direct product, metabelian, soluble, monomial, A-group
Aliases: C2×C32⋊2C16, C62.3C8, (C3×C6)⋊2C16, (C6×C12).2C4, (C3×C12).5C8, C32⋊6(C2×C16), C32⋊4C8.9C4, C4.3(C32⋊2C8), C32⋊4C8.34C22, C22.2(C32⋊2C8), (C3×C6).22(C2×C8), C4.19(C2×C32⋊C4), (C3×C12).16(C2×C4), (C2×C4).9(C32⋊C4), C2.1(C2×C32⋊2C8), (C2×C32⋊4C8).19C2, SmallGroup(288,420)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C2×C32⋊2C16 |
Generators and relations for C2×C32⋊2C16
G = < a,b,c,d | a2=b3=c3=d16=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, dbd-1=b-1c >
Smallest permutation representation of C2×C32⋊2C16
►On 96 points
Generators in S96
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(15 57)(16 58)(17 85)(18 86)(19 87)(20 88)(21 89)(22 90)(23 91)(24 92)(25 93)(26 94)(27 95)(28 96)(29 81)(30 82)(31 83)(32 84)(33 67)(34 68)(35 69)(36 70)(37 71)(38 72)(39 73)(40 74)(41 75)(42 76)(43 77)(44 78)(45 79)(46 80)(47 65)(48 66)
(2 48 95)(4 81 34)(6 36 83)(8 85 38)(10 40 87)(12 89 42)(14 44 91)(16 93 46)(17 72 50)(19 52 74)(21 76 54)(23 56 78)(25 80 58)(27 60 66)(29 68 62)(31 64 70)
(1 47 94)(2 48 95)(3 96 33)(4 81 34)(5 35 82)(6 36 83)(7 84 37)(8 85 38)(9 39 86)(10 40 87)(11 88 41)(12 89 42)(13 43 90)(14 44 91)(15 92 45)(16 93 46)(17 72 50)(18 51 73)(19 52 74)(20 75 53)(21 76 54)(22 55 77)(23 56 78)(24 79 57)(25 80 58)(26 59 65)(27 60 66)(28 67 61)(29 68 62)(30 63 69)(31 64 70)(32 71 49)
(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)
G:=sub<Sym(96)| (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,57)(16,58)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,91)(24,92)(25,93)(26,94)(27,95)(28,96)(29,81)(30,82)(31,83)(32,84)(33,67)(34,68)(35,69)(36,70)(37,71)(38,72)(39,73)(40,74)(41,75)(42,76)(43,77)(44,78)(45,79)(46,80)(47,65)(48,66), (2,48,95)(4,81,34)(6,36,83)(8,85,38)(10,40,87)(12,89,42)(14,44,91)(16,93,46)(17,72,50)(19,52,74)(21,76,54)(23,56,78)(25,80,58)(27,60,66)(29,68,62)(31,64,70), (1,47,94)(2,48,95)(3,96,33)(4,81,34)(5,35,82)(6,36,83)(7,84,37)(8,85,38)(9,39,86)(10,40,87)(11,88,41)(12,89,42)(13,43,90)(14,44,91)(15,92,45)(16,93,46)(17,72,50)(18,51,73)(19,52,74)(20,75,53)(21,76,54)(22,55,77)(23,56,78)(24,79,57)(25,80,58)(26,59,65)(27,60,66)(28,67,61)(29,68,62)(30,63,69)(31,64,70)(32,71,49), (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)>;
G:=Group( (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,57)(16,58)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,91)(24,92)(25,93)(26,94)(27,95)(28,96)(29,81)(30,82)(31,83)(32,84)(33,67)(34,68)(35,69)(36,70)(37,71)(38,72)(39,73)(40,74)(41,75)(42,76)(43,77)(44,78)(45,79)(46,80)(47,65)(48,66), (2,48,95)(4,81,34)(6,36,83)(8,85,38)(10,40,87)(12,89,42)(14,44,91)(16,93,46)(17,72,50)(19,52,74)(21,76,54)(23,56,78)(25,80,58)(27,60,66)(29,68,62)(31,64,70), (1,47,94)(2,48,95)(3,96,33)(4,81,34)(5,35,82)(6,36,83)(7,84,37)(8,85,38)(9,39,86)(10,40,87)(11,88,41)(12,89,42)(13,43,90)(14,44,91)(15,92,45)(16,93,46)(17,72,50)(18,51,73)(19,52,74)(20,75,53)(21,76,54)(22,55,77)(23,56,78)(24,79,57)(25,80,58)(26,59,65)(27,60,66)(28,67,61)(29,68,62)(30,63,69)(31,64,70)(32,71,49), (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) );
