Copied to
clipboard

## G = D13⋊M4(2)  order 416 = 25·13

### The semidirect product of D13 and M4(2) acting via M4(2)/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — D13⋊M4(2)
 Chief series C1 — C13 — C26 — Dic13 — C13⋊C8 — D13⋊C8 — D13⋊M4(2)
 Lower central C13 — C26 — D13⋊M4(2)
 Upper central C1 — C4 — C2×C4

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)

Smallest permutation representation of 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`

׿
×
𝔽