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## G = C2×C4×D13order 208 = 24·13

### Direct product of C2×C4 and D13

Aliases: C2×C4×D13, C523C22, C26.2C23, C22.9D26, D26.8C22, Dic133C22, C262(C2×C4), (C2×C52)⋊5C2, C132(C22×C4), (C2×Dic13)⋊5C2, (C2×C26).9C22, C2.1(C22×D13), (C22×D13).4C2, SmallGroup(208,36)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×C4×D13
G = < a,b,c,d | a2=b4=c13=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 282 in 54 conjugacy classes, 35 normal (11 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C23, C13, C22×C4, D13, C26, C26, Dic13, C52, D26, C2×C26, C4×D13, C2×Dic13, C2×C52, C22×D13, C2×C4×D13
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, D13, D26, C4×D13, C22×D13, C2×C4×D13

Smallest permutation representation of C2×C4×D13
On 104 points
Generators in S104
(1 57)(2 58)(3 59)(4 60)(5 61)(6 62)(7 63)(8 64)(9 65)(10 53)(11 54)(12 55)(13 56)(14 68)(15 69)(16 70)(17 71)(18 72)(19 73)(20 74)(21 75)(22 76)(23 77)(24 78)(25 66)(26 67)(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)(40 100)(41 101)(42 102)(43 103)(44 104)(45 92)(46 93)(47 94)(48 95)(49 96)(50 97)(51 98)(52 99)
(1 51 16 28)(2 52 17 29)(3 40 18 30)(4 41 19 31)(5 42 20 32)(6 43 21 33)(7 44 22 34)(8 45 23 35)(9 46 24 36)(10 47 25 37)(11 48 26 38)(12 49 14 39)(13 50 15 27)(53 94 66 85)(54 95 67 86)(55 96 68 87)(56 97 69 88)(57 98 70 89)(58 99 71 90)(59 100 72 91)(60 101 73 79)(61 102 74 80)(62 103 75 81)(63 104 76 82)(64 92 77 83)(65 93 78 84)
(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 13)(2 12)(3 11)(4 10)(5 9)(6 8)(14 17)(15 16)(18 26)(19 25)(20 24)(21 23)(27 28)(29 39)(30 38)(31 37)(32 36)(33 35)(40 48)(41 47)(42 46)(43 45)(49 52)(50 51)(53 60)(54 59)(55 58)(56 57)(61 65)(62 64)(66 73)(67 72)(68 71)(69 70)(74 78)(75 77)(79 85)(80 84)(81 83)(86 91)(87 90)(88 89)(92 103)(93 102)(94 101)(95 100)(96 99)(97 98)

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

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

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

C2×C4×D13 is a maximal subgroup of
D261C8  D26⋊C8  D26.Q8  C42⋊D13  Dic134D4  D26.12D4  D26⋊D4  C4⋊C47D13  D528C4  D26.13D4  C42D52  D26⋊Q8  D262Q8  C522D4  D263Q8  D13⋊M4(2)  D26.C23
C2×C4×D13 is a maximal quotient of
C42⋊D13  C23.11D26  Dic134D4  Dic133Q8  C4⋊C47D13  D528C4  D52.3C4  D52.2C4

64 conjugacy classes

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

64 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C4 D13 D26 D26 C4×D13 kernel C2×C4×D13 C4×D13 C2×Dic13 C2×C52 C22×D13 D26 C2×C4 C4 C22 C2 # reps 1 4 1 1 1 8 6 12 6 24

Matrix representation of C2×C4×D13 in GL3(𝔽53) generated by

 1 0 0 0 52 0 0 0 52
,
 30 0 0 0 1 0 0 0 1
,
 1 0 0 0 0 1 0 52 13
,
 52 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(53))| [1,0,0,0,52,0,0,0,52],[30,0,0,0,1,0,0,0,1],[1,0,0,0,0,52,0,1,13],[52,0,0,0,0,1,0,1,0] >;

C2×C4×D13 in GAP, Magma, Sage, TeX

C_2\times C_4\times D_{13}
% in TeX

G:=Group("C2xC4xD13");
// GroupNames label

G:=SmallGroup(208,36);
// by ID

G=gap.SmallGroup(208,36);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-13,42,4804]);
// Polycyclic

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

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