direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C11×C7⋊C3, C7⋊C33, C77⋊C3, SmallGroup(231,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C11×C7⋊C3 |
Generators and relations for C11×C7⋊C3
G = < a,b,c | a11=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >
Smallest permutation representation of C11×C7⋊C3
►On 77 points
Generators in S77
(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)
(1 38 48 28 15 60 70)(2 39 49 29 16 61 71)(3 40 50 30 17 62 72)(4 41 51 31 18 63 73)(5 42 52 32 19 64 74)(6 43 53 33 20 65 75)(7 44 54 23 21 66 76)(8 34 55 24 22 56 77)(9 35 45 25 12 57 67)(10 36 46 26 13 58 68)(11 37 47 27 14 59 69)
(12 35 45)(13 36 46)(14 37 47)(15 38 48)(16 39 49)(17 40 50)(18 41 51)(19 42 52)(20 43 53)(21 44 54)(22 34 55)(23 76 66)(24 77 56)(25 67 57)(26 68 58)(27 69 59)(28 70 60)(29 71 61)(30 72 62)(31 73 63)(32 74 64)(33 75 65)
G:=sub<Sym(77)| (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), (1,38,48,28,15,60,70)(2,39,49,29,16,61,71)(3,40,50,30,17,62,72)(4,41,51,31,18,63,73)(5,42,52,32,19,64,74)(6,43,53,33,20,65,75)(7,44,54,23,21,66,76)(8,34,55,24,22,56,77)(9,35,45,25,12,57,67)(10,36,46,26,13,58,68)(11,37,47,27,14,59,69), (12,35,45)(13,36,46)(14,37,47)(15,38,48)(16,39,49)(17,40,50)(18,41,51)(19,42,52)(20,43,53)(21,44,54)(22,34,55)(23,76,66)(24,77,56)(25,67,57)(26,68,58)(27,69,59)(28,70,60)(29,71,61)(30,72,62)(31,73,63)(32,74,64)(33,75,65)>;
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), (1,38,48,28,15,60,70)(2,39,49,29,16,61,71)(3,40,50,30,17,62,72)(4,41,51,31,18,63,73)(5,42,52,32,19,64,74)(6,43,53,33,20,65,75)(7,44,54,23,21,66,76)(8,34,55,24,22,56,77)(9,35,45,25,12,57,67)(10,36,46,26,13,58,68)(11,37,47,27,14,59,69), (12,35,45)(13,36,46)(14,37,47)(15,38,48)(16,39,49)(17,40,50)(18,41,51)(19,42,52)(20,43,53)(21,44,54)(22,34,55)(23,76,66)(24,77,56)(25,67,57)(26,68,58)(27,69,59)(28,70,60)(29,71,61)(30,72,62)(31,73,63)(32,74,64)(33,75,65) );
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)], [(1,38,48,28,15,60,70),(2,39,49,29,16,61,71),(3,40,50,30,17,62,72),(4,41,51,31,18,63,73),(5,42,52,32,19,64,74),(6,43,53,33,20,65,75),(7,44,54,23,21,66,76),(8,34,55,24,22,56,77),(9,35,45,25,12,57,67),(10,36,46,26,13,58,68),(11,37,47,27,14,59,69)], [(12,35,45),(13,36,46),(14,37,47),(15,38,48),(16,39,49),(17,40,50),(18,41,51),(19,42,52),(20,43,53),(21,44,54),(22,34,55),(23,76,66),(24,77,56),(25,67,57),(26,68,58),(27,69,59),(28,70,60),(29,71,61),(30,72,62),(31,73,63),(32,74,64),(33,75,65)]])
C11×C7⋊C3 is a maximal subgroup of
C11⋊F7
55 conjugacy classes
| class | 1 | 3A | 3B | 7A | 7B | 11A | ··· | 11J | 33A | ··· | 33T | 77A | ··· | 77T |
| order | 1 | 3 | 3 | 7 | 7 | 11 | ··· | 11 | 33 | ··· | 33 | 77 | ··· | 77 |
| size | 1 | 7 | 7 | 3 | 3 | 1 | ··· | 1 | 7 | ··· | 7 | 3 | ··· | 3 |
55 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | |||||
| image | C1 | C3 | C11 | C33 | C7⋊C3 | C11×C7⋊C3 |
| kernel | C11×C7⋊C3 | C77 | C7⋊C3 | C7 | C11 | C1 |
| # reps | 1 | 2 | 10 | 20 | 2 | 20 |
Matrix representation of C11×C7⋊C3 ►in GL3(𝔽463) generated by
| 158 | 0 | 0 |
| 0 | 158 | 0 |
| 0 | 0 | 158 |
| 0 | 0 | 1 |
| 1 | 0 | 383 |
| 0 | 1 | 382 |
| 1 | 0 | 382 |
| 0 | 0 | 462 |
| 0 | 1 | 462 |
G:=sub<GL(3,GF(463))| [158,0,0,0,158,0,0,0,158],[0,1,0,0,0,1,1,383,382],[1,0,0,0,0,1,382,462,462] >;
C11×C7⋊C3 in GAP, Magma, Sage, TeX
C_{11}\times C_7\rtimes C_3 % in TeX
G:=Group("C11xC7:C3"); // GroupNames label
G:=SmallGroup(231,1);
// by ID
G=gap.SmallGroup(231,1);
# by ID
G:=PCGroup([3,-3,-11,-7,596]);
// Polycyclic
G:=Group<a,b,c|a^11=b^7=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations
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