direct product, metacyclic, supersoluble, monomial, Z-group, 5-hyperelementary
Aliases: C7×C11⋊C5, C77⋊C5, C11⋊C35, SmallGroup(385,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C7×C11⋊C5 |
Generators and relations for C7×C11⋊C5
G = < a,b,c | a7=b11=c5=1, ab=ba, ac=ca, cbc-1=b3 >
Smallest permutation representation of C7×C11⋊C5
►On 77 points
Generators in S77
(1 67 56 45 34 23 12)(2 68 57 46 35 24 13)(3 69 58 47 36 25 14)(4 70 59 48 37 26 15)(5 71 60 49 38 27 16)(6 72 61 50 39 28 17)(7 73 62 51 40 29 18)(8 74 63 52 41 30 19)(9 75 64 53 42 31 20)(10 76 65 54 43 32 21)(11 77 66 55 44 33 22)
(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)
(2 5 6 10 4)(3 9 11 8 7)(13 16 17 21 15)(14 20 22 19 18)(24 27 28 32 26)(25 31 33 30 29)(35 38 39 43 37)(36 42 44 41 40)(46 49 50 54 48)(47 53 55 52 51)(57 60 61 65 59)(58 64 66 63 62)(68 71 72 76 70)(69 75 77 74 73)
G:=sub<Sym(77)| (1,67,56,45,34,23,12)(2,68,57,46,35,24,13)(3,69,58,47,36,25,14)(4,70,59,48,37,26,15)(5,71,60,49,38,27,16)(6,72,61,50,39,28,17)(7,73,62,51,40,29,18)(8,74,63,52,41,30,19)(9,75,64,53,42,31,20)(10,76,65,54,43,32,21)(11,77,66,55,44,33,22), (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), (2,5,6,10,4)(3,9,11,8,7)(13,16,17,21,15)(14,20,22,19,18)(24,27,28,32,26)(25,31,33,30,29)(35,38,39,43,37)(36,42,44,41,40)(46,49,50,54,48)(47,53,55,52,51)(57,60,61,65,59)(58,64,66,63,62)(68,71,72,76,70)(69,75,77,74,73)>;
G:=Group( (1,67,56,45,34,23,12)(2,68,57,46,35,24,13)(3,69,58,47,36,25,14)(4,70,59,48,37,26,15)(5,71,60,49,38,27,16)(6,72,61,50,39,28,17)(7,73,62,51,40,29,18)(8,74,63,52,41,30,19)(9,75,64,53,42,31,20)(10,76,65,54,43,32,21)(11,77,66,55,44,33,22), (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), (2,5,6,10,4)(3,9,11,8,7)(13,16,17,21,15)(14,20,22,19,18)(24,27,28,32,26)(25,31,33,30,29)(35,38,39,43,37)(36,42,44,41,40)(46,49,50,54,48)(47,53,55,52,51)(57,60,61,65,59)(58,64,66,63,62)(68,71,72,76,70)(69,75,77,74,73) );
G=PermutationGroup([[(1,67,56,45,34,23,12),(2,68,57,46,35,24,13),(3,69,58,47,36,25,14),(4,70,59,48,37,26,15),(5,71,60,49,38,27,16),(6,72,61,50,39,28,17),(7,73,62,51,40,29,18),(8,74,63,52,41,30,19),(9,75,64,53,42,31,20),(10,76,65,54,43,32,21),(11,77,66,55,44,33,22)], [(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)], [(2,5,6,10,4),(3,9,11,8,7),(13,16,17,21,15),(14,20,22,19,18),(24,27,28,32,26),(25,31,33,30,29),(35,38,39,43,37),(36,42,44,41,40),(46,49,50,54,48),(47,53,55,52,51),(57,60,61,65,59),(58,64,66,63,62),(68,71,72,76,70),(69,75,77,74,73)]])
49 conjugacy classes
| class | 1 | 5A | 5B | 5C | 5D | 7A | ··· | 7F | 11A | 11B | 35A | ··· | 35X | 77A | ··· | 77L |
| order | 1 | 5 | 5 | 5 | 5 | 7 | ··· | 7 | 11 | 11 | 35 | ··· | 35 | 77 | ··· | 77 |
| size | 1 | 11 | 11 | 11 | 11 | 1 | ··· | 1 | 5 | 5 | 11 | ··· | 11 | 5 | ··· | 5 |
49 irreducible representations
| dim | 1 | 1 | 1 | 1 | 5 | 5 |
| type | + | |||||
| image | C1 | C5 | C7 | C35 | C11⋊C5 | C7×C11⋊C5 |
| kernel | C7×C11⋊C5 | C77 | C11⋊C5 | C11 | C7 | C1 |
| # reps | 1 | 4 | 6 | 24 | 2 | 12 |
Matrix representation of C7×C11⋊C5 ►in GL5(𝔽2311)
| 159 | 0 | 0 | 0 | 0 |
| 0 | 159 | 0 | 0 | 0 |
| 0 | 0 | 159 | 0 | 0 |
| 0 | 0 | 0 | 159 | 0 |
| 0 | 0 | 0 | 0 | 159 |
| 1711 | 2 | 597 | 1712 | 1 |
| 1712 | 2 | 597 | 1712 | 1 |
| 1711 | 3 | 597 | 1712 | 1 |
| 1711 | 2 | 598 | 1712 | 1 |
| 1711 | 2 | 597 | 1713 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 1713 | 600 | 2309 | 1714 | 599 |
| 1714 | 599 | 1711 | 2 | 598 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(2311))| [159,0,0,0,0,0,159,0,0,0,0,0,159,0,0,0,0,0,159,0,0,0,0,0,159],[1711,1712,1711,1711,1711,2,2,3,2,2,597,597,597,598,597,1712,1712,1712,1712,1713,1,1,1,1,1],[0,1713,1714,1,0,0,600,599,0,0,1,2309,1711,0,0,0,1714,2,0,1,0,599,598,0,0] >;
C7×C11⋊C5 in GAP, Magma, Sage, TeX
C_7\times C_{11}\rtimes C_5 % in TeX
G:=Group("C7xC11:C5"); // GroupNames label
G:=SmallGroup(385,1);
// by ID
G=gap.SmallGroup(385,1);
# by ID
G:=PCGroup([3,-5,-7,-11,1262]);
// Polycyclic
G:=Group<a,b,c|a^7=b^11=c^5=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations
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