direct product, abelian, monomial, 2-elementary
Aliases: C4×C28, SmallGroup(112,19)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C4×C28 |
| C1 — C4×C28 |
| C1 — C4×C28 |
Generators and relations for C4×C28
G = < a,b | a4=b28=1, ab=ba >
Smallest permutation representation of C4×C28
►Regular action on 112 points
Generators in S112
(1 77 93 36)(2 78 94 37)(3 79 95 38)(4 80 96 39)(5 81 97 40)(6 82 98 41)(7 83 99 42)(8 84 100 43)(9 57 101 44)(10 58 102 45)(11 59 103 46)(12 60 104 47)(13 61 105 48)(14 62 106 49)(15 63 107 50)(16 64 108 51)(17 65 109 52)(18 66 110 53)(19 67 111 54)(20 68 112 55)(21 69 85 56)(22 70 86 29)(23 71 87 30)(24 72 88 31)(25 73 89 32)(26 74 90 33)(27 75 91 34)(28 76 92 35)
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112)
G:=sub<Sym(112)| (1,77,93,36)(2,78,94,37)(3,79,95,38)(4,80,96,39)(5,81,97,40)(6,82,98,41)(7,83,99,42)(8,84,100,43)(9,57,101,44)(10,58,102,45)(11,59,103,46)(12,60,104,47)(13,61,105,48)(14,62,106,49)(15,63,107,50)(16,64,108,51)(17,65,109,52)(18,66,110,53)(19,67,111,54)(20,68,112,55)(21,69,85,56)(22,70,86,29)(23,71,87,30)(24,72,88,31)(25,73,89,32)(26,74,90,33)(27,75,91,34)(28,76,92,35), (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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)>;
G:=Group( (1,77,93,36)(2,78,94,37)(3,79,95,38)(4,80,96,39)(5,81,97,40)(6,82,98,41)(7,83,99,42)(8,84,100,43)(9,57,101,44)(10,58,102,45)(11,59,103,46)(12,60,104,47)(13,61,105,48)(14,62,106,49)(15,63,107,50)(16,64,108,51)(17,65,109,52)(18,66,110,53)(19,67,111,54)(20,68,112,55)(21,69,85,56)(22,70,86,29)(23,71,87,30)(24,72,88,31)(25,73,89,32)(26,74,90,33)(27,75,91,34)(28,76,92,35), (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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112) );
G=PermutationGroup([[(1,77,93,36),(2,78,94,37),(3,79,95,38),(4,80,96,39),(5,81,97,40),(6,82,98,41),(7,83,99,42),(8,84,100,43),(9,57,101,44),(10,58,102,45),(11,59,103,46),(12,60,104,47),(13,61,105,48),(14,62,106,49),(15,63,107,50),(16,64,108,51),(17,65,109,52),(18,66,110,53),(19,67,111,54),(20,68,112,55),(21,69,85,56),(22,70,86,29),(23,71,87,30),(24,72,88,31),(25,73,89,32),(26,74,90,33),(27,75,91,34),(28,76,92,35)], [(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),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)]])
C4×C28 is a maximal subgroup of
C42.D7 C28⋊C8 Dic14⋊C4 C28⋊2Q8 C28.6Q8 C42⋊D7 C28⋊4D4 C4.D28 C42⋊2D7 C42⋊(C7⋊C3)
112 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4L | 7A | ··· | 7F | 14A | ··· | 14R | 28A | ··· | 28BT |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
112 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||
| image | C1 | C2 | C4 | C7 | C14 | C28 |
| kernel | C4×C28 | C2×C28 | C28 | C42 | C2×C4 | C4 |
| # reps | 1 | 3 | 12 | 6 | 18 | 72 |
Matrix representation of C4×C28 ►in GL2(𝔽29) generated by
| 17 | 0 |
| 0 | 1 |
| 26 | 0 |
| 0 | 14 |
G:=sub<GL(2,GF(29))| [17,0,0,1],[26,0,0,14] >;
C4×C28 in GAP, Magma, Sage, TeX
C_4\times C_{28} % in TeX
G:=Group("C4xC28"); // GroupNames label
G:=SmallGroup(112,19);
// by ID
G=gap.SmallGroup(112,19);
# by ID
G:=PCGroup([5,-2,-2,-7,-2,-2,140,286]);
// Polycyclic
G:=Group<a,b|a^4=b^28=1,a*b=b*a>;
// generators/relations
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