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## G = C2×D13.D4order 416 = 25·13

### Direct product of C2 and D13.D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — C2×D13.D4
 Chief series C1 — C13 — D13 — D26 — C2×C13⋊C4 — C22×C13⋊C4 — C2×D13.D4
 Lower central C13 — C26 — C2×D13.D4
 Upper central C1 — C22 — C23

Generators and relations for C2×D13.D4
G = < a,b,c,d,e | a2=b13=c2=d4=1, e2=b-1c, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, dbd-1=ebe-1=b5, dcd-1=ece-1=b4c, ede-1=b-1cd-1 >

Subgroups: 1124 in 132 conjugacy classes, 46 normal (14 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C23, C23, C13, C22⋊C4, C22×C4, C24, D13, D13, C26, C26, C26, C2×C22⋊C4, C13⋊C4, D26, D26, D26, C2×C26, C2×C26, C2×C26, C2×C13⋊C4, C2×C13⋊C4, C22×D13, C22×D13, C22×D13, C22×C26, D13.D4, C22×C13⋊C4, C23×D13, C2×D13.D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C2×C22⋊C4, C13⋊C4, C2×C13⋊C4, D13.D4, C22×C13⋊C4, C2×D13.D4

Smallest permutation representation of C2×D13.D4
On 104 points
Generators in S104
(1 53)(2 54)(3 55)(4 56)(5 57)(6 58)(7 59)(8 60)(9 61)(10 62)(11 63)(12 64)(13 65)(14 66)(15 67)(16 68)(17 69)(18 70)(19 71)(20 72)(21 73)(22 74)(23 75)(24 76)(25 77)(26 78)(27 79)(28 80)(29 81)(30 82)(31 83)(32 84)(33 85)(34 86)(35 87)(36 88)(37 89)(38 90)(39 91)(40 92)(41 93)(42 94)(43 95)(44 96)(45 97)(46 98)(47 99)(48 100)(49 101)(50 102)(51 103)(52 104)
(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 52)(28 51)(29 50)(30 49)(31 48)(32 47)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 40)(53 78)(54 77)(55 76)(56 75)(57 74)(58 73)(59 72)(60 71)(61 70)(62 69)(63 68)(64 67)(65 66)(79 104)(80 103)(81 102)(82 101)(83 100)(84 99)(85 98)(86 97)(87 96)(88 95)(89 94)(90 93)(91 92)
(1 66)(2 74 13 71)(3 69 12 76)(4 77 11 68)(5 72 10 73)(6 67 9 78)(7 75 8 70)(14 53)(15 61 26 58)(16 56 25 63)(17 64 24 55)(18 59 23 60)(19 54 22 65)(20 62 21 57)(27 79)(28 87 39 84)(29 82 38 89)(30 90 37 81)(31 85 36 86)(32 80 35 91)(33 88 34 83)(40 92)(41 100 52 97)(42 95 51 102)(43 103 50 94)(44 98 49 99)(45 93 48 104)(46 101 47 96)
(1 40 14 27)(2 48 26 32)(3 43 25 37)(4 51 24 29)(5 46 23 34)(6 41 22 39)(7 49 21 31)(8 44 20 36)(9 52 19 28)(10 47 18 33)(11 42 17 38)(12 50 16 30)(13 45 15 35)(53 92 66 79)(54 100 78 84)(55 95 77 89)(56 103 76 81)(57 98 75 86)(58 93 74 91)(59 101 73 83)(60 96 72 88)(61 104 71 80)(62 99 70 85)(63 94 69 90)(64 102 68 82)(65 97 67 87)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 4A ··· 4H 13A 13B 13C 26A ··· 26U order 1 2 2 2 2 2 2 2 2 2 2 2 4 ··· 4 13 13 13 26 ··· 26 size 1 1 1 1 2 2 13 13 13 13 26 26 26 ··· 26 4 4 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 2 4 4 4 type + + + + + + + + image C1 C2 C2 C2 C4 C4 D4 C13⋊C4 C2×C13⋊C4 D13.D4 kernel C2×D13.D4 D13.D4 C22×C13⋊C4 C23×D13 C22×D13 C22×C26 D26 C23 C22 C2 # reps 1 4 2 1 6 2 4 3 9 12

Matrix representation of C2×D13.D4 in GL6(𝔽53)

 52 0 0 0 0 0 0 52 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 39 31 39 52 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 52 0 0 0 0 0 0 52 0 0 0 0 0 0 14 22 14 1 0 0 38 24 17 39 0 0 22 29 37 15 0 0 39 30 25 31
,
 1 0 0 0 0 0 52 52 0 0 0 0 0 0 52 0 0 0 0 0 39 17 24 38 0 0 22 29 37 15 0 0 0 0 52 0
,
 52 51 0 0 0 0 1 1 0 0 0 0 0 0 50 44 47 45 0 0 20 18 50 22 0 0 7 11 30 31 0 0 0 8 3 8

G:=sub<GL(6,GF(53))| [52,0,0,0,0,0,0,52,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,39,1,0,0,0,0,31,0,1,0,0,0,39,0,0,1,0,0,52,0,0,0],[52,0,0,0,0,0,0,52,0,0,0,0,0,0,14,38,22,39,0,0,22,24,29,30,0,0,14,17,37,25,0,0,1,39,15,31],[1,52,0,0,0,0,0,52,0,0,0,0,0,0,52,39,22,0,0,0,0,17,29,0,0,0,0,24,37,52,0,0,0,38,15,0],[52,1,0,0,0,0,51,1,0,0,0,0,0,0,50,20,7,0,0,0,44,18,11,8,0,0,47,50,30,3,0,0,45,22,31,8] >;

C2×D13.D4 in GAP, Magma, Sage, TeX

C_2\times D_{13}.D_4
% in TeX

G:=Group("C2xD13.D4");
// GroupNames label

G:=SmallGroup(416,211);
// by ID

G=gap.SmallGroup(416,211);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,362,9221,1751]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^13=c^2=d^4=1,e^2=b^-1*c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,d*b*d^-1=e*b*e^-1=b^5,d*c*d^-1=e*c*e^-1=b^4*c,e*d*e^-1=b^-1*c*d^-1>;
// generators/relations

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