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## G = C2×D4×D13order 416 = 25·13

### Direct product of C2, D4 and D13

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×D4×D13
G = < a,b,c,d,e | a2=b4=c2=d13=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1792 in 236 conjugacy classes, 97 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×12], C4 [×2], C4 [×2], C22, C22 [×4], C22 [×34], C2×C4, C2×C4 [×5], D4 [×4], D4 [×12], C23 [×2], C23 [×19], C13, C22×C4, C2×D4, C2×D4 [×11], C24 [×2], D13 [×4], D13 [×4], C26, C26 [×2], C26 [×4], C22×D4, Dic13 [×2], C52 [×2], D26 [×10], D26 [×20], C2×C26, C2×C26 [×4], C2×C26 [×4], C4×D13 [×4], D52 [×4], C2×Dic13, C13⋊D4 [×8], C2×C52, D4×C13 [×4], C22×D13, C22×D13 [×10], C22×D13 [×8], C22×C26 [×2], C2×C4×D13, C2×D52, D4×D13 [×8], C2×C13⋊D4 [×2], D4×C26, C23×D13 [×2], C2×D4×D13
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, D13, C22×D4, D26 [×7], C22×D13 [×7], D4×D13 [×2], C23×D13, C2×D4×D13

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

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 4A 4B 4C 4D 13A ··· 13F 26A ··· 26R 26S ··· 26AP 52A ··· 52L order 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 13 ··· 13 26 ··· 26 26 ··· 26 52 ··· 52 size 1 1 1 1 2 2 2 2 13 13 13 13 26 26 26 26 2 2 26 26 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D4 D13 D26 D26 D26 D4×D13 kernel C2×D4×D13 C2×C4×D13 C2×D52 D4×D13 C2×C13⋊D4 D4×C26 C23×D13 D26 C2×D4 C2×C4 D4 C23 C2 # reps 1 1 1 8 2 1 2 4 6 6 24 12 12

Matrix representation of C2×D4×D13 in GL4(𝔽53) generated by

 52 0 0 0 0 52 0 0 0 0 52 0 0 0 0 52
,
 52 0 0 0 0 52 0 0 0 0 0 52 0 0 1 0
,
 52 0 0 0 0 52 0 0 0 0 0 52 0 0 52 0
,
 41 1 0 0 24 20 0 0 0 0 1 0 0 0 0 1
,
 0 45 0 0 33 0 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(53))| [52,0,0,0,0,52,0,0,0,0,52,0,0,0,0,52],[52,0,0,0,0,52,0,0,0,0,0,1,0,0,52,0],[52,0,0,0,0,52,0,0,0,0,0,52,0,0,52,0],[41,24,0,0,1,20,0,0,0,0,1,0,0,0,0,1],[0,33,0,0,45,0,0,0,0,0,1,0,0,0,0,1] >;

C2×D4×D13 in GAP, Magma, Sage, TeX

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

G:=Group("C2xD4xD13");
// GroupNames label

G:=SmallGroup(416,216);
// by ID

G=gap.SmallGroup(416,216);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,159,13829]);
// Polycyclic

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

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