metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D13⋊M4(2), Dic13.11C23, D13⋊C8⋊4C2, C13⋊C8⋊2C22, (C2×C52).8C4, C52.20(C2×C4), (C4×D13).8C4, C13⋊2(C2×M4(2)), C52.C4⋊5C2, D26.14(C2×C4), C26.3(C22×C4), C13⋊M4(2)⋊3C2, (C22×D13).9C4, Dic13.16(C2×C4), (C4×D13).34C22, (C2×Dic13).56C22, C4.21(C2×C13⋊C4), (C2×C4).8(C13⋊C4), (C2×C4×D13).15C2, C2.5(C22×C13⋊C4), C22.6(C2×C13⋊C4), (C2×C26).15(C2×C4), SmallGroup(416,201)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D13⋊M4(2)
G = < a,b,c,d | a13=b2=c8=d2=1, bab=a-1, cac-1=a5, ad=da, cbc-1=a4b, bd=db, dcd=c5 >
Subgroups: 436 in 68 conjugacy classes, 36 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C23, C13, C2×C8, M4(2), C22×C4, D13, D13, C26, C26, C2×M4(2), Dic13, C52, D26, D26, C2×C26, C13⋊C8, C4×D13, C2×Dic13, C2×C52, C22×D13, D13⋊C8, C52.C4, C13⋊M4(2), C2×C4×D13, D13⋊M4(2)
Quotients: C1, C2, C4, C22, C2×C4, C23, M4(2), C22×C4, C2×M4(2), C13⋊C4, C2×C13⋊C4, C22×C13⋊C4, D13⋊M4(2)
►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 13)(2 12)(3 11)(4 10)(5 9)(6 8)(14 17)(15 16)(18 26)(19 25)(20 24)(21 23)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(53 60)(54 59)(55 58)(56 57)(61 65)(62 64)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)(79 90)(80 89)(81 88)(82 87)(83 86)(84 85)(92 97)(93 96)(94 95)(98 104)(99 103)(100 102)
(1 95 27 57 16 85 40 72)(2 103 39 62 17 80 52 77)(3 98 38 54 18 88 51 69)(4 93 37 59 19 83 50 74)(5 101 36 64 20 91 49 66)(6 96 35 56 21 86 48 71)(7 104 34 61 22 81 47 76)(8 99 33 53 23 89 46 68)(9 94 32 58 24 84 45 73)(10 102 31 63 25 79 44 78)(11 97 30 55 26 87 43 70)(12 92 29 60 14 82 42 75)(13 100 28 65 15 90 41 67)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 14)(13 15)(27 40)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 49)(37 50)(38 51)(39 52)
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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,17)(15,16)(18,26)(19,25)(20,24)(21,23)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(53,60)(54,59)(55,58)(56,57)(61,65)(62,64)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(79,90)(80,89)(81,88)(82,87)(83,86)(84,85)(92,97)(93,96)(94,95)(98,104)(99,103)(100,102), (1,95,27,57,16,85,40,72)(2,103,39,62,17,80,52,77)(3,98,38,54,18,88,51,69)(4,93,37,59,19,83,50,74)(5,101,36,64,20,91,49,66)(6,96,35,56,21,86,48,71)(7,104,34,61,22,81,47,76)(8,99,33,53,23,89,46,68)(9,94,32,58,24,84,45,73)(10,102,31,63,25,79,44,78)(11,97,30,55,26,87,43,70)(12,92,29,60,14,82,42,75)(13,100,28,65,15,90,41,67), (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,14)(13,15)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(37,50)(38,51)(39,52)>;
