluble, monomial
Aliases: C2×C32⋊2D8, D12⋊19D6, C62.40D4, (C3×C6)⋊2D8, C32⋊5(C2×D8), C6⋊2(D4⋊S3), (C2×D12)⋊7S3, (C6×D12)⋊1C2, (C3×C12).60D4, (C2×C12).110D6, C4.4(D6⋊S3), (C3×D12)⋊19C22, C12.42(C3⋊D4), C12.77(C22×S3), (C6×C12).70C22, (C3×C12).57C23, C32⋊4C8⋊18C22, C22.12(D6⋊S3), C4.72(C2×S32), C3⋊3(C2×D4⋊S3), (C2×C4).107S32, (C3×C6).61(C2×D4), C6.69(C2×C3⋊D4), (C2×C32⋊4C8)⋊5C2, C2.4(C2×D6⋊S3), (C2×C6).54(C3⋊D4), SmallGroup(288,469)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C32⋊2D8
G = < a,b,c,d,e | a2=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1, be=eb, dcd-1=ece=c-1, ede=d-1 >
Subgroups: 626 in 163 conjugacy classes, 52 normal (14 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C8, C2×C4, D4, C23, C32, C12, C12, D6, C2×C6, C2×C6, C2×C8, D8, C2×D4, C3×S3, C3×C6, C3×C6, C3⋊C8, D12, D12, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C2×D8, C3×C12, S3×C6, C62, C2×C3⋊C8, D4⋊S3, C2×D12, C6×D4, C32⋊4C8, C3×D12, C3×D12, C6×C12, S3×C2×C6, C2×D4⋊S3, C32⋊2D8, C2×C32⋊4C8, C6×D12, C2×C32⋊2D8
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C3⋊D4, C22×S3, C2×D8, S32, D4⋊S3, C2×C3⋊D4, D6⋊S3, C2×S32, C2×D4⋊S3, C32⋊2D8, C2×D6⋊S3, C2×C32⋊2D8
►On 96 points
Generators in S96
(1 44)(2 45)(3 46)(4 47)(5 48)(6 41)(7 42)(8 43)(9 87)(10 88)(11 81)(12 82)(13 83)(14 84)(15 85)(16 86)(17 30)(18 31)(19 32)(20 25)(21 26)(22 27)(23 28)(24 29)(33 74)(34 75)(35 76)(36 77)(37 78)(38 79)(39 80)(40 73)(49 89)(50 90)(51 91)(52 92)(53 93)(54 94)(55 95)(56 96)(57 71)(58 72)(59 65)(60 66)(61 67)(62 68)(63 69)(64 70)
(1 83 94)(2 95 84)(3 85 96)(4 89 86)(5 87 90)(6 91 88)(7 81 92)(8 93 82)(9 50 48)(10 41 51)(11 52 42)(12 43 53)(13 54 44)(14 45 55)(15 56 46)(16 47 49)(17 79 57)(18 58 80)(19 73 59)(20 60 74)(21 75 61)(22 62 76)(23 77 63)(24 64 78)(25 66 33)(26 34 67)(27 68 35)(28 36 69)(29 70 37)(30 38 71)(31 72 39)(32 40 65)
(1 83 94)(2 95 84)(3 85 96)(4 89 86)(5 87 90)(6 91 88)(7 81 92)(8 93 82)(9 50 48)(10 41 51)(11 52 42)(12 43 53)(13 54 44)(14 45 55)(15 56 46)(16 47 49)(17 57 79)(18 80 58)(19 59 73)(20 74 60)(21 61 75)(22 76 62)(23 63 77)(24 78 64)(25 33 66)(26 67 34)(27 35 68)(28 69 36)(29 37 70)(30 71 38)(31 39 72)(32 65 40)
(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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 66)(10 65)(11 72)(12 71)(13 70)(14 69)(15 68)(16 67)(25 48)(26 47)(27 46)(28 45)(29 44)(30 43)(31 42)(32 41)(33 50)(34 49)(35 56)(36 55)(37 54)(38 53)(39 52)(40 51)(57 82)(58 81)(59 88)(60 87)(61 86)(62 85)(63 84)(64 83)(73 91)(74 90)(75 89)(76 96)(77 95)(78 94)(79 93)(80 92)
