metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D26.11C23, (C2×C52)⋊4C4, (C4×D13)⋊6C4, C52⋊C4⋊3C2, C13⋊(C42⋊C2), C52.21(C2×C4), D13.D4.C2, (C2×Dic13)⋊9C4, D26.17(C2×C4), C26.6(C22×C4), D13.1(C4○D4), Dic13.17(C2×C4), (C4×D13).36C22, (C22×D13).38C22, (C4×C13⋊C4)⋊4C2, (C2×C4)⋊4(C13⋊C4), C4.13(C2×C13⋊C4), (C2×C4×D13).16C2, C22.7(C2×C13⋊C4), C2.7(C22×C13⋊C4), (C2×C26).18(C2×C4), (C2×C13⋊C4).2C22, SmallGroup(416,204)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D26.C23
G = < a,b,c,d,e | a26=b2=e2=1, c2=a-1b, d2=a13, bab=a-1, cac-1=a5, ad=da, ae=ea, cbc-1=a4b, bd=db, be=eb, cd=dc, ece=a13c, de=ed >
Subgroups: 548 in 76 conjugacy classes, 36 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C23, C13, C42, C22⋊C4, C4⋊C4, C22×C4, D13, D13, C26, C26, C42⋊C2, Dic13, C52, C13⋊C4, D26, D26, C2×C26, C4×D13, C2×Dic13, C2×C52, C2×C13⋊C4, C22×D13, C4×C13⋊C4, C52⋊C4, D13.D4, C2×C4×D13, D26.C23
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C4○D4, C42⋊C2, C13⋊C4, C2×C13⋊C4, C22×C13⋊C4, D26.C23
►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 26)(2 25)(3 24)(4 23)(5 22)(6 21)(7 20)(8 19)(9 18)(10 17)(11 16)(12 15)(13 14)(27 50)(28 49)(29 48)(30 47)(31 46)(32 45)(33 44)(34 43)(35 42)(36 41)(37 40)(38 39)(51 52)(53 70)(54 69)(55 68)(56 67)(57 66)(58 65)(59 64)(60 63)(61 62)(71 78)(72 77)(73 76)(74 75)(79 86)(80 85)(81 84)(82 83)(87 104)(88 103)(89 102)(90 101)(91 100)(92 99)(93 98)(94 97)(95 96)
(2 22 26 6)(3 17 25 11)(4 12 24 16)(5 7 23 21)(8 18 20 10)(9 13 19 15)(27 34 51 44)(28 29 50 49)(30 45 48 33)(31 40 47 38)(32 35 46 43)(36 41 42 37)(39 52)(53 55 71 69)(54 76 70 74)(56 66 68 58)(57 61 67 63)(59 77 65 73)(60 72 64 78)(79 90 87 102)(80 85 86 81)(82 101 84 91)(83 96)(88 97 104 95)(89 92 103 100)(93 98 99 94)
(1 75 14 62)(2 76 15 63)(3 77 16 64)(4 78 17 65)(5 53 18 66)(6 54 19 67)(7 55 20 68)(8 56 21 69)(9 57 22 70)(10 58 23 71)(11 59 24 72)(12 60 25 73)(13 61 26 74)(27 84 40 97)(28 85 41 98)(29 86 42 99)(30 87 43 100)(31 88 44 101)(32 89 45 102)(33 90 46 103)(34 91 47 104)(35 92 48 79)(36 93 49 80)(37 94 50 81)(38 95 51 82)(39 96 52 83)
(1 39)(2 40)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 49)(12 50)(13 51)(14 52)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)(25 37)(26 38)(53 100)(54 101)(55 102)(56 103)(57 104)(58 79)(59 80)(60 81)(61 82)(62 83)(63 84)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 91)(71 92)(72 93)(73 94)(74 95)(75 96)(76 97)(77 98)(78 99)
