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## G = C2×C4×C13⋊C4order 416 = 25·13

### Direct product of C2×C4 and C13⋊C4

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×C4×C13⋊C4
G = < a,b,c,d | a2=b4=c13=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >

Subgroups: 660 in 108 conjugacy classes, 62 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C23, C13, C42, C22×C4, D13, C26, C26, C2×C42, Dic13, C52, C13⋊C4, D26, D26, C2×C26, C4×D13, C2×Dic13, C2×C52, C2×C13⋊C4, C22×D13, C4×C13⋊C4, C2×C4×D13, C22×C13⋊C4, C2×C4×C13⋊C4
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, C22×C4, C2×C42, C13⋊C4, C2×C13⋊C4, C4×C13⋊C4, C22×C13⋊C4, C2×C4×C13⋊C4

Smallest permutation representation of C2×C4×C13⋊C4
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 40 14 27)(2 41 15 28)(3 42 16 29)(4 43 17 30)(5 44 18 31)(6 45 19 32)(7 46 20 33)(8 47 21 34)(9 48 22 35)(10 49 23 36)(11 50 24 37)(12 51 25 38)(13 52 26 39)(53 92 66 79)(54 93 67 80)(55 94 68 81)(56 95 69 82)(57 96 70 83)(58 97 71 84)(59 98 72 85)(60 99 73 86)(61 100 74 87)(62 101 75 88)(63 102 76 89)(64 103 77 90)(65 104 78 91)
(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 53)(2 61 13 58)(3 56 12 63)(4 64 11 55)(5 59 10 60)(6 54 9 65)(7 62 8 57)(14 66)(15 74 26 71)(16 69 25 76)(17 77 24 68)(18 72 23 73)(19 67 22 78)(20 75 21 70)(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)

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4X 13A 13B 13C 26A ··· 26I 52A ··· 52L order 1 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 13 13 13 26 ··· 26 52 ··· 52 size 1 1 1 1 13 13 13 13 1 1 1 1 13 ··· 13 4 4 4 4 ··· 4 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 4 4 4 4 type + + + + + + + image C1 C2 C2 C2 C4 C4 C4 C4 C13⋊C4 C2×C13⋊C4 C2×C13⋊C4 C4×C13⋊C4 kernel C2×C4×C13⋊C4 C4×C13⋊C4 C2×C4×D13 C22×C13⋊C4 C4×D13 C2×Dic13 C2×C52 C2×C13⋊C4 C2×C4 C4 C22 C2 # reps 1 4 1 2 4 2 2 16 3 6 3 12

Matrix representation of C2×C4×C13⋊C4 in GL6(𝔽53)

 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 0 0 0 0 0 0 1
,
 30 0 0 0 0 0 0 30 0 0 0 0 0 0 52 0 0 0 0 0 0 52 0 0 0 0 0 0 52 0 0 0 0 0 0 52
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 46 12 46 52 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 30 0 0 0 0 0 0 23 0 0 0 0 0 0 1 0 0 0 0 0 7 48 15 8 0 0 12 38 4 45 0 0 0 0 1 0

G:=sub<GL(6,GF(53))| [52,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[30,0,0,0,0,0,0,30,0,0,0,0,0,0,52,0,0,0,0,0,0,52,0,0,0,0,0,0,52,0,0,0,0,0,0,52],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,1,0,0,0,0,12,0,1,0,0,0,46,0,0,1,0,0,52,0,0,0],[30,0,0,0,0,0,0,23,0,0,0,0,0,0,1,7,12,0,0,0,0,48,38,0,0,0,0,15,4,1,0,0,0,8,45,0] >;

C2×C4×C13⋊C4 in GAP, Magma, Sage, TeX

C_2\times C_4\times C_{13}\rtimes C_4
% in TeX

G:=Group("C2xC4xC13:C4");
// GroupNames label

G:=SmallGroup(416,202);
// by ID

G=gap.SmallGroup(416,202);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,86,9221,1751]);
// Polycyclic

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

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