direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4○D4×D13, D4⋊7D26, Q8⋊6D26, D52⋊10C22, C52.25C23, C26.11C24, D26.6C23, Dic26⋊10C22, Dic13.17C23, (C2×C4)⋊7D26, (D4×D13)⋊6C2, (Q8×D13)⋊6C2, (C2×C52)⋊4C22, D52⋊5C2⋊7C2, D52⋊C2⋊6C2, D4⋊2D13⋊6C2, (D4×C13)⋊8C22, (C4×D13)⋊7C22, C13⋊D4⋊4C22, (C2×C26).3C23, (Q8×C13)⋊7C22, C2.12(C23×D13), C4.25(C22×D13), C22.2(C22×D13), (C2×Dic13)⋊10C22, (C22×D13).34C22, (C2×C4×D13)⋊6C2, C13⋊4(C2×C4○D4), (C13×C4○D4)⋊3C2, SmallGroup(416,222)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4○D4×D13
G = < a,b,c,d,e | a4=c2=d13=e2=1, b2=a2, ab=ba, ac=ca, ad=da, ae=ea, cbc=a2b, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 1040 in 164 conjugacy classes, 87 normal (16 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C13, C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, D13, D13, C26, C26, C2×C4○D4, Dic13, Dic13, C52, C52, D26, D26, D26, C2×C26, Dic26, C4×D13, C4×D13, D52, C2×Dic13, C13⋊D4, C2×C52, D4×C13, Q8×C13, C22×D13, C2×C4×D13, D52⋊5C2, D4×D13, D4⋊2D13, Q8×D13, D52⋊C2, C13×C4○D4, C4○D4×D13
Quotients: C1, C2, C22, C23, C4○D4, C24, D13, C2×C4○D4, D26, C22×D13, C23×D13, C4○D4×D13
►On 104 points
Generators in S104
(1 52 16 36)(2 40 17 37)(3 41 18 38)(4 42 19 39)(5 43 20 27)(6 44 21 28)(7 45 22 29)(8 46 23 30)(9 47 24 31)(10 48 25 32)(11 49 26 33)(12 50 14 34)(13 51 15 35)(53 95 77 87)(54 96 78 88)(55 97 66 89)(56 98 67 90)(57 99 68 91)(58 100 69 79)(59 101 70 80)(60 102 71 81)(61 103 72 82)(62 104 73 83)(63 92 74 84)(64 93 75 85)(65 94 76 86)
(1 61 16 72)(2 62 17 73)(3 63 18 74)(4 64 19 75)(5 65 20 76)(6 53 21 77)(7 54 22 78)(8 55 23 66)(9 56 24 67)(10 57 25 68)(11 58 26 69)(12 59 14 70)(13 60 15 71)(27 86 43 94)(28 87 44 95)(29 88 45 96)(30 89 46 97)(31 90 47 98)(32 91 48 99)(33 79 49 100)(34 80 50 101)(35 81 51 102)(36 82 52 103)(37 83 40 104)(38 84 41 92)(39 85 42 93)
(53 77)(54 78)(55 66)(56 67)(57 68)(58 69)(59 70)(60 71)(61 72)(62 73)(63 74)(64 75)(65 76)(79 100)(80 101)(81 102)(82 103)(83 104)(84 92)(85 93)(86 94)(87 95)(88 96)(89 97)(90 98)(91 99)
(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 31)(28 30)(32 39)(33 38)(34 37)(35 36)(40 50)(41 49)(42 48)(43 47)(44 46)(51 52)(53 55)(56 65)(57 64)(58 63)(59 62)(60 61)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)(79 84)(80 83)(81 82)(85 91)(86 90)(87 89)(92 100)(93 99)(94 98)(95 97)(101 104)(102 103)
G:=sub<Sym(104)| (1,52,16,36)(2,40,17,37)(3,41,18,38)(4,42,19,39)(5,43,20,27)(6,44,21,28)(7,45,22,29)(8,46,23,30)(9,47,24,31)(10,48,25,32)(11,49,26,33)(12,50,14,34)(13,51,15,35)(53,95,77,87)(54,96,78,88)(55,97,66,89)(56,98,67,90)(57,99,68,91)(58,100,69,79)(59,101,70,80)(60,102,71,81)(61,103,72,82)(62,104,73,83)(63,92,74,84)(64,93,75,85)(65,94,76,86), (1,61,16,72)(2,62,17,73)(3,63,18,74)(4,64,19,75)(5,65,20,76)(6,53,21,77)(7,54,22,78)(8,55,23,66)(9,56,24,67)(10,57,25,68)(11,58,26,69)(12,59,14,70)(13,60,15,71)(27,86,43,94)(28,87,44,95)(29,88,45,96)(30,89,46,97)(31,90,47,98)(32,91,48,99)(33,79,49,100)(34,80,50,101)(35,81,51,102)(36,82,52,103)(37,83,40,104)(38,84,41,92)(39,85,42,93), (53,77)(54,78)(55,66)(56,67)(57,68)(58,69)(59,70)(60,71)(61,72)(62,73)(63,74)(64,75)(65,76)(79,100)(80,101)(81,102)(82,103)(83,104)(84,92)(85,93)(86,94)(87,95)(88,96)(89,97)(90,98)(91,99), (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)(27,31)(28,30)(32,39)(33,38)(34,37)(35,36)(40,50)(41,49)(42,48)(43,47)(44,46)(51,52)(53,55)(56,65)(57,64)(58,63)(59,62)(60,61)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(79,84)(80,83)(81,82)(85,91)(86,90)(87,89)(92,100)(93,99)(94,98)(95,97)(101,104)(102,103)>;