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), (1,26)(2,25)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)(51,52)(53,70)(54,69)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(61,62)(71,78)(72,77)(73,76)(74,75)(79,86)(80,85)(81,84)(82,83)(87,104)(88,103)(89,102)(90,101)(91,100)(92,99)(93,98)(94,97)(95,96), (2,22,26,6)(3,17,25,11)(4,12,24,16)(5,7,23,21)(8,18,20,10)(9,13,19,15)(27,34,51,44)(28,29,50,49)(30,45,48,33)(31,40,47,38)(32,35,46,43)(36,41,42,37)(39,52)(53,55,71,69)(54,76,70,74)(56,66,68,58)(57,61,67,63)(59,77,65,73)(60,72,64,78)(79,90,87,102)(80,85,86,81)(82,101,84,91)(83,96)(88,97,104,95)(89,92,103,100)(93,98,99,94), (1,75,14,62)(2,76,15,63)(3,77,16,64)(4,78,17,65)(5,53,18,66)(6,54,19,67)(7,55,20,68)(8,56,21,69)(9,57,22,70)(10,58,23,71)(11,59,24,72)(12,60,25,73)(13,61,26,74)(27,84,40,97)(28,85,41,98)(29,86,42,99)(30,87,43,100)(31,88,44,101)(32,89,45,102)(33,90,46,103)(34,91,47,104)(35,92,48,79)(36,93,49,80)(37,94,50,81)(38,95,51,82)(39,96,52,83), (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(53,100)(54,101)(55,102)(56,103)(57,104)(58,79)(59,80)(60,81)(61,82)(62,83)(63,84)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,92)(72,93)(73,94)(74,95)(75,96)(76,97)(77,98)(78,99)>;
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), (1,26)(2,25)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)(51,52)(53,70)(54,69)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(61,62)(71,78)(72,77)(73,76)(74,75)(79,86)(80,85)(81,84)(82,83)(87,104)(88,103)(89,102)(90,101)(91,100)(92,99)(93,98)(94,97)(95,96), (2,22,26,6)(3,17,25,11)(4,12,24,16)(5,7,23,21)(8,18,20,10)(9,13,19,15)(27,34,51,44)(28,29,50,49)(30,45,48,33)(31,40,47,38)(32,35,46,43)(36,41,42,37)(39,52)(53,55,71,69)(54,76,70,74)(56,66,68,58)(57,61,67,63)(59,77,65,73)(60,72,64,78)(79,90,87,102)(80,85,86,81)(82,101,84,91)(83,96)(88,97,104,95)(89,92,103,100)(93,98,99,94), (1,75,14,62)(2,76,15,63)(3,77,16,64)(4,78,17,65)(5,53,18,66)(6,54,19,67)(7,55,20,68)(8,56,21,69)(9,57,22,70)(10,58,23,71)(11,59,24,72)(12,60,25,73)(13,61,26,74)(27,84,40,97)(28,85,41,98)(29,86,42,99)(30,87,43,100)(31,88,44,101)(32,89,45,102)(33,90,46,103)(34,91,47,104)(35,92,48,79)(36,93,49,80)(37,94,50,81)(38,95,51,82)(39,96,52,83), (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(53,100)(54,101)(55,102)(56,103)(57,104)(58,79)(59,80)(60,81)(61,82)(62,83)(63,84)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,92)(72,93)(73,94)(74,95)(75,96)(76,97)(77,98)(78,99) );
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)], [(1,26),(2,25),(3,24),(4,23),(5,22),(6,21),(7,20),(8,19),(9,18),(10,17),(11,16),(12,15),(13,14),(27,50),(28,49),(29,48),(30,47),(31,46),(32,45),(33,44),(34,43),(35,42),(36,41),(37,40),(38,39),(51,52),(53,70),(54,69),(55,68),(56,67),(57,66),(58,65),(59,64),(60,63),(61,62),(71,78),(72,77),(73,76),(74,75),(79,86),(80,85),(81,84),(82,83),(87,104),(88,103),(89,102),(90,101),(91,100),(92,99),(93,98),(94,97),(95,96)], [(2,22,26,6),(3,17,25,11),(4,12,24,16),(5,7,23,21),(8,18,20,10),(9,13,19,15),(27,34,51,44),(28,29,50,49),(30,45,48,33