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## G = D26.C23order 416 = 25·13

### 11st non-split extension by D26 of C23 acting via C23/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — D26.C23
 Chief series C1 — C13 — D13 — D26 — C2×C13⋊C4 — C4×C13⋊C4 — D26.C23
 Lower central C13 — C26 — D26.C23
 Upper central C1 — C4 — C2×C4

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 [×4], C4 [×2], C4 [×6], C22, C22 [×4], C2×C4, C2×C4 [×9], C23, C13, C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, D13 [×2], D13, C26, C26, C42⋊C2, Dic13 [×2], C52 [×2], C13⋊C4 [×4], D26 [×2], D26 [×2], C2×C26, C4×D13 [×4], C2×Dic13, C2×C52, C2×C13⋊C4 [×4], C22×D13, C4×C13⋊C4 [×2], C52⋊C4 [×2], D13.D4 [×2], C2×C4×D13, D26.C23
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, C22×C4, C4○D4 [×2], C42⋊C2, C13⋊C4, C2×C13⋊C4 [×3], C22×C13⋊C4, D26.C23

Smallest permutation representation of 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 44)(28 43)(29 42)(30 41)(31 40)(32 39)(33 38)(34 37)(35 36)(45 52)(46 51)(47 50)(48 49)(53 62)(54 61)(55 60)(56 59)(57 58)(63 78)(64 77)(65 76)(66 75)(67 74)(68 73)(69 72)(70 71)(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)
(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 42 45 30)(28 37 44 35)(29 32 43 40)(31 48 41 50)(33 38 39 34)(36 49)(46 51 52 47)(53 57 63 59)(54 78 62 64)(55 73 61 69)(56 68 60 74)(65 75 77 67)(66 70 76 72)(79 80 101 100)(81 96 99 84)(82 91 98 89)(83 86 97 94)(85 102 95 104)(87 92 93 88)(90 103)
(1 58 14 71)(2 59 15 72)(3 60 16 73)(4 61 17 74)(5 62 18 75)(6 63 19 76)(7 64 20 77)(8 65 21 78)(9 66 22 53)(10 67 23 54)(11 68 24 55)(12 69 25 56)(13 70 26 57)(27 81 40 94)(28 82 41 95)(29 83 42 96)(30 84 43 97)(31 85 44 98)(32 86 45 99)(33 87 46 100)(34 88 47 101)(35 89 48 102)(36 90 49 103)(37 91 50 104)(38 92 51 79)(39 93 52 80)
(1 36)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 49)(15 50)(16 51)(17 52)(18 27)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(53 85)(54 86)(55 87)(56 88)(57 89)(58 90)(59 91)(60 92)(61 93)(62 94)(63 95)(64 96)(65 97)(66 98)(67 99)(68 100)(69 101)(70 102)(71 103)(72 104)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)```

`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)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(45,52)(46,51)(47,50)(48,49)(53,62)(54,61)(55,60)(56,59)(57,58)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(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), (2,22,26,6)(3,17,25,11)(4,12,24,16)(5,7,23,21)(8,18,20,10)(9,13,19,15)(27,42,45,30)(28,37,44,35)(29,32,43,40)(31,48,41,50)(33,38,39,34)(36,49)(46,51,52,47)(53,57,63,59)(54,78,62,64)(55,73,61,69)(56,68,60,74)(65,75,77,67)(66,70,76,72)(79,80,101,100)(81,96,99,84)(82,91,98,89)(83,86,97,94)(85,102,95,104)(87,92,93,88)(90,103), (1,58,14,71)(2,59,15,72)(3,60,16,73)(4,61,17,74)(5,62,18,75)(6,63,19,76)(7,64,20,77)(8,65,21,78)(9,66,22,53)(10,67,23,54)(11,68,24,55)(12,69,25,56)(13,70,26,57)(27,81,40,94)(28,82,41,95)(29,83,42,96)(30,84,43,97)(31,85,44,98)(32,86,45,99)(33,87,46,100)(34,88,47,101)(35,89,48,102)(36,90,49,103)(37,91,50,104)(38,92,51,79)(39,93,52,80), (1,36)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,27)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(53,85)(54,86)(55,87)(56,88)(57,89)(58,90)(59,91)(60,92)(61,93)(62,94)(63,95)(64,96)(65,97)(66,98)(67,99)(68,100)(69,101)(70,102)(71,103)(72,104)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)>;`