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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,17)(15,16)(18,26)(19,25)(20,24)(21,23)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(53,60)(54,59)(55,58)(56,57)(61,65)(62,64)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(79,90)(80,89)(81,88)(82,87)(83,86)(84,85)(92,97)(93,96)(94,95)(98,104)(99,103)(100,102), (1,95,27,57,16,85,40,72)(2,103,39,62,17,80,52,77)(3,98,38,54,18,88,51,69)(4,93,37,59,19,83,50,74)(5,101,36,64,20,91,49,66)(6,96,35,56,21,86,48,71)(7,104,34,61,22,81,47,76)(8,99,33,53,23,89,46,68)(9,94,32,58,24,84,45,73)(10,102,31,63,25,79,44,78)(11,97,30,55,26,87,43,70)(12,92,29,60,14,82,42,75)(13,100,28,65,15,90,41,67), (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,14)(13,15)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(37,50)(38,51)(39,52) );
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)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,8),(14,17),(15,16),(18,26),(19,25),(20,24),(21,23),(27,39),(28,38),(29,37),(30,36),(31,35),(32,34),(40,52),(41,51),(42,50),(43,49),(44,48),(45,47),(53,60),(54,59),(55,58),(56,57),(61,65),(62,64),(66,77),(67,76),(68,75),(69,74),(70,73),(71,72),(79,90),(80,89),(81,88),(82,87),(83,86),(84,85),(92,97),(93,96),(94,95),(98,104),(99,103),(100,102)], [(1,95,27,57,16,85,40,72),(2,103,39,62,17,80,52,77),(3,98,38,54,18,88,51,69),(4,93,37,59,19,83,50,74),(5,101,36,64,20,91,49,66),(6,96,35,56,21,86,48,71),(7,104,34,61,22,81,47,76),(8,99,33,53,23,89,46,68),(9,94,32,58,24,84,45,73),(10,102,31,63,25,79,44,78),(11,97,30,55,26,87,43,70),(12,92,29,60,14,82,42,75),(13,100,28,65,15,90,41,67)], [(1,16),(2,17),(3,18),(4,19),(5,20),(6,21),(7,22),(8,23),(9,24),(10,25),(11,26),(12,14),(13,15),(27,40),(28,41),(29,42),(30,43),(31,44),(32,45),(33,46),(34,47),(35,48),(36,49),(37,50),(38,51),(39,52)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 8A | ··· | 8H | 13A | 13B | 13C | 26A | ··· | 26I | 52A | ··· | 52L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 13 | 13 | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 2 | 13 | 13 | 26 | 1 | 1 | 2 | 13 | 13 | 26 | 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 | M4(2) | C13⋊C4 | C2×C13⋊C4 | C2×C13⋊C4 | D13⋊M4(2) |
| kernel | D13⋊M4(2) | D13⋊C8 | C52.C4 | C13⋊M4(2) | C2×C4×D13 | C4×D13 | C2×C52 | C22×D13 | D13 | C2×C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 2 | 2 | 1 | 4 | 2 | 2 | 4 | 3 | 6 | 3 | 12 |
Matrix representation of D13⋊M4(2) ►in GL4(𝔽313) generated by
| 0 | 1 | 0 | 0 |
| 312 | 77 | 0 | 0 |
| 0 | 0 | 49 | 247 |
| 0 | 0 | 66 | 288 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 49 | 247 |
| 0 | 0 | 264 | 264 |
| 0 | 0 | 312 | 0 |
| 0 | 0 | 0 | 312 |
| 288 | 0 | 0 | 0 |
| 266 | 25 | 0 | 0 |
| 312 | 0 | 0 | 0 |
| 0 | 312 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(313))| [0,312,0,0,1,77,0,0,0,0,49,66,0,0,247,288],[0,1,0,0,1,0,0,0,0,0,49,264,0,0,247,264],[0,0,288,266,0,0,0,25,312,0,0,0,0,312,0,0],[312,0,0,0,0,312,0,0,0,0,1,0,0,0,0,1] >;
D13⋊M4(2) in GAP, Magma, Sage, TeX
D_{13}\rtimes M_4(2) % in TeX
G:=Group("D13:M4(2)"); // GroupNames label
G:=SmallGroup(416,201);
// by ID
G=gap.SmallGroup(416,201);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,103,362,69,9221,1751]);
// Polycyclic
G:=Group<a,b,c,d|a^13=b^2=c^8=d^2=1,b*a*b=a^-1,c*a*c^-1=a^5,a*d=d*a,c*b*c^-1=a^4*b,b*d=d*b,d*c*d=c^5>;
// generators/relations