direct product, metacyclic, supersoluble, monomial, Z-group
Aliases: C7⋊C3×D11, C77⋊2C6, (C7×D11)⋊C3, C7⋊2(C3×D11), C11⋊(C2×C7⋊C3), (C11×C7⋊C3)⋊2C2, SmallGroup(462,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C77 — C7⋊C3×D11 |
Generators and relations for C7⋊C3×D11
G = < a,b,c,d | a7=b3=c11=d2=1, bab-1=a4, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Smallest permutation representation of C7⋊C3×D11
►On 77 points
Generators in S77
(1 21 32 43 54 65 76)(2 22 33 44 55 66 77)(3 12 23 34 45 56 67)(4 13 24 35 46 57 68)(5 14 25 36 47 58 69)(6 15 26 37 48 59 70)(7 16 27 38 49 60 71)(8 17 28 39 50 61 72)(9 18 29 40 51 62 73)(10 19 30 41 52 63 74)(11 20 31 42 53 64 75)
(12 23 45)(13 24 46)(14 25 47)(15 26 48)(16 27 49)(17 28 50)(18 29 51)(19 30 52)(20 31 53)(21 32 54)(22 33 55)(34 67 56)(35 68 57)(36 69 58)(37 70 59)(38 71 60)(39 72 61)(40 73 62)(41 74 63)(42 75 64)(43 76 65)(44 77 66)
(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 11)(2 10)(3 9)(4 8)(5 7)(12 18)(13 17)(14 16)(19 22)(20 21)(23 29)(24 28)(25 27)(30 33)(31 32)(34 40)(35 39)(36 38)(41 44)(42 43)(45 51)(46 50)(47 49)(52 55)(53 54)(56 62)(57 61)(58 60)(63 66)(64 65)(67 73)(68 72)(69 71)(74 77)(75 76)
G:=sub<Sym(77)| (1,21,32,43,54,65,76)(2,22,33,44,55,66,77)(3,12,23,34,45,56,67)(4,13,24,35,46,57,68)(5,14,25,36,47,58,69)(6,15,26,37,48,59,70)(7,16,27,38,49,60,71)(8,17,28,39,50,61,72)(9,18,29,40,51,62,73)(10,19,30,41,52,63,74)(11,20,31,42,53,64,75), (12,23,45)(13,24,46)(14,25,47)(15,26,48)(16,27,49)(17,28,50)(18,29,51)(19,30,52)(20,31,53)(21,32,54)(22,33,55)(34,67,56)(35,68,57)(36,69,58)(37,70,59)(38,71,60)(39,72,61)(40,73,62)(41,74,63)(42,75,64)(43,76,65)(44,77,66), (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), (1,11)(2,10)(3,9)(4,8)(5,7)(12,18)(13,17)(14,16)(19,22)(20,21)(23,29)(24,28)(25,27)(30,33)(31,32)(34,40)(35,39)(36,38)(41,44)(42,43)(45,51)(46,50)(47,49)(52,55)(53,54)(56,62)(57,61)(58,60)(63,66)(64,65)(67,73)(68,72)(69,71)(74,77)(75,76)>;
G:=Group( (1,21,32,43,54,65,76)(2,22,33,44,55,66,77)(3,12,23,34,45,56,67)(4,13,24,35,46,57,68)(5,14,25,36,47,58,69)(6,15,26,37,48,59,70)(7,16,27,38,49,60,71)(8,17,28,39,50,61,72)(9,18,29,40,51,62,73)(10,19,30,41,52,63,74)(11,20,31,42,53,64,75), (12,23,45)(13,24,46)(14,25,47)(15,26,48)(16,27,49)(17,28,50)(18,29,51)(19,30,52)(20,31,53)(21,32,54)(22,33,55)(34,67,56)(35,68,57)(36,69,58)(37,70,59)(38,71,60)(39,72,61)(40,73,62)(41,74,63)(42,75,64)(43,76,65)(44,77,66), (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), (1,11)(2,10)(3,9)(4,8)(5,7)(12,18)(13,17)(14,16)(19,22)(20,21)(23,29)(24,28)(25,27)(30,33)(31,32)(34,40)(35,39)(36,38)(41,44)(42,43)(45,51)(46,50)(47,49)(52,55)(53,54)(56,62)(57,61)(58,60)(63,66)(64,65)(67,73)(68,72)(69,71)(74,77)(75,76) );
