Copied to
clipboard

G = C2×Dic13order 104 = 23·13

Direct product of C2 and Dic13

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

 Derived series C1 — C13 — C2×Dic13
 Chief series C1 — C13 — C26 — Dic13 — C2×Dic13
 Lower central C13 — C2×Dic13
 Upper central C1 — C22

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

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

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

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

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

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 2A 2B 2C 4A 4B 4C 4D 13A ··· 13F 26A ··· 26R order 1 2 2 2 4 4 4 4 13 ··· 13 26 ··· 26 size 1 1 1 1 13 13 13 13 2 ··· 2 2 ··· 2

32 irreducible representations

 dim 1 1 1 1 2 2 2 type + + + + - + image C1 C2 C2 C4 D13 Dic13 D26 kernel C2×Dic13 Dic13 C2×C26 C26 C22 C2 C2 # reps 1 2 1 4 6 12 6

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

 52 0 0 0 1 0 0 0 1
,
 1 0 0 0 0 52 0 1 27
,
 1 0 0 0 22 35 0 24 31
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

׿
×
𝔽