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G = D26⋊C4order 208 = 24·13

1st semidirect product of D26 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D261C4, C26.6D4, C2.2D52, C22.6D26, (C2×C52)⋊1C2, (C2×C4)⋊1D13, C2.5(C4×D13), C132(C22⋊C4), C26.12(C2×C4), (C2×Dic13)⋊1C2, C2.2(C13⋊D4), (C2×C26).6C22, (C22×D13).1C2, SmallGroup(208,14)

Series: Derived Chief Lower central Upper central

C1C26 — D26⋊C4
C1C13C26C2×C26C22×D13 — D26⋊C4
C13C26 — D26⋊C4
C1C22C2×C4

Generators and relations for D26⋊C4
 G = < a,b,c | a26=b2=c4=1, bab=a-1, ac=ca, cbc-1=a13b >

26C2
26C2
2C4
13C22
13C22
26C4
26C22
26C22
2D13
2D13
13C2×C4
13C23
2D26
2D26
2Dic13
2C52
13C22⋊C4

Smallest permutation representation of D26⋊C4
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 47)(2 46)(3 45)(4 44)(5 43)(6 42)(7 41)(8 40)(9 39)(10 38)(11 37)(12 36)(13 35)(14 34)(15 33)(16 32)(17 31)(18 30)(19 29)(20 28)(21 27)(22 52)(23 51)(24 50)(25 49)(26 48)(53 104)(54 103)(55 102)(56 101)(57 100)(58 99)(59 98)(60 97)(61 96)(62 95)(63 94)(64 93)(65 92)(66 91)(67 90)(68 89)(69 88)(70 87)(71 86)(72 85)(73 84)(74 83)(75 82)(76 81)(77 80)(78 79)
(1 103 48 68)(2 104 49 69)(3 79 50 70)(4 80 51 71)(5 81 52 72)(6 82 27 73)(7 83 28 74)(8 84 29 75)(9 85 30 76)(10 86 31 77)(11 87 32 78)(12 88 33 53)(13 89 34 54)(14 90 35 55)(15 91 36 56)(16 92 37 57)(17 93 38 58)(18 94 39 59)(19 95 40 60)(20 96 41 61)(21 97 42 62)(22 98 43 63)(23 99 44 64)(24 100 45 65)(25 101 46 66)(26 102 47 67)

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

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

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

D26⋊C4 is a maximal subgroup of
C42⋊D13  C4×D52  C4.D52  C422D13  C22⋊C4×D13  Dic134D4  C22⋊D52  D26.12D4  D26⋊D4  C23.6D26  C22.D52  C4⋊C47D13  D528C4  D26.13D4  C42D52  D26⋊Q8  D262Q8  C4⋊C4⋊D13  C4×C13⋊D4  C23.23D26  C527D4  C23⋊D26  Dic13⋊D4  D263Q8  C52.23D4
D26⋊C4 is a maximal quotient of
D524C4  C22.2D52  D526C4  C26.Q16  C52.44D4  D261C8  D525C4  C52.46D4  C4.12D52  D527C4  C26.10C42

58 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D13A···13F26A···26R52A···52X
order122222444413···1326···2652···52
size111126262226262···22···22···2

58 irreducible representations

dim11111222222
type++++++++
imageC1C2C2C2C4D4D13D26C4×D13D52C13⋊D4
kernelD26⋊C4C2×Dic13C2×C52C22×D13D26C26C2×C4C22C2C2C2
# reps11114266121212

Matrix representation of D26⋊C4 in GL3(𝔽53) generated by

100
0152
01012
,
100
01341
01440
,
2300
02617
03227
G:=sub<GL(3,GF(53))| [1,0,0,0,15,10,0,2,12],[1,0,0,0,13,14,0,41,40],[23,0,0,0,26,32,0,17,27] >;

D26⋊C4 in GAP, Magma, Sage, TeX

D_{26}\rtimes C_4
% in TeX

G:=Group("D26:C4");
// GroupNames label

G:=SmallGroup(208,14);
// by ID

G=gap.SmallGroup(208,14);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-13,101,26,4804]);
// Polycyclic

G:=Group<a,b,c|a^26=b^2=c^4=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^13*b>;
// generators/relations

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Subgroup lattice of D26⋊C4 in TeX

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