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## G = C2×C52⋊C4order 416 = 25·13

### Direct product of C2 and C52⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — C2×C52⋊C4
 Chief series C1 — C13 — D13 — D26 — C2×C13⋊C4 — C22×C13⋊C4 — C2×C52⋊C4
 Lower central C13 — C26 — C2×C52⋊C4
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×C52⋊C4
G = < a,b,c | a2=b52=c4=1, ab=ba, ac=ca, cbc-1=b31 >

Subgroups: 660 in 92 conjugacy classes, 46 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×6], C22, C22 [×6], C2×C4, C2×C4 [×13], C23, C13, C4⋊C4 [×4], C22×C4 [×3], D13 [×2], D13 [×2], C26, C26 [×2], C2×C4⋊C4, Dic13 [×2], C52 [×2], C13⋊C4 [×4], D26 [×2], D26 [×4], C2×C26, C4×D13 [×4], C2×Dic13, C2×C52, C2×C13⋊C4 [×4], C2×C13⋊C4 [×4], C22×D13, C52⋊C4 [×4], C2×C4×D13, C22×C13⋊C4 [×2], C2×C52⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C2×C4⋊C4, C13⋊C4, C2×C13⋊C4 [×3], C52⋊C4 [×2], C22×C13⋊C4, C2×C52⋊C4

Smallest permutation representation of C2×C52⋊C4
On 104 points
Generators in S104
(1 96)(2 97)(3 98)(4 99)(5 100)(6 101)(7 102)(8 103)(9 104)(10 53)(11 54)(12 55)(13 56)(14 57)(15 58)(16 59)(17 60)(18 61)(19 62)(20 63)(21 64)(22 65)(23 66)(24 67)(25 68)(26 69)(27 70)(28 71)(29 72)(30 73)(31 74)(32 75)(33 76)(34 77)(35 78)(36 79)(37 80)(38 81)(39 82)(40 83)(41 84)(42 85)(43 86)(44 87)(45 88)(46 89)(47 90)(48 91)(49 92)(50 93)(51 94)(52 95)
(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 57)(2 104 26 88)(3 99 51 67)(4 94 24 98)(5 89 49 77)(6 84 22 56)(7 79 47 87)(8 74 20 66)(9 69 45 97)(10 64 18 76)(11 59 43 55)(12 54 16 86)(13 101 41 65)(14 96)(15 91 39 75)(17 81 37 85)(19 71 35 95)(21 61 33 53)(23 103 31 63)(25 93 29 73)(27 83)(28 78 52 62)(30 68 50 72)(32 58 48 82)(34 100 46 92)(36 90 44 102)(38 80 42 60)(40 70)

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 4 4 4 4 type + + + + + - + + + image C1 C2 C2 C2 C4 C4 C4 D4 Q8 C13⋊C4 C2×C13⋊C4 C2×C13⋊C4 C52⋊C4 kernel C2×C52⋊C4 C52⋊C4 C2×C4×D13 C22×C13⋊C4 C4×D13 C2×Dic13 C2×C52 D26 D26 C2×C4 C4 C22 C2 # reps 1 4 1 2 4 2 2 2 2 3 6 3 12

Matrix representation of C2×C52⋊C4 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
,
 42 13 0 0 0 0 11 11 0 0 0 0 0 0 46 37 32 16 0 0 17 52 47 52 0 0 24 46 39 23 0 0 1 33 0 21
,
 12 34 0 0 0 0 16 41 0 0 0 0 0 0 52 38 15 1 0 0 32 4 45 30 0 0 41 28 18 28 0 0 15 4 9 32

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],[42,11,0,0,0,0,13,11,0,0,0,0,0,0,46,17,24,1,0,0,37,52,46,33,0,0,32,47,39,0,0,0,16,52,23,21],[12,16,0,0,0,0,34,41,0,0,0,0,0,0,52,32,41,15,0,0,38,4,28,4,0,0,15,45,18,9,0,0,1,30,28,32] >;

C2×C52⋊C4 in GAP, Magma, Sage, TeX

C_2\times C_{52}\rtimes C_4
% in TeX

G:=Group("C2xC52:C4");
// GroupNames label

G:=SmallGroup(416,203);
// by ID

G=gap.SmallGroup(416,203);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,362,86,9221,1751]);
// Polycyclic

G:=Group<a,b,c|a^2=b^52=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^31>;
// generators/relations

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