metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊3D26, Q8⋊2D26, D104⋊6C2, D4.3D26, C104⋊3C22, SD16⋊1D13, D26.15D4, D52⋊2C22, C52.5C23, Dic13.17D4, D4⋊D13⋊3C2, (D4×D13)⋊3C2, Q8⋊D13⋊2C2, C8⋊D13⋊1C2, C13⋊3(C8⋊C22), C2.19(D4×D13), C26.31(C2×D4), D52⋊C2⋊1C2, C13⋊2C8⋊2C22, (C13×SD16)⋊1C2, (Q8×C13)⋊2C22, C4.5(C22×D13), (C4×D13).2C22, (D4×C13).3C22, SmallGroup(416,135)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊D26
G = < a,b,c,d | a4=c26=d2=1, b2=a2, bab-1=cac-1=dad=a-1, cbc-1=a-1b, dbd=ab, dcd=c-1 >
Subgroups: 632 in 68 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C8, C8, C2×C4, D4, D4, Q8, C23, C13, M4(2), D8, SD16, SD16, C2×D4, C4○D4, D13, C26, C26, C8⋊C22, Dic13, C52, C52, D26, D26, C2×C26, C13⋊2C8, C104, C4×D13, C4×D13, D52, D52, C13⋊D4, D4×C13, Q8×C13, C22×D13, C8⋊D13, D104, D4⋊D13, Q8⋊D13, C13×SD16, D4×D13, D52⋊C2, Q8⋊D26
Quotients: C1, C2, C22, D4, C23, C2×D4, D13, C8⋊C22, D26, C22×D13, D4×D13, Q8⋊D26
►On 104 points
Generators in S104
(1 36 16 49)(2 50 17 37)(3 38 18 51)(4 52 19 39)(5 40 20 27)(6 28 21 41)(7 42 22 29)(8 30 23 43)(9 44 24 31)(10 32 25 45)(11 46 26 33)(12 34 14 47)(13 48 15 35)(53 66 96 83)(54 84 97 67)(55 68 98 85)(56 86 99 69)(57 70 100 87)(58 88 101 71)(59 72 102 89)(60 90 103 73)(61 74 104 91)(62 92 79 75)(63 76 80 93)(64 94 81 77)(65 78 82 95)
(1 65 16 82)(2 53 17 96)(3 67 18 84)(4 55 19 98)(5 69 20 86)(6 57 21 100)(7 71 22 88)(8 59 23 102)(9 73 24 90)(10 61 25 104)(11 75 26 92)(12 63 14 80)(13 77 15 94)(27 56 40 99)(28 87 41 70)(29 58 42 101)(30 89 43 72)(31 60 44 103)(32 91 45 74)(33 62 46 79)(34 93 47 76)(35 64 48 81)(36 95 49 78)(37 66 50 83)(38 97 51 54)(39 68 52 85)
(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 18)(2 17)(3 16)(4 15)(5 14)(6 26)(7 25)(8 24)(9 23)(10 22)(11 21)(12 20)(13 19)(27 47)(28 46)(29 45)(30 44)(31 43)(32 42)(33 41)(34 40)(35 39)(36 38)(48 52)(49 51)(53 66)(54 65)(55 64)(56 63)(57 62)(58 61)(59 60)(67 78)(68 77)(69 76)(70 75)(71 74)(72 73)(79 100)(80 99)(81 98)(82 97)(83 96)(84 95)(85 94)(86 93)(87 92)(88 91)(89 90)(101 104)(102 103)
G:=sub<Sym(104)| (1,36,16,49)(2,50,17,37)(3,38,18,51)(4,52,19,39)(5,40,20,27)(6,28,21,41)(7,42,22,29)(8,30,23,43)(9,44,24,31)(10,32,25,45)(11,46,26,33)(12,34,14,47)(13,48,15,35)(53,66,96,83)(54,84,97,67)(55,68,98,85)(56,86,99,69)(57,70,100,87)(58,88,101,71)(59,72,102,89)(60,90,103,73)(61,74,104,91)(62,92,79,75)(63,76,80,93)(64,94,81,77)(65,78,82,95), (1,65,16,82)(2,53,17,96)(3,67,18,84)(4,55,19,98)(5,69,20,86)(6,57,21,100)(7,71,22,88)(8,59,23,102)(9,73,24,90)(10,61,25,104)(11,75,26,92)(12,63,14,80)(13,77,15,94)(27,56,40,99)(28,87,41,70)(29,58,42,101)(30,89,43,72)(31,60,44,103)(32,91,45,74)(33,62,46,79)(34,93,47,76)(35,64,48,81)(36,95,49,78)(37,66,50,83)(38,97,51,54)(39,68,52,85), (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), (1,18)(2,17)(3,16)(4,15)(5,14)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(27,47)(28,46)(29,45)(30,44)(31,43)(32,42)(33,41)(34,40)(35,39)(36,38)(48,52)(49,51)(53,66)(54,65)(55,64)(56,63)(57,62)(58,61)(59,60)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)(79,100)(80,99)(81,98)(82,97)(83,96)(84,95)(85,94)(86,93)(87,92)(88,91)(89,90)(101,104)(102,103)>;
