direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×C17⋊C4, C17⋊C42, C68⋊2C4, Dic17⋊2C4, D34.4C22, C34.3(C2×C4), (C4×D17).6C2, D17.2(C2×C4), C2.2(C2×C17⋊C4), (C2×C17⋊C4).3C2, SmallGroup(272,31)
Series: Derived ►Chief ►Lower central ►Upper central
| C17 — C4×C17⋊C4 |
Generators and relations for C4×C17⋊C4
G = < a,b,c | a4=b17=c4=1, ab=ba, ac=ca, cbc-1=b4 >
Smallest permutation representation of C4×C17⋊C4
►On 68 points
Generators in S68
(1 52 18 35)(2 53 19 36)(3 54 20 37)(4 55 21 38)(5 56 22 39)(6 57 23 40)(7 58 24 41)(8 59 25 42)(9 60 26 43)(10 61 27 44)(11 62 28 45)(12 63 29 46)(13 64 30 47)(14 65 31 48)(15 66 32 49)(16 67 33 50)(17 68 34 51)
(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)
(1 52 18 35)(2 65 34 39)(3 61 33 43)(4 57 32 47)(5 53 31 51)(6 66 30 38)(7 62 29 42)(8 58 28 46)(9 54 27 50)(10 67 26 37)(11 63 25 41)(12 59 24 45)(13 55 23 49)(14 68 22 36)(15 64 21 40)(16 60 20 44)(17 56 19 48)
G:=sub<Sym(68)| (1,52,18,35)(2,53,19,36)(3,54,20,37)(4,55,21,38)(5,56,22,39)(6,57,23,40)(7,58,24,41)(8,59,25,42)(9,60,26,43)(10,61,27,44)(11,62,28,45)(12,63,29,46)(13,64,30,47)(14,65,31,48)(15,66,32,49)(16,67,33,50)(17,68,34,51), (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), (1,52,18,35)(2,65,34,39)(3,61,33,43)(4,57,32,47)(5,53,31,51)(6,66,30,38)(7,62,29,42)(8,58,28,46)(9,54,27,50)(10,67,26,37)(11,63,25,41)(12,59,24,45)(13,55,23,49)(14,68,22,36)(15,64,21,40)(16,60,20,44)(17,56,19,48)>;
G:=Group( (1,52,18,35)(2,53,19,36)(3,54,20,37)(4,55,21,38)(5,56,22,39)(6,57,23,40)(7,58,24,41)(8,59,25,42)(9,60,26,43)(10,61,27,44)(11,62,28,45)(12,63,29,46)(13,64,30,47)(14,65,31,48)(15,66,32,49)(16,67,33,50)(17,68,34,51), (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), (1,52,18,35)(2,65,34,39)(3,61,33,43)(4,57,32,47)(5,53,31,51)(6,66,30,38)(7,62,29,42)(8,58,28,46)(9,54,27,50)(10,67,26,37)(11,63,25,41)(12,59,24,45)(13,55,23,49)(14,68,22,36)(15,64,21,40)(16,60,20,44)(17,56,19,48) );
G=PermutationGroup([[(1,52,18,35),(2,53,19,36),(3,54,20,37),(4,55,21,38),(5,56,22,39),(6,57,23,40),(7,58,24,41),(8,59,25,42),(9,60,26,43),(10,61,27,44),(11,62,28,45),(12,63,29,46),(13,64,30,47),(14,65,31,48),(15,66,32,49),(16,67,33,50),(17,68,34,51)], [(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)], [(1,52,18,35),(2,65,34,39),(3,61,33,43),(4,57,32,47),(5,53,31,51),(6,66,30,38),(7,62,29,42),(8,58,28,46),(9,54,27,50),(10,67,26,37),(11,63,25,41),(12,59,24,45),(13,55,23,49),(14,68,22,36),(15,64,21,40),(16,60,20,44),(17,56,19,48)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | ··· | 4L | 17A | 17B | 17C | 17D | 34A | 34B | 34C | 34D | 68A | ··· | 68H |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 17 | 17 | 17 | 17 | 34 | 34 | 34 | 34 | 68 | ··· | 68 |
| size | 1 | 1 | 17 | 17 | 1 | 1 | 17 | ··· | 17 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 |
| type | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | C17⋊C4 | C2×C17⋊C4 | C4×C17⋊C4 |
| kernel | C4×C17⋊C4 | C4×D17 | C2×C17⋊C4 | Dic17 | C68 | C17⋊C4 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 8 | 4 | 4 | 8 |
Matrix representation of C4×C17⋊C4 ►in GL5(𝔽137)
| 37 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 63 | 27 | 63 | 136 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 136 | 63 | 27 | 63 |
| 0 | 136 | 1 | 74 | 109 |
| 0 | 109 | 74 | 1 | 136 |
G:=sub<GL(5,GF(137))| [37,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,63,1,0,0,0,27,0,1,0,0,63,0,0,1,0,136,0,0,0],[1,0,0,0,0,0,1,136,136,109,0,0,63,1,74,0,0,27,74,1,0,0,63,109,136] >;
C4×C17⋊C4 in GAP, Magma, Sage, TeX
C_4\times C_{17}\rtimes C_4 % in TeX
G:=Group("C4xC17:C4"); // GroupNames label
G:=SmallGroup(272,31);
// by ID
G=gap.SmallGroup(272,31);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-17,20,46,5204,1614]);
// Polycyclic
G:=Group<a,b,c|a^4=b^17=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations
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