metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D26:1C4, C26.6D4, C2.2D52, C22.6D26, (C2xC52):1C2, (C2xC4):1D13, C2.5(C4xD13), C13:2(C22:C4), C26.12(C2xC4), (C2xDic13):1C2, C2.2(C13:D4), (C2xC26).6C22, (C22xD13).1C2, SmallGroup(208,14)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D26:C4
G = < a,b,c | a26=b2=c4=1, bab=a-1, ac=ca, cbc-1=a13b >
Quotients: C1, C2, C4, C22, C2xC4, D4, C22:C4, D13, D26, C4xD13, D52, C13:D4, D26:C4
Smallest permutation representation of D26:C4
►On 104 points
Generators in S104
(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)
(1 44)(2 43)(3 42)(4 41)(5 40)(6 39)(7 38)(8 37)(9 36)(10 35)(11 34)(12 33)(13 32)(14 31)(15 30)(16 29)(17 28)(18 27)(19 52)(20 51)(21 50)(22 49)(23 48)(24 47)(25 46)(26 45)(53 84)(54 83)(55 82)(56 81)(57 80)(58 79)(59 104)(60 103)(61 102)(62 101)(63 100)(64 99)(65 98)(66 97)(67 96)(68 95)(69 94)(70 93)(71 92)(72 91)(73 90)(74 89)(75 88)(76 87)(77 86)(78 85)
(1 91 45 60)(2 92 46 61)(3 93 47 62)(4 94 48 63)(5 95 49 64)(6 96 50 65)(7 97 51 66)(8 98 52 67)(9 99 27 68)(10 100 28 69)(11 101 29 70)(12 102 30 71)(13 103 31 72)(14 104 32 73)(15 79 33 74)(16 80 34 75)(17 81 35 76)(18 82 36 77)(19 83 37 78)(20 84 38 53)(21 85 39 54)(22 86 40 55)(23 87 41 56)(24 88 42 57)(25 89 43 58)(26 90 44 59)
G:=sub<Sym(104)| (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), (1,44)(2,43)(3,42)(4,41)(5,40)(6,39)(7,38)(8,37)(9,36)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,52)(20,51)(21,50)(22,49)(23,48)(24,47)(25,46)(26,45)(53,84)(54,83)(55,82)(56,81)(57,80)(58,79)(59,104)(60,103)(61,102)(62,101)(63,100)(64,99)(65,98)(66,97)(67,96)(68,95)(69,94)(70,93)(71,92)(72,91)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85), (1,91,45,60)(2,92,46,61)(3,93,47,62)(4,94,48,63)(5,95,49,64)(6,96,50,65)(7,97,51,66)(8,98,52,67)(9,99,27,68)(10,100,28,69)(11,101,29,70)(12,102,30,71)(13,103,31,72)(14,104,32,73)(15,79,33,74)(16,80,34,75)(17,81,35,76)(18,82,36,77)(19,83,37,78)(20,84,38,53)(21,85,39,54)(22,86,40,55)(23,87,41,56)(24,88,42,57)(25,89,43,58)(26,90,44,59)>;
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), (1,44)(2,43)(3,42)(4,41)(5,40)(6,39)(7,38)(8,37)(9,36)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,52)(20,51)(21,50)(22,49)(23,48)(24,47)(25,46)(26,45)(53,84)(54,83)(55,82)(56,81)(57,80)(58,79)(59,104)(60,103)(61,102)(62,101)(63,100)(64,99)(65,98)(66,97)(67,96)(68,95)(69,94)(70,93)(71,92)(72,91)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85), (1,91,45,60)(2,92,46,61)(3,93,47,62)(4,94,48,63)(5,95,49,64)(6,96,50,65)(7,97,51,66)(8,98,52,67)(9,99,27,68)(10,100,28,69)(11,101,29,70)(12,102,30,71)(13,103,31,72)(14,104,32,73)(15,79,33,74)(16,80,34,75)(17,81,35,76)(18,82,36,77)(19,83,37,78)(20,84,38,53)(21,85,39,54)(22,86,40,55)(23,87,41,56)(24,88,42,57)(25,89,43,58)(26,90,44,59) );
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)], [(1,44),(2,43),(3,42),(4,41),(5,40),(6,39),(7,38),(8,37),(9,36),(10,35),(11,34),(12,33),(13,32),(14,31),(15,30),(16,29),(17,28),(18,27),(19,52),(20,51),(21,50),(22,49),(23,48),(24,47),(25,46),(26,45),(53,84),(54,83),(55,82),(56,81),(57,80),(58,79),(59,104),(60,103),(61,102),(62,101),(63,100),(64,99),(65,98),(66,97),(67,96),(68,95),(69,94),(70,93),(71,92),(72,91),(73,90),(74,89),(75,88),(76,87),(77,86),(78,85)], [(1,91,45,60),(2,92,46,61),(3,93,47,62),(4,94,48,63),(5,95,49,64),(6,96,50,65),(7,97,51,66),(8,98,52,67),(9,99,27,68),(10,100,28,69),(11,101,29,70),(12,102,30,71),(13,103,31,72),(14,104,32,73),(15,79,33,74),(16,80,34,75),(17,81,35,76),(18,82,36,77),(19,83,37,78),(20,84,38,53),(21,85,39,54),(22,86,40,55),(23,87,41,56),(24,88,42,57),(25,89,43,58),(26,90,44,59)]])
D26:C4 is a maximal subgroup of
C42:D13 C4xD52 C4.D52 C42:2D13 C22:C4xD13 Dic13:4D4 C22:D52 D26.12D4 D26:D4 C23.6D26 C22.D52 C4:C4:7D13 D52:8C4 D26.13D4 C4:2D52 D26:Q8 D26:2Q8 C4:C4:D13 C4xC13:D4 C23.23D26 C52:7D4 C23:D26 Dic13:D4 D26:3Q8 C52.23D4
D26:C4 is a maximal quotient of
D52:4C4 C22.2D52 D52:6C4 C26.Q16 C52.44D4 D26:1C8 D52:5C4 C52.46D4 C4.12D52 D52:7C4 C26.10C42
58 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 13A | ··· | 13F | 26A | ··· | 26R | 52A | ··· | 52X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 26 | 26 | 2 | 2 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D13 | D26 | C4xD13 | D52 | C13:D4 |
| kernel | D26:C4 | C2xDic13 | C2xC52 | C22xD13 | D26 | C26 | C2xC4 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 2 | 6 | 6 | 12 | 12 | 12 |
Matrix representation of D26:C4 ►in GL3(F53) generated by
| 1 | 0 | 0 |
| 0 | 15 | 2 |
| 0 | 10 | 12 |
| 1 | 0 | 0 |
| 0 | 13 | 41 |
| 0 | 14 | 40 |
| 23 | 0 | 0 |
| 0 | 26 | 17 |
| 0 | 32 | 27 |
G:=sub<GL(3,GF(53))| [1,0,0,0,15,10,0,2,12],[1,0,0,0,13,14,0,41,40],[23,0,0,0,26,32,0,17,27] >;
D26:C4 in GAP, Magma, Sage, TeX
D_{26}\rtimes C_4 % in TeX
G:=Group("D26:C4"); // GroupNames label
G:=SmallGroup(208,14);
// by ID
G=gap.SmallGroup(208,14);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-13,101,26,4804]);
// Polycyclic
G:=Group<a,b,c|a^26=b^2=c^4=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^13*b>;
// generators/relations
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