G:=sub<Sym(96)| (1,44)(2,45)(3,46)(4,47)(5,48)(6,41)(7,42)(8,43)(9,87)(10,88)(11,81)(12,82)(13,83)(14,84)(15,85)(16,86)(17,30)(18,31)(19,32)(20,25)(21,26)(22,27)(23,28)(24,29)(33,74)(34,75)(35,76)(36,77)(37,78)(38,79)(39,80)(40,73)(49,89)(50,90)(51,91)(52,92)(53,93)(54,94)(55,95)(56,96)(57,71)(58,72)(59,65)(60,66)(61,67)(62,68)(63,69)(64,70), (1,83,94)(2,95,84)(3,85,96)(4,89,86)(5,87,90)(6,91,88)(7,81,92)(8,93,82)(9,50,48)(10,41,51)(11,52,42)(12,43,53)(13,54,44)(14,45,55)(15,56,46)(16,47,49)(17,79,57)(18,58,80)(19,73,59)(20,60,74)(21,75,61)(22,62,76)(23,77,63)(24,64,78)(25,66,33)(26,34,67)(27,68,35)(28,36,69)(29,70,37)(30,38,71)(31,72,39)(32,40,65), (1,83,94)(2,95,84)(3,85,96)(4,89,86)(5,87,90)(6,91,88)(7,81,92)(8,93,82)(9,50,48)(10,41,51)(11,52,42)(12,43,53)(13,54,44)(14,45,55)(15,56,46)(16,47,49)(17,57,79)(18,80,58)(19,59,73)(20,74,60)(21,61,75)(22,76,62)(23,63,77)(24,78,64)(25,33,66)(26,67,34)(27,35,68)(28,69,36)(29,37,70)(30,71,38)(31,39,72)(32,65,40), (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), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,66)(10,65)(11,72)(12,71)(13,70)(14,69)(15,68)(16,67)(25,48)(26,47)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,50)(34,49)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(57,82)(58,81)(59,88)(60,87)(61,86)(62,85)(63,84)(64,83)(73,91)(74,90)(75,89)(76,96)(77,95)(78,94)(79,93)(80,92)>;
G:=Group( (1,44)(2,45)(3,46)(4,47)(5,48)(6,41)(7,42)(8,43)(9,87)(10,88)(11,81)(12,82)(13,83)(14,84)(15,85)(16,86)(17,30)(18,31)(19,32)(20,25)(21,26)(22,27)(23,28)(24,29)(33,74)(34,75)(35,76)(36,77)(37,78)(38,79)(39,80)(40,73)(49,89)(50,90)(51,91)(52,92)(53,93)(54,94)(55,95)(56,96)(57,71)(58,72)(59,65)(60,66)(61,67)(62,68)(63,69)(64,70), (1,83,94)(2,95,84)(3,85,96)(4,89,86)(5,87,90)(6,91,88)(7,81,92)(8,93,82)(9,50,48)(10,41,51)(11,52,42)(12,43,53)(13,54,44)(14,45,55)(15,56,46)(16,47,49)(17,79,57)(18,58,80)(19,73,59)(20,60,74)(21,75,61)(22,62,76)(23,77,63)(24,64,78)(25,66,33)(26,34,67)(27,68,35)(28,36,69)(29,70,37)(30,38,71)(31,72,39)(32,40,65), (1,83,94)(2,95,84)(3,85,96)(4,89,86)(5,87,90)(6,91,88)(7,81,92)(8,93,82)(9,50,48)(10,41,51)(11,52,42)(12,43,53)(13,54,44)(14,45,55)(15,56,46)(16,47,49)(17,57,79)(18,80,58)(19,59,73)(20,74,60)(21,61,75)(22,76,62)(23,63,77)(24,78,64)(25,33,66)(26,67,34)(27,35,68)(28,69,36)(29,37,70)(30,71,38)(31,39,72)(32,65,40), (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), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,66)(10,65)(11,72)(12,71)(13,70)(14,69)(15,68)(16,67)(25,48)(26,47)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,50)(34,49)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(57,82)(58,81)(59,88)(60,87)(61,86)(62,85)(63,84)(64,83)(73,91)(74,90)(75,89)(76,96)(77,95)(78,94)(79,93)(80,92) );