G=PermutationGroup([[(1,59),(2,60),(3,61),(4,62),(5,63),(6,64),(7,49),(8,50),(9,51),(10,52),(11,53),(12,54),(13,55),(14,56),(15,57),(16,58),(17,85),(18,86),(19,87),(20,88),(21,89),(22,90),(23,91),(24,92),(25,93),(26,94),(27,95),(28,96),(29,81),(30,82),(31,83),(32,84),(33,67),(34,68),(35,69),(36,70),(37,71),(38,72),(39,73),(40,74),(41,75),(42,76),(43,77),(44,78),(45,79),(46,80),(47,65),(48,66)], [(2,48,95),(4,81,34),(6,36,83),(8,85,38),(10,40,87),(12,89,42),(14,44,91),(16,93,46),(17,72,50),(19,52,74),(21,76,54),(23,56,78),(25,80,58),(27,60,66),(29,68,62),(31,64,70)], [(1,47,94),(2,48,95),(3,96,33),(4,81,34),(5,35,82),(6,36,83),(7,84,37),(8,85,38),(9,39,86),(10,40,87),(11,88,41),(12,89,42),(13,43,90),(14,44,91),(15,92,45),(16,93,46),(17,72,50),(18,51,73),(19,52,74),(20,75,53),(21,76,54),(22,55,77),(23,56,78),(24,79,57),(25,80,58),(26,59,65),(27,60,66),(28,67,61),(29,68,62),(30,63,69),(31,64,70),(32,71,49)], [(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)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 8A | ··· | 8H | 12A | ··· | 12H | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 9 | ··· | 9 | 4 | ··· | 4 | 9 | ··· | 9 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | - | ||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | C32⋊C4 | C32⋊2C8 | C2×C32⋊C4 | C32⋊2C8 | C32⋊2C16 |
| kernel | C2×C32⋊2C16 | C32⋊2C16 | C2×C32⋊4C8 | C32⋊4C8 | C6×C12 | C3×C12 | C62 | C3×C6 | C2×C4 | C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 16 | 2 | 2 | 2 | 2 | 8 |
Matrix representation of C2×C32⋊2C16 ►in GL5(𝔽97)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 96 | 0 | 0 | 0 |
| 0 | 0 | 96 | 0 | 0 |
| 0 | 0 | 0 | 96 | 0 |
| 0 | 0 | 0 | 0 | 96 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 13 | 29 | 96 | 96 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 96 | 0 | 0 |
| 0 | 1 | 96 | 0 | 0 |
| 0 | 70 | 41 | 0 | 1 |
| 0 | 83 | 70 | 96 | 96 |
| 85 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 96 | 1 |
| 0 | 13 | 29 | 95 | 96 |
| 0 | 34 | 44 | 41 | 27 |
| 0 | 51 | 91 | 41 | 27 |
G:=sub<GL(5,GF(97))| [1,0,0,0,0,0,96,0,0,0,0,0,96,0,0,0,0,0,96,0,0,0,0,0,96],[1,0,0,0,0,0,1,0,0,13,0,0,1,0,29,0,0,0,0,96,0,0,0,1,96],[1,0,0,0,0,0,0,1,70,83,0,96,96,41,70,0,0,0,0,96,0,0,0,1,96],[85,0,0,0,0,0,0,13,34,51,0,0,29,44,91,0,96,95,41,41,0,1,96,27,27] >;
C2×C32⋊2C16 in GAP, Magma, Sage, TeX
C_2\times C_3^2\rtimes_2C_{16} % in TeX
G:=Group("C2xC3^2:2C16"); // GroupNames label
G:=SmallGroup(288,420);
// by ID
G=gap.SmallGroup(288,420);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,58,80,9413,691,12550,2372]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^3=c^3=d^16=1,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,d*b*d^-1=b^-1*c>;
// generators/relations
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