metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C109⋊C4, D109.C2, SmallGroup(436,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C109 — C109⋊C4 |
Generators and relations for C109⋊C4
G = < a,b | a109=b4=1, bab-1=a76 >
Smallest permutation representation of C109⋊C4
►On 109 points: primitive
Generators in S109
(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)
(2 34 109 77)(3 67 108 44)(4 100 107 11)(5 24 106 87)(6 57 105 54)(7 90 104 21)(8 14 103 97)(9 47 102 64)(10 80 101 31)(12 37 99 74)(13 70 98 41)(15 27 96 84)(16 60 95 51)(17 93 94 18)(19 50 92 61)(20 83 91 28)(22 40 89 71)(23 73 88 38)(25 30 86 81)(26 63 85 48)(29 53 82 58)(32 43 79 68)(33 76 78 35)(36 66 75 45)(39 56 72 55)(42 46 69 65)(49 59 62 52)
G:=sub<Sym(109)| (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), (2,34,109,77)(3,67,108,44)(4,100,107,11)(5,24,106,87)(6,57,105,54)(7,90,104,21)(8,14,103,97)(9,47,102,64)(10,80,101,31)(12,37,99,74)(13,70,98,41)(15,27,96,84)(16,60,95,51)(17,93,94,18)(19,50,92,61)(20,83,91,28)(22,40,89,71)(23,73,88,38)(25,30,86,81)(26,63,85,48)(29,53,82,58)(32,43,79,68)(33,76,78,35)(36,66,75,45)(39,56,72,55)(42,46,69,65)(49,59,62,52)>;
G:=Group( (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), (2,34,109,77)(3,67,108,44)(4,100,107,11)(5,24,106,87)(6,57,105,54)(7,90,104,21)(8,14,103,97)(9,47,102,64)(10,80,101,31)(12,37,99,74)(13,70,98,41)(15,27,96,84)(16,60,95,51)(17,93,94,18)(19,50,92,61)(20,83,91,28)(22,40,89,71)(23,73,88,38)(25,30,86,81)(26,63,85,48)(29,53,82,58)(32,43,79,68)(33,76,78,35)(36,66,75,45)(39,56,72,55)(42,46,69,65)(49,59,62,52) );
G=PermutationGroup([[(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)], [(2,34,109,77),(3,67,108,44),(4,100,107,11),(5,24,106,87),(6,57,105,54),(7,90,104,21),(8,14,103,97),(9,47,102,64),(10,80,101,31),(12,37,99,74),(13,70,98,41),(15,27,96,84),(16,60,95,51),(17,93,94,18),(19,50,92,61),(20,83,91,28),(22,40,89,71),(23,73,88,38),(25,30,86,81),(26,63,85,48),(29,53,82,58),(32,43,79,68),(33,76,78,35),(36,66,75,45),(39,56,72,55),(42,46,69,65),(49,59,62,52)]])
31 conjugacy classes
| class | 1 | 2 | 4A | 4B | 109A | ··· | 109AA |
| order | 1 | 2 | 4 | 4 | 109 | ··· | 109 |
| size | 1 | 109 | 109 | 109 | 4 | ··· | 4 |
31 irreducible representations
| dim | 1 | 1 | 1 | 4 |
| type | + | + | + | |
| image | C1 | C2 | C4 | C109⋊C4 |
| kernel | C109⋊C4 | D109 | C109 | C1 |
| # reps | 1 | 1 | 2 | 27 |
Matrix representation of C109⋊C4 ►in GL4(𝔽2617) generated by
| 1387 | 1 | 0 | 0 |
| 1775 | 0 | 1 | 0 |
| 2275 | 0 | 0 | 1 |
| 1267 | 33 | 405 | 365 |
| 1240 | 1736 | 767 | 2008 |
| 633 | 1460 | 151 | 1064 |
| 2199 | 1624 | 2388 | 1699 |
| 818 | 2515 | 2229 | 146 |
G:=sub<GL(4,GF(2617))| [1387,1775,2275,1267,1,0,0,33,0,1,0,405,0,0,1,365],[1240,633,2199,818,1736,1460,1624,2515,767,151,2388,2229,2008,1064,1699,146] >;
C109⋊C4 in GAP, Magma, Sage, TeX
C_{109}\rtimes C_4 % in TeX
G:=Group("C109:C4"); // GroupNames label
G:=SmallGroup(436,3);
// by ID
G=gap.SmallGroup(436,3);
# by ID
G:=PCGroup([3,-2,-2,-109,6,1190,1949]);
// Polycyclic
G:=Group<a,b|a^109=b^4=1,b*a*b^-1=a^76>;
// generators/relations
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