direct product, metabelian, soluble, monomial, A-group
Aliases: C2×C13⋊A4, C26⋊A4, C13⋊2(C2×A4), C23⋊(C13⋊C3), (C2×C26)⋊7C6, (C22×C26)⋊3C3, C22⋊(C2×C13⋊C3), SmallGroup(312,57)
Series: Derived ►Chief ►Lower central ►Upper central
| C2×C26 — C2×C13⋊A4 |
Generators and relations for C2×C13⋊A4
G = < a,b,c,d,e | a2=b13=c2=d2=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b9, ece-1=cd=dc, ede-1=c >
Smallest permutation representation of C2×C13⋊A4
►On 78 points
Generators in S78
(1 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 25)(13 26)(27 48)(28 49)(29 50)(30 51)(31 52)(32 40)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(53 78)(54 66)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)(61 73)(62 74)(63 75)(64 76)(65 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 78)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 25)(13 26)(27 48)(28 49)(29 50)(30 51)(31 52)(32 40)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 25)(13 26)(53 78)(54 66)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)(61 73)(62 74)(63 75)(64 76)(65 77)
(1 53 27)(2 56 36)(3 59 32)(4 62 28)(5 65 37)(6 55 33)(7 58 29)(8 61 38)(9 64 34)(10 54 30)(11 57 39)(12 60 35)(13 63 31)(14 78 48)(15 68 44)(16 71 40)(17 74 49)(18 77 45)(19 67 41)(20 70 50)(21 73 46)(22 76 42)(23 66 51)(24 69 47)(25 72 43)(26 75 52)
G:=sub<Sym(78)| (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(27,48)(28,49)(29,50)(30,51)(31,52)(32,40)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(53,78)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(61,73)(62,74)(63,75)(64,76)(65,77), (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)(66,67,68,69,70,71,72,73,74,75,76,77,78), (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(27,48)(28,49)(29,50)(30,51)(31,52)(32,40)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47), (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(53,78)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(61,73)(62,74)(63,75)(64,76)(65,77), (1,53,27)(2,56,36)(3,59,32)(4,62,28)(5,65,37)(6,55,33)(7,58,29)(8,61,38)(9,64,34)(10,54,30)(11,57,39)(12,60,35)(13,63,31)(14,78,48)(15,68,44)(16,71,40)(17,74,49)(18,77,45)(19,67,41)(20,70,50)(21,73,46)(22,76,42)(23,66,51)(24,69,47)(25,72,43)(26,75,52)>;
G:=Group( (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(27,48)(28,49)(29,50)(30,51)(31,52)(32,40)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(53,78)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(61,73)(62,74)(63,75)(64,76)(65,77), (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)(66,67,68,69,70,71,72,73,74,75,76,77,78), (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(27,48)(28,49)(29,50)(30,51)(31,52)(32,40)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47), (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(53,78)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(61,73)(62,74)(63,75)(64,76)(65,77), (1,53,27)(2,56,36)(3,59,32)(4,62,28)(5,65,37)(6,55,33)(7,58,29)(8,61,38)(9,64,34)(10,54,30)(11,57,39)(12,60,35)(13,63,31)(14,78,48)(15,68,44)(16,71,40)(17,74,49)(18,77,45)(19,67,41)(20,70,50)(21,73,46)(22,76,42)(23,66,51)(24,69,47)(25,72,43)(26,75,52) );
G=PermutationGroup([[(1,14),(2,15),(3,16),(4,17),(5,18),(6,19),(7,20),(8,21),(9,22),(10,23),(11,24),(12,25),(13,26),(27,48),(28,49),(29,50),(30,51),(31,52),(32,40),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(53,78),(54,66),(55,67),(56,68),(57,69),(58,70),(59,71),(60,72),(61,73),(62,74),(63,75),(64,76),(65,77)], [(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),(66,67,68,69,70,71,72,73,74,75,76,77,78)], [(1,14),(2,15),(3,16),(4,17),(5,18),(6,19),(7,20),(8,21),(9,22),(10,23),(11,24),(12,25),(13,26),(27,48),(28,49),(29,50),(30,51),(31,52),(32,40),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47)], [(1,14),(2,15),(3,16),(4,17),(5,18),(6,19),(7,20),(8,21),(9,22),(10,23),(11,24),(12,25),(13,26),(53,78),(54,66),(55,67),(56,68),(57,69),(58,70),(59,71),(60,72),(61,73),(62,74),(63,75),(64,76),(65,77)], [(1,53,27),(2,56,36),(3,59,32),(4,62,28),(5,65,37),(6,55,33),(7,58,29),(8,61,38),(9,64,34),(10,54,30),(11,57,39),(12,60,35),(13,63,31),(14,78,48),(15,68,44),(16,71,40),(17,74,49),(18,77,45),(19,67,41),(20,70,50),(21,73,46),(22,76,42),(23,66,51),(24,69,47),(25,72,43),(26,75,52)]])
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | 6B | 13A | 13B | 13C | 13D | 26A | ··· | 26AB |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 13 | 13 | 13 | 13 | 26 | ··· | 26 |
| size | 1 | 1 | 3 | 3 | 52 | 52 | 52 | 52 | 3 | 3 | 3 | 3 | 3 | ··· | 3 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | ||||||
| image | C1 | C2 | C3 | C6 | A4 | C2×A4 | C13⋊C3 | C2×C13⋊C3 | C13⋊A4 | C2×C13⋊A4 |
| kernel | C2×C13⋊A4 | C13⋊A4 | C22×C26 | C2×C26 | C26 | C13 | C23 | C22 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 4 | 12 | 12 |
Matrix representation of C2×C13⋊A4 ►in GL3(𝔽79) generated by
| 78 | 0 | 0 |
| 0 | 78 | 0 |
| 0 | 0 | 78 |
| 52 | 0 | 0 |
| 0 | 67 | 0 |
| 0 | 0 | 10 |
| 78 | 0 | 0 |
| 0 | 78 | 0 |
| 0 | 0 | 1 |
| 78 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 78 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
G:=sub<GL(3,GF(79))| [78,0,0,0,78,0,0,0,78],[52,0,0,0,67,0,0,0,10],[78,0,0,0,78,0,0,0,1],[78,0,0,0,1,0,0,0,78],[0,0,1,1,0,0,0,1,0] >;
C2×C13⋊A4 in GAP, Magma, Sage, TeX
C_2\times C_{13}\rtimes A_4 % in TeX
G:=Group("C2xC13:A4"); // GroupNames label
G:=SmallGroup(312,57);
// by ID
G=gap.SmallGroup(312,57);
# by ID
G:=PCGroup([5,-2,-3,-2,2,-13,97,188,909]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^13=c^2=d^2=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b^9,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations
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