`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)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(45,52)(46,51)(47,50)(48,49)(53,62)(54,61)(55,60)(56,59)(57,58)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(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), (2,22,26,6)(3,17,25,11)(4,12,24,16)(5,7,23,21)(8,18,20,10)(9,13,19,15)(27,42,45,30)(28,37,44,35)(29,32,43,40)(31,48,41,50)(33,38,39,34)(36,49)(46,51,52,47)(53,57,63,59)(54,78,62,64)(55,73,61,69)(56,68,60,74)(65,75,77,67)(66,70,76,72)(79,80,101,100)(81,96,99,84)(82,91,98,89)(83,86,97,94)(85,102,95,104)(87,92,93,88)(90,103), (1,58,14,71)(2,59,15,72)(3,60,16,73)(4,61,17,74)(5,62,18,75)(6,63,19,76)(7,64,20,77)(8,65,21,78)(9,66,22,53)(10,67,23,54)(11,68,24,55)(12,69,25,56)(13,70,26,57)(27,81,40,94)(28,82,41,95)(29,83,42,96)(30,84,43,97)(31,85,44,98)(32,86,45,99)(33,87,46,100)(34,88,47,101)(35,89,48,102)(36,90,49,103)(37,91,50,104)(38,92,51,79)(39,93,52,80), (1,36)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,27)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(53,85)(54,86)(55,87)(56,88)(57,89)(58,90)(59,91)(60,92)(61,93)(62,94)(63,95)(64,96)(65,97)(66,98)(67,99)(68,100)(69,101)(70,102)(71,103)(72,104)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84) );`

`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),(27,44),(28,43),(29,42),(30,41),(31,40),(32,39),(33,38),(34,37),(35,36),(45,52),(46,51),(47,50),(48,49),(53,62),(54,61),(55,60),(56,59),(57,58),(63,78),(64,77),(65,76),(66,75),(67,74),(68,73),(69,72),(70,71),(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)], [(2,22,26,6),(3,17,25,11),(4,12,24,16),(5,7,23,21),(8,18,20,10),(9,13,19,15),(27,42,45,30),(28,37,44,35),(29,32,43,40),(31,48,41,50),(33,38,39,34),(36,49),(46,51,52,47),(53,57,63,59),(54,78,62,64),(55,73,61,69),(56,68,60,74),(65,75,77,67),(66,70,76,72),(79,80,101,100),(81,96,99,84),(82,91,98,89),(83,86,97,94),(85,102,95,104),(87,92,93,88),(90,103)], [(1,58,14,71),(2,59,15,72),(3,60,16,73),(4,61,17,74),(5,62,18,75),(6,63,19,76),(7,64,20,77),(8,65,21,78),(9,66,22,53),(10,67,23,54),(11,68,24,55),(12,69,25,56),(13,70,26,57),(27,81,40,94),(28,82,41,95),(29,83,42,96),(30,84,43,97),(31,85,44,98),(32,86,45,99),(33,87,46,100),(34,88,47,101),(35,89,48,102),(36,90,49,103),(37,91,50,104),(38,92,51,79),(39,93,52,80)], [(1,36),(2,37),(3,38),(4,39),(5,40),(6,41),(7,42),(8,43),(9,44),(10,45),(11,46),(12,47),(13,48),(14,49),(15,50),(16,51),(17,52),(18,27),(19,28),(20,29),(21,30),(22,31),(23,32),(24,33),(25,34),(26,35),(53,85),(54,86),(55,87),(56,88),(57,89),(58,90),(59,91),(60,92),(61,93),(62,94),(63,95),(64,96),(65,97),(66,98),(67,99),(68,100),(69,101),(70,102),(71,103),(72,104),(73,79),(74,80),(75,81),(76,82),(77,83),(78,84)])`

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`

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