G=PermutationGroup([[(1,21,32,43,54,65,76),(2,22,33,44,55,66,77),(3,12,23,34,45,56,67),(4,13,24,35,46,57,68),(5,14,25,36,47,58,69),(6,15,26,37,48,59,70),(7,16,27,38,49,60,71),(8,17,28,39,50,61,72),(9,18,29,40,51,62,73),(10,19,30,41,52,63,74),(11,20,31,42,53,64,75)], [(12,23,45),(13,24,46),(14,25,47),(15,26,48),(16,27,49),(17,28,50),(18,29,51),(19,30,52),(20,31,53),(21,32,54),(22,33,55),(34,67,56),(35,68,57),(36,69,58),(37,70,59),(38,71,60),(39,72,61),(40,73,62),(41,74,63),(42,75,64),(43,76,65),(44,77,66)], [(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)], [(1,11),(2,10),(3,9),(4,8),(5,7),(12,18),(13,17),(14,16),(19,22),(20,21),(23,29),(24,28),(25,27),(30,33),(31,32),(34,40),(35,39),(36,38),(41,44),(42,43),(45,51),(46,50),(47,49),(52,55),(53,54),(56,62),(57,61),(58,60),(63,66),(64,65),(67,73),(68,72),(69,71),(74,77),(75,76)]])
35 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 7A | 7B | 11A | ··· | 11E | 14A | 14B | 33A | ··· | 33J | 77A | ··· | 77J |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 7 | 7 | 11 | ··· | 11 | 14 | 14 | 33 | ··· | 33 | 77 | ··· | 77 |
| size | 1 | 11 | 7 | 7 | 77 | 77 | 3 | 3 | 2 | ··· | 2 | 33 | 33 | 14 | ··· | 14 | 6 | ··· | 6 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 6 |
| type | + | + | + | ||||||
| image | C1 | C2 | C3 | C6 | D11 | C3×D11 | C7⋊C3 | C2×C7⋊C3 | C7⋊C3×D11 |
| kernel | C7⋊C3×D11 | C11×C7⋊C3 | C7×D11 | C77 | C7⋊C3 | C7 | D11 | C11 | C1 |
| # reps | 1 | 1 | 2 | 2 | 5 | 10 | 2 | 2 | 10 |
Matrix representation of C7⋊C3×D11 ►in GL5(𝔽463)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 181 |
| 0 | 0 | 347 | 462 | 382 |
| 0 | 0 | 0 | 1 | 382 |
| 21 | 0 | 0 | 0 | 0 |
| 0 | 21 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 336 |
| 0 | 0 | 0 | 0 | 462 |
| 0 | 0 | 0 | 1 | 462 |
| 215 | 1 | 0 | 0 | 0 |
| 442 | 366 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 379 | 334 | 0 | 0 | 0 |
| 349 | 84 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(463))| [1,0,0,0,0,0,1,0,0,0,0,0,1,347,0,0,0,0,462,1,0,0,181,382,382],[21,0,0,0,0,0,21,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,336,462,462],[215,442,0,0,0,1,366,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[379,349,0,0,0,334,84,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1] >;
C7⋊C3×D11 in GAP, Magma, Sage, TeX
C_7\rtimes C_3\times D_{11} % in TeX
G:=Group("C7:C3xD11"); // GroupNames label
G:=SmallGroup(462,1);
// by ID
G=gap.SmallGroup(462,1);
# by ID
G:=PCGroup([4,-2,-3,-7,-11,78,6723]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^3=c^11=d^2=1,b*a*b^-1=a^4,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
Export