G=PermutationGroup([[(1,44),(2,45),(3,46),(4,47),(5,48),(6,41),(7,42),(8,43),(9,87),(10,88),(11,81),(12,82),(13,83),(14,84),(15,85),(16,86),(17,30),(18,31),(19,32),(20,25),(21,26),(22,27),(23,28),(24,29),(33,74),(34,75),(35,76),(36,77),(37,78),(38,79),(39,80),(40,73),(49,89),(50,90),(51,91),(52,92),(53,93),(54,94),(55,95),(56,96),(57,71),(58,72),(59,65),(60,66),(61,67),(62,68),(63,69),(64,70)], [(1,83,94),(2,95,84),(3,85,96),(4,89,86),(5,87,90),(6,91,88),(7,81,92),(8,93,82),(9,50,48),(10,41,51),(11,52,42),(12,43,53),(13,54,44),(14,45,55),(15,56,46),(16,47,49),(17,79,57),(18,58,80),(19,73,59),(20,60,74),(21,75,61),(22,62,76),(23,77,63),(24,64,78),(25,66,33),(26,34,67),(27,68,35),(28,36,69),(29,70,37),(30,38,71),(31,72,39),(32,40,65)], [(1,83,94),(2,95,84),(3,85,96),(4,89,86),(5,87,90),(6,91,88),(7,81,92),(8,93,82),(9,50,48),(10,41,51),(11,52,42),(12,43,53),(13,54,44),(14,45,55),(15,56,46),(16,47,49),(17,57,79),(18,80,58),(19,59,73),(20,74,60),(21,61,75),(22,76,62),(23,63,77),(24,78,64),(25,33,66),(26,67,34),(27,35,68),(28,69,36),(29,37,70),(30,71,38),(31,39,72),(32,65,40)], [(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)], [(1,24),(2,23),(3,22),(4,21),(5,20),(6,19),(7,18),(8,17),(9,66),(10,65),(11,72),(12,71),(13,70),(14,69),(15,68),(16,67),(25,48),(26,47),(27,46),(28,45),(29,44),(30,43),(31,42),(32,41),(33,50),(34,49),(35,56),(36,55),(37,54),(38,53),(39,52),(40,51),(57,82),(58,81),(59,88),(60,87),(61,86),(62,85),(63,84),(64,83),(73,91),(74,90),(75,89),(76,96),(77,95),(78,94),(79,93),(80,92)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 4A | 4B | 6A | ··· | 6F | 6G | 6H | 6I | 6J | ··· | 6Q | 8A | 8B | 8C | 8D | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 12 | 12 | 2 | 2 | 4 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 12 | ··· | 12 | 18 | 18 | 18 | 18 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D8 | C3⋊D4 | C3⋊D4 | S32 | D4⋊S3 | D6⋊S3 | C2×S32 | D6⋊S3 | C32⋊2D8 |
| kernel | C2×C32⋊2D8 | C32⋊2D8 | C2×C32⋊4C8 | C6×D12 | C2×D12 | C3×C12 | C62 | D12 | C2×C12 | C3×C6 | C12 | C2×C6 | C2×C4 | C6 | C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 2 | 2 | 1 | 1 | 4 | 2 | 4 | 4 | 4 | 1 | 4 | 1 | 1 | 1 | 4 |
Matrix representation of C2×C32⋊2D8 ►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 | 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 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 51 | 48 | 0 | 0 | 0 | 0 |
| 0 | 63 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 48 |
| 0 | 0 | 0 | 0 | 11 | 37 |
| 6 | 1 | 0 | 0 | 0 | 0 |
| 38 | 67 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 50 |
| 0 | 0 | 0 | 0 | 68 | 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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[51,0,0,0,0,0,48,63,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,36,11,0,0,0,0,48,37],[6,38,0,0,0,0,1,67,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,68,0,0,0,0,50,55] >;
C2×C32⋊2D8 in GAP, Magma, Sage, TeX
C_2\times C_3^2\rtimes_2D_8
% in TeX
G:=Group("C2xC3^2:2D8"); // GroupNames label
G:=SmallGroup(288,469);
// by ID
G=gap.SmallGroup(288,469);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,675,346,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations