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G = C2×Dic13order 104 = 23·13

Direct product of C2 and Dic13

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C2×Dic13, C262C4, C2.2D26, C22.D13, C26.4C22, C133(C2×C4), (C2×C26).C2, SmallGroup(104,7)

Series: Derived Chief Lower central Upper central

C1C13 — C2×Dic13
C1C13C26Dic13 — C2×Dic13
C13 — C2×Dic13
C1C22

Generators and relations for C2×Dic13
 G = < a,b,c | a2=b26=1, c2=b13, ab=ba, ac=ca, cbc-1=b-1 >

13C4
13C4
13C2×C4

Smallest permutation representation of C2×Dic13
Regular action on 104 points
Generators in S104
(1 32)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(17 48)(18 49)(19 50)(20 51)(21 52)(22 27)(23 28)(24 29)(25 30)(26 31)(53 79)(54 80)(55 81)(56 82)(57 83)(58 84)(59 85)(60 86)(61 87)(62 88)(63 89)(64 90)(65 91)(66 92)(67 93)(68 94)(69 95)(70 96)(71 97)(72 98)(73 99)(74 100)(75 101)(76 102)(77 103)(78 104)
(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 66 14 53)(2 65 15 78)(3 64 16 77)(4 63 17 76)(5 62 18 75)(6 61 19 74)(7 60 20 73)(8 59 21 72)(9 58 22 71)(10 57 23 70)(11 56 24 69)(12 55 25 68)(13 54 26 67)(27 97 40 84)(28 96 41 83)(29 95 42 82)(30 94 43 81)(31 93 44 80)(32 92 45 79)(33 91 46 104)(34 90 47 103)(35 89 48 102)(36 88 49 101)(37 87 50 100)(38 86 51 99)(39 85 52 98)

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

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

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

C2×Dic13 is a maximal subgroup of   C26.D4  C523C4  D26⋊C4  C23.D13  C13⋊M4(2)  C2×C4×D13  D42D13
C2×Dic13 is a maximal quotient of   C52.4C4  C523C4  C23.D13

32 conjugacy classes

class 1 2A2B2C4A4B4C4D13A···13F26A···26R
order1222444413···1326···26
size1111131313132···22···2

32 irreducible representations

dim1111222
type++++-+
imageC1C2C2C4D13Dic13D26
kernelC2×Dic13Dic13C2×C26C26C22C2C2
# reps12146126

Matrix representation of C2×Dic13 in GL3(𝔽53) generated by

5200
010
001
,
100
0052
0127
,
100
02235
02431
G:=sub<GL(3,GF(53))| [52,0,0,0,1,0,0,0,1],[1,0,0,0,0,1,0,52,27],[1,0,0,0,22,24,0,35,31] >;

C2×Dic13 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_{13}
% in TeX

G:=Group("C2xDic13");
// GroupNames label

G:=SmallGroup(104,7);
// by ID

G=gap.SmallGroup(104,7);
# by ID

G:=PCGroup([4,-2,-2,-2,-13,16,1539]);
// Polycyclic

G:=Group<a,b,c|a^2=b^26=1,c^2=b^13,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C2×Dic13 in TeX

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