G:=Group( (1,36,16,49)(2,50,17,37)(3,38,18,51)(4,52,19,39)(5,40,20,27)(6,28,21,41)(7,42,22,29)(8,30,23,43)(9,44,24,31)(10,32,25,45)(11,46,26,33)(12,34,14,47)(13,48,15,35)(53,66,96,83)(54,84,97,67)(55,68,98,85)(56,86,99,69)(57,70,100,87)(58,88,101,71)(59,72,102,89)(60,90,103,73)(61,74,104,91)(62,92,79,75)(63,76,80,93)(64,94,81,77)(65,78,82,95), (1,65,16,82)(2,53,17,96)(3,67,18,84)(4,55,19,98)(5,69,20,86)(6,57,21,100)(7,71,22,88)(8,59,23,102)(9,73,24,90)(10,61,25,104)(11,75,26,92)(12,63,14,80)(13,77,15,94)(27,56,40,99)(28,87,41,70)(29,58,42,101)(30,89,43,72)(31,60,44,103)(32,91,45,74)(33,62,46,79)(34,93,47,76)(35,64,48,81)(36,95,49,78)(37,66,50,83)(38,97,51,54)(39,68,52,85), (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), (1,18)(2,17)(3,16)(4,15)(5,14)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(27,47)(28,46)(29,45)(30,44)(31,43)(32,42)(33,41)(34,40)(35,39)(36,38)(48,52)(49,51)(53,66)(54,65)(55,64)(56,63)(57,62)(58,61)(59,60)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)(79,100)(80,99)(81,98)(82,97)(83,96)(84,95)(85,94)(86,93)(87,92)(88,91)(89,90)(101,104)(102,103) );
G=PermutationGroup([[(1,36,16,49),(2,50,17,37),(3,38,18,51),(4,52,19,39),(5,40,20,27),(6,28,21,41),(7,42,22,29),(8,30,23,43),(9,44,24,31),(10,32,25,45),(11,46,26,33),(12,34,14,47),(13,48,15,35),(53,66,96,83),(54,84,97,67),(55,68,98,85),(56,86,99,69),(57,70,100,87),(58,88,101,71),(59,72,102,89),(60,90,103,73),(61,74,104,91),(62,92,79,75),(63,76,80,93),(64,94,81,77),(65,78,82,95)], [(1,65,16,82),(2,53,17,96),(3,67,18,84),(4,55,19,98),(5,69,20,86),(6,57,21,100),(7,71,22,88),(8,59,23,102),(9,73,24,90),(10,61,25,104),(11,75,26,92),(12,63,14,80),(13,77,15,94),(27,56,40,99),(28,87,41,70),(29,58,42,101),(30,89,43,72),(31,60,44,103),(32,91,45,74),(33,62,46,79),(34,93,47,76),(35,64,48,81),(36,95,49,78),(37,66,50,83),(38,97,51,54),(39,68,52,85)], [(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)], [(1,18),(2,17),(3,16),(4,15),(5,14),(6,26),(7,25),(8,24),(9,23),(10,22),(11,21),(12,20),(13,19),(27,47),(28,46),(29,45),(30,44),(31,43),(32,42),(33,41),(34,40),(35,39),(36,38),(48,52),(49,51),(53,66),(54,65),(55,64),(56,63),(57,62),(58,61),(59,60),(67,78),(68,77),(69,76),(70,75),(71,74),(72,73),(79,100),(80,99),(81,98),(82,97),(83,96),(84,95),(85,94),(86,93),(87,92),(88,91),(89,90),(101,104),(102,103)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 8A | 8B | 13A | ··· | 13F | 26A | ··· | 26F | 26G | ··· | 26L | 52A | ··· | 52F | 52G | ··· | 52L | 104A | ··· | 104L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 4 | 26 | 52 | 52 | 2 | 4 | 26 | 4 | 52 | 2 | ··· | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D13 | D26 | D26 | D26 | C8⋊C22 | D4×D13 | Q8⋊D26 |
| kernel | Q8⋊D26 | C8⋊D13 | D104 | D4⋊D13 | Q8⋊D13 | C13×SD16 | D4×D13 | D52⋊C2 | Dic13 | D26 | SD16 | C8 | D4 | Q8 | C13 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 1 | 6 | 12 |
Matrix representation of Q8⋊D26 ►in GL4(𝔽313) generated by
| 1 | 0 | 108 | 79 |
| 0 | 1 | 306 | 161 |
| 291 | 108 | 312 | 0 |
| 244 | 98 | 0 | 312 |
| 0 | 0 | 186 | 44 |
| 0 | 0 | 86 | 61 |
| 293 | 76 | 0 | 0 |
| 177 | 134 | 0 | 0 |
| 208 | 187 | 184 | 215 |
| 179 | 182 | 217 | 249 |
| 0 | 0 | 304 | 126 |
| 0 | 0 | 216 | 245 |
| 221 | 110 | 249 | 113 |
| 182 | 92 | 232 | 81 |
| 0 | 0 | 86 | 1 |
| 0 | 0 | 117 | 227 |
G:=sub<GL(4,GF(313))| [1,0,291,244,0,1,108,98,108,306,312,0,79,161,0,312],[0,0,293,177,0,0,76,134,186,86,0,0,44,61,0,0],[208,179,0,0,187,182,0,0,184,217,304,216,215,249,126,245],[221,182,0,0,110,92,0,0,249,232,86,117,113,81,1,227] >;
Q8⋊D26 in GAP, Magma, Sage, TeX
Q_8\rtimes D_{26} % in TeX
G:=Group("Q8:D26"); // GroupNames label
G:=SmallGroup(416,135);
// by ID
G=gap.SmallGroup(416,135);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,362,116,86,297,159,69,13829]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^26=d^2=1,b^2=a^2,b*a*b^-1=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^-1*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations