direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C4×D13, C52⋊3C22, C26.2C23, C22.9D26, D26.8C22, Dic13⋊3C22, C26⋊2(C2×C4), (C2×C52)⋊5C2, C13⋊2(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
| C13 — C2×C4×D13 |
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
►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
D26⋊1C8 D26⋊C8 D26.Q8 C42⋊D13 Dic13⋊4D4 D26.12D4 D26⋊D4 C4⋊C4⋊7D13 D52⋊8C4 D26.13D4 C4⋊2D52 D26⋊Q8 D26⋊2Q8 C52⋊2D4 D26⋊3Q8 D13⋊M4(2) D26.C23
C2×C4×D13 is a maximal quotient of
C42⋊D13 C23.11D26 Dic13⋊4D4 Dic13⋊3Q8 C4⋊C4⋊7D13 D52⋊8C4 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