G:=Group( (1,52,16,36)(2,40,17,37)(3,41,18,38)(4,42,19,39)(5,43,20,27)(6,44,21,28)(7,45,22,29)(8,46,23,30)(9,47,24,31)(10,48,25,32)(11,49,26,33)(12,50,14,34)(13,51,15,35)(53,95,77,87)(54,96,78,88)(55,97,66,89)(56,98,67,90)(57,99,68,91)(58,100,69,79)(59,101,70,80)(60,102,71,81)(61,103,72,82)(62,104,73,83)(63,92,74,84)(64,93,75,85)(65,94,76,86), (1,61,16,72)(2,62,17,73)(3,63,18,74)(4,64,19,75)(5,65,20,76)(6,53,21,77)(7,54,22,78)(8,55,23,66)(9,56,24,67)(10,57,25,68)(11,58,26,69)(12,59,14,70)(13,60,15,71)(27,86,43,94)(28,87,44,95)(29,88,45,96)(30,89,46,97)(31,90,47,98)(32,91,48,99)(33,79,49,100)(34,80,50,101)(35,81,51,102)(36,82,52,103)(37,83,40,104)(38,84,41,92)(39,85,42,93), (53,77)(54,78)(55,66)(56,67)(57,68)(58,69)(59,70)(60,71)(61,72)(62,73)(63,74)(64,75)(65,76)(79,100)(80,101)(81,102)(82,103)(83,104)(84,92)(85,93)(86,94)(87,95)(88,96)(89,97)(90,98)(91,99), (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)(27,31)(28,30)(32,39)(33,38)(34,37)(35,36)(40,50)(41,49)(42,48)(43,47)(44,46)(51,52)(53,55)(56,65)(57,64)(58,63)(59,62)(60,61)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(79,84)(80,83)(81,82)(85,91)(86,90)(87,89)(92,100)(93,99)(94,98)(95,97)(101,104)(102,103) );
G=PermutationGroup([[(1,52,16,36),(2,40,17,37),(3,41,18,38),(4,42,19,39),(5,43,20,27),(6,44,21,28),(7,45,22,29),(8,46,23,30),(9,47,24,31),(10,48,25,32),(11,49,26,33),(12,50,14,34),(13,51,15,35),(53,95,77,87),(54,96,78,88),(55,97,66,89),(56,98,67,90),(57,99,68,91),(58,100,69,79),(59,101,70,80),(60,102,71,81),(61,103,72,82),(62,104,73,83),(63,92,74,84),(64,93,75,85),(65,94,76,86)], [(1,61,16,72),(2,62,17,73),(3,63,18,74),(4,64,19,75),(5,65,20,76),(6,53,21,77),(7,54,22,78),(8,55,23,66),(9,56,24,67),(10,57,25,68),(11,58,26,69),(12,59,14,70),(13,60,15,71),(27,86,43,94),(28,87,44,95),(29,88,45,96),(30,89,46,97),(31,90,47,98),(32,91,48,99),(33,79,49,100),(34,80,50,101),(35,81,51,102),(36,82,52,103),(37,83,40,104),(38,84,41,92),(39,85,42,93)], [(53,77),(54,78),(55,66),(56,67),(57,68),(58,69),(59,70),(60,71),(61,72),(62,73),(63,74),(64,75),(65,76),(79,100),(80,101),(81,102),(82,103),(83,104),(84,92),(85,93),(86,94),(87,95),(88,96),(89,97),(90,98),(91,99)], [(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),(27,31),(28,30),(32,39),(33,38),(34,37),(35,36),(40,50),(41,49),(42,48),(43,47),(44,46),(51,52),(53,55),(56,65),(57,64),(58,63),(59,62),(60,61),(66,77),(67,76),(68,75),(69,74),(70,73),(71,72),(79,84),(80,83),(81,82),(85,91),(86,90),(87,89),(92,100),(93,99),(94,98),(95,97),(101,104),(102,103)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 13A | ··· | 13F | 26A | ··· | 26F | 26G | ··· | 26X | 52A | ··· | 52L | 52M | ··· | 52AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 |
| size | 1 | 1 | 2 | 2 | 2 | 13 | 13 | 26 | 26 | 26 | 1 | 1 | 2 | 2 | 2 | 13 | 13 | 26 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4○D4 | D13 | D26 | D26 | D26 | C4○D4×D13 |
| kernel | C4○D4×D13 | C2×C4×D13 | D52⋊5C2 | D4×D13 | D4⋊2D13 | Q8×D13 | D52⋊C2 | C13×C4○D4 | D13 | C4○D4 | C2×C4 | D4 | Q8 | C1 |
| # reps | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 4 | 6 | 18 | 18 | 6 | 12 |
Matrix representation of C4○D4×D13 ►in GL4(𝔽53) generated by
| 52 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 0 | 30 |
| 52 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 |
| 0 | 0 | 52 | 30 |
| 0 | 0 | 7 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 46 | 52 |
| 32 | 1 | 0 | 0 |
| 16 | 32 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 13 | 39 | 0 | 0 |
| 12 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(53))| [52,0,0,0,0,52,0,0,0,0,30,0,0,0,0,30],[52,0,0,0,0,52,0,0,0,0,52,7,0,0,30,1],[1,0,0,0,0,1,0,0,0,0,1,46,0,0,0,52],[32,16,0,0,1,32,0,0,0,0,1,0,0,0,0,1],[13,12,0,0,39,40,0,0,0,0,1,0,0,0,0,1] >;
C4○D4×D13 in GAP, Magma, Sage, TeX
C_4\circ D_4\times D_{13} % in TeX
G:=Group("C4oD4xD13"); // GroupNames label
G:=SmallGroup(416,222);
// by ID
G=gap.SmallGroup(416,222);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,86,297,13829]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=c^2=d^13=e^2=1,b^2=a^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=a^2*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations