metabelian, soluble, monomial, A-group
Aliases: C42⋊C27, (C4×C36).C3, (C4×C12).C9, C9.(C42⋊C3), C3.(C42⋊C9), (C2×C18).1A4, C22.(C9.A4), (C2×C6).1(C3.A4), SmallGroup(432,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C42 — C42⋊C27 |
Generators and relations for C42⋊C27
G = < a,b,c | a4=b4=c27=1, ab=ba, cac-1=ab-1, cbc-1=a-1b2 >
Smallest permutation representation of C42⋊C27
►On 108 points
Generators in S108
(1 31)(2 100 32 67)(3 101 33 68)(4 34)(5 103 35 70)(6 104 36 71)(7 37)(8 106 38 73)(9 107 39 74)(10 40)(11 82 41 76)(12 83 42 77)(13 43)(14 85 44 79)(15 86 45 80)(16 46)(17 88 47 55)(18 89 48 56)(19 49)(20 91 50 58)(21 92 51 59)(22 52)(23 94 53 61)(24 95 54 62)(25 28)(26 97 29 64)(27 98 30 65)(57 90)(60 93)(63 96)(66 99)(69 102)(72 105)(75 108)(78 84)(81 87)
(1 99 31 66)(3 68 33 101)(4 102 34 69)(6 71 36 104)(7 105 37 72)(9 74 39 107)(10 108 40 75)(12 77 42 83)(13 84 43 78)(15 80 45 86)(16 87 46 81)(18 56 48 89)(19 90 49 57)(21 59 51 92)(22 93 52 60)(24 62 54 95)(25 96 28 63)(27 65 30 98)
(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 97 98 99 100 101 102 103 104 105 106 107 108)
G:=sub<Sym(108)| (1,31)(2,100,32,67)(3,101,33,68)(4,34)(5,103,35,70)(6,104,36,71)(7,37)(8,106,38,73)(9,107,39,74)(10,40)(11,82,41,76)(12,83,42,77)(13,43)(14,85,44,79)(15,86,45,80)(16,46)(17,88,47,55)(18,89,48,56)(19,49)(20,91,50,58)(21,92,51,59)(22,52)(23,94,53,61)(24,95,54,62)(25,28)(26,97,29,64)(27,98,30,65)(57,90)(60,93)(63,96)(66,99)(69,102)(72,105)(75,108)(78,84)(81,87), (1,99,31,66)(3,68,33,101)(4,102,34,69)(6,71,36,104)(7,105,37,72)(9,74,39,107)(10,108,40,75)(12,77,42,83)(13,84,43,78)(15,80,45,86)(16,87,46,81)(18,56,48,89)(19,90,49,57)(21,59,51,92)(22,93,52,60)(24,62,54,95)(25,96,28,63)(27,65,30,98), (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)(82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108)>;
G:=Group( (1,31)(2,100,32,67)(3,101,33,68)(4,34)(5,103,35,70)(6,104,36,71)(7,37)(8,106,38,73)(9,107,39,74)(10,40)(11,82,41,76)(12,83,42,77)(13,43)(14,85,44,79)(15,86,45,80)(16,46)(17,88,47,55)(18,89,48,56)(19,49)(20,91,50,58)(21,92,51,59)(22,52)(23,94,53,61)(24,95,54,62)(25,28)(26,97,29,64)(27,98,30,65)(57,90)(60,93)(63,96)(66,99)(69,102)(72,105)(75,108)(78,84)(81,87), (1,99,31,66)(3,68,33,101)(4,102,34,69)(6,71,36,104)(7,105,37,72)(9,74,39,107)(10,108,40,75)(12,77,42,83)(13,84,43,78)(15,80,45,86)(16,87,46,81)(18,56,48,89)(19,90,49,57)(21,59,51,92)(22,93,52,60)(24,62,54,95)(25,96,28,63)(27,65,30,98), (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)(82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108) );
G=PermutationGroup([[(1,31),(2,100,32,67),(3,101,33,68),(4,34),(5,103,35,70),(6,104,36,71),(7,37),(8,106,38,73),(9,107,39,74),(10,40),(11,82,41,76),(12,83,42,77),(13,43),(14,85,44,79),(15,86,45,80),(16,46),(17,88,47,55),(18,89,48,56),(19,49),(20,91,50,58),(21,92,51,59),(22,52),(23,94,53,61),(24,95,54,62),(25,28),(26,97,29,64),(27,98,30,65),(57,90),(60,93),(63,96),(66,99),(69,102),(72,105),(75,108),(78,84),(81,87)], [(1,99,31,66),(3,68,33,101),(4,102,34,69),(6,71,36,104),(7,105,37,72),(9,74,39,107),(10,108,40,75),(12,77,42,83),(13,84,43,78),(15,80,45,86),(16,87,46,81),(18,56,48,89),(19,90,49,57),(21,59,51,92),(22,93,52,60),(24,62,54,95),(25,96,28,63),(27,65,30,98)], [(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),(82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108)]])
72 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 4C | 4D | 6A | 6B | 9A | ··· | 9F | 12A | ··· | 12H | 18A | ··· | 18F | 27A | ··· | 27R | 36A | ··· | 36X |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 27 | ··· | 27 | 36 | ··· | 36 |
| size | 1 | 3 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 3 | ··· | 3 | 16 | ··· | 16 | 3 | ··· | 3 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||
| image | C1 | C3 | C9 | C27 | A4 | C3.A4 | C42⋊C3 | C9.A4 | C42⋊C9 | C42⋊C27 |
| kernel | C42⋊C27 | C4×C36 | C4×C12 | C42 | C2×C18 | C2×C6 | C9 | C22 | C3 | C1 |
| # reps | 1 | 2 | 6 | 18 | 1 | 2 | 4 | 6 | 8 | 24 |
Matrix representation of C42⋊C27 ►in GL3(𝔽109) generated by
| 108 | 0 | 0 |
| 0 | 33 | 0 |
| 0 | 0 | 33 |
| 33 | 0 | 0 |
| 0 | 76 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 66 | 0 | 0 |
G:=sub<GL(3,GF(109))| [108,0,0,0,33,0,0,0,33],[33,0,0,0,76,0,0,0,1],[0,0,66,1,0,0,0,1,0] >;
C42⋊C27 in GAP, Magma, Sage, TeX
C_4^2\rtimes C_{27} % in TeX
G:=Group("C4^2:C27"); // GroupNames label
G:=SmallGroup(432,3);
// by ID
G=gap.SmallGroup(432,3);
# by ID
G:=PCGroup([7,-3,-3,-3,-2,2,-2,2,21,50,1515,360,10399,102,9077,15882]);
// Polycyclic
G:=Group<a,b,c|a^4=b^4=c^27=1,a*b=b*a,c*a*c^-1=a*b^-1,c*b*c^-1=a^-1*b^2>;
// generators/relations
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