),(31,40,47,38),(32,35,46,43),(36,41,42,37),(39,52),(53,55,71,69),(54,76,70,74),(56,66,68,58),(57,61,67,63),(59,77,65,73),(60,72,64,78),(79,90,87,102),(80,85,86,81),(82,101,84,91),(83,96),(88,97,104,95),(89,92,103,100),(93,98,99,94)], [(1,75,14,62),(2,76,15,63),(3,77,16,64),(4,78,17,65),(5,53,18,66),(6,54,19,67),(7,55,20,68),(8,56,21,69),(9,57,22,70),(10,58,23,71),(11,59,24,72),(12,60,25,73),(13,61,26,74),(27,84,40,97),(28,85,41,98),(29,86,42,99),(30,87,43,100),(31,88,44,101),(32,89,45,102),(33,90,46,103),(34,91,47,104),(35,92,48,79),(36,93,49,80),(37,94,50,81),(38,95,51,82),(39,96,52,83)], [(1,39),(2,40),(3,41),(4,42),(5,43),(6,44),(7,45),(8,46),(9,47),(10,48),(11,49),(12,50),(13,51),(14,52),(15,27),(16,28),(17,29),(18,30),(19,31),(20,32),(21,33),(22,34),(23,35),(24,36),(25,37),(26,38),(53,100),(54,101),(55,102),(56,103),(57,104),(58,79),(59,80),(60,81),(61,82),(62,83),(63,84),(64,85),(65,86),(66,87),(67,88),(68,89),(69,90),(70,91),(71,92),(72,93),(73,94),(74,95),(75,96),(76,97),(77,98),(78,99)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | ··· | 4N | 13A | 13B | 13C | 26A | ··· | 26I | 52A | ··· | 52L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 13 | 13 | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 2 | 13 | 13 | 26 | 1 | 1 | 2 | 13 | 13 | 26 | ··· | 26 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4○D4 | C13⋊C4 | C2×C13⋊C4 | C2×C13⋊C4 | D26.C23 |
| kernel | D26.C23 | C4×C13⋊C4 | C52⋊C4 | D13.D4 | C2×C4×D13 | C4×D13 | C2×Dic13 | C2×C52 | D13 | C2×C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 2 | 2 | 1 | 4 | 2 | 2 | 4 | 3 | 6 | 3 | 12 |
Matrix representation of D26.C23 ►in GL4(𝔽53) generated by
| 5 | 40 | 4 | 20 |
| 33 | 34 | 34 | 33 |
| 20 | 4 | 40 | 5 |
| 48 | 14 | 29 | 34 |
| 5 | 40 | 4 | 20 |
| 34 | 29 | 14 | 48 |
| 20 | 25 | 39 | 19 |
| 1 | 33 | 48 | 33 |
| 1 | 0 | 0 | 0 |
| 33 | 28 | 14 | 34 |
| 5 | 39 | 24 | 19 |
| 0 | 0 | 1 | 0 |
| 30 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 0 | 30 |
| 43 | 47 | 33 | 1 |
| 52 | 10 | 52 | 0 |
| 0 | 52 | 10 | 52 |
| 1 | 33 | 47 | 43 |
G:=sub<GL(4,GF(53))| [5,33,20,48,40,34,4,14,4,34,40,29,20,33,5,34],[5,34,20,1,40,29,25,33,4,14,39,48,20,48,19,33],[1,33,5,0,0,28,39,0,0,14,24,1,0,34,19,0],[30,0,0,0,0,30,0,0,0,0,30,0,0,0,0,30],[43,52,0,1,47,10,52,33,33,52,10,47,1,0,52,43] >;
D26.C23 in GAP, Magma, Sage, TeX
D_{26}.C_2^3 % in TeX
G:=Group("D26.C2^3"); // GroupNames label
G:=SmallGroup(416,204);
// by ID
G=gap.SmallGroup(416,204);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,103,362,9221,1751]);
// Polycyclic
G:=Group<a,b,c,d,e|a^26=b^2=e^2=1,c^2=a^-1*b,d^2=a^13,b*a*b=a^-1,c*a*c^-1=a^5,a*d=d*a,a*e=e*a,c*b*c^-1=a^4*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=a^13*c,d*e=e*d>;
// generators/relations