metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22.2D52, C23.1D26, C13⋊3(C23⋊C4), C22⋊C4⋊1D13, (C2×C26).27D4, (C2×Dic13)⋊1C4, (C22×D13)⋊1C4, C23.D13⋊1C2, C22.3(C4×D13), C2.4(D26⋊C4), C26.13(C22⋊C4), C22.8(C13⋊D4), (C22×C26).5C22, (C13×C22⋊C4)⋊1C2, (C2×C26).21(C2×C4), (C2×C13⋊D4).1C2, SmallGroup(416,13)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22.2D52
G = < a,b,c,d,e | a2=b2=c2=1, d26=a, e2=abc, ab=ba, eae-1=ac=ca, ad=da, dbd-1=ebe-1=bc=cb, cd=dc, ce=ec, ede-1=bcd25 >
2C2
2C2
2C2
52C2
4C4
4C22
26C22
26C4
52C4
52C22
2C26
2C26
2C26
4D13
13C23
13C2×C4
26C2×C4
26D4
26D4
2D26
4D26
4C52
13C22⋊C4
13C2×D4
13C23⋊C4
Smallest permutation representation of C22.2D52
►On 104 points
Generators in S104
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 37)(12 38)(13 39)(14 40)(15 41)(16 42)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 49)(24 50)(25 51)(26 52)(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 83)(2 28)(3 85)(4 30)(5 87)(6 32)(7 89)(8 34)(9 91)(10 36)(11 93)(12 38)(13 95)(14 40)(15 97)(16 42)(17 99)(18 44)(19 101)(20 46)(21 103)(22 48)(23 53)(24 50)(25 55)(26 52)(27 57)(29 59)(31 61)(33 63)(35 65)(37 67)(39 69)(41 71)(43 73)(45 75)(47 77)(49 79)(51 81)(54 80)(56 82)(58 84)(60 86)(62 88)(64 90)(66 92)(68 94)(70 96)(72 98)(74 100)(76 102)(78 104)
(1 57)(2 58)(3 59)(4 60)(5 61)(6 62)(7 63)(8 64)(9 65)(10 66)(11 67)(12 68)(13 69)(14 70)(15 71)(16 72)(17 73)(18 74)(19 75)(20 76)(21 77)(22 78)(23 79)(24 80)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 97)(42 98)(43 99)(44 100)(45 101)(46 102)(47 103)(48 104)(49 53)(50 54)(51 55)(52 56)
(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 68 58 12)(3 67)(4 10 60 66)(5 9)(6 64 62 8)(7 63)(11 59)(14 56 70 52)(15 55)(16 50 72 54)(17 49)(18 104 74 48)(19 103)(20 46 76 102)(21 45)(22 100 78 44)(23 99)(24 42 80 98)(25 41)(26 96 82 40)(27 95)(28 38 84 94)(29 37)(30 92 86 36)(31 91)(32 34 88 90)(35 87)(39 83)(43 79)(47 75)(51 71)(53 73)(57 69)(61 65)(77 101)(81 97)(85 93)
G:=sub<Sym(104)| (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,37)(12,38)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52)(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), (1,83)(2,28)(3,85)(4,30)(5,87)(6,32)(7,89)(8,34)(9,91)(10,36)(11,93)(12,38)(13,95)(14,40)(15,97)(16,42)(17,99)(18,44)(19,101)(20,46)(21,103)(22,48)(23,53)(24,50)(25,55)(26,52)(27,57)(29,59)(31,61)(33,63)(35,65)(37,67)(39,69)(41,71)(43,73)(45,75)(47,77)(49,79)(51,81)(54,80)(56,82)(58,84)(60,86)(62,88)(64,90)(66,92)(68,94)(70,96)(72,98)(74,100)(76,102)(78,104), (1,57)(2,58)(3,59)(4,60)(5,61)(6,62)(7,63)(8,64)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,73)(18,74)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,53)(50,54)(51,55)(52,56), (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)(2,68,58,12)(3,67)(4,10,60,66)(5,9)(6,64,62,8)(7,63)(11,59)(14,56,70,52)(15,55)(16,50,72,54)(17,49)(18,104,74,48)(19,103)(20,46,76,102)(21,45)(22,100,78,44)(23,99)(24,42,80,98)(25,41)(26,96,82,40)(27,95)(28,38,84,94)(29,37)(30,92,86,36)(31,91)(32,34,88,90)(35,87)(39,83)(43,79)(47,75)(51,71)(53,73)(57,69)(61,65)(77,101)(81,97)(85,93)>;
G:=Group( (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,37)(12,38)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52)(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), (1,83)(2,28)(3,85)(4,30)(5,87)(6,32)(7,89)(8,34)(9,91)(10,36)(11,93)(12,38)(13,95)(14,40)(15,97)(16,42)(17,99)(18,44)(19,101)(20,46)(21,103)(22,48)(23,53)(24,50)(25,55)(26,52)(27,57)(29,59)(31,61)(33,63)(35,65)(37,67)(39,69)(41,71)(43,73)(45,75)(47,77)(49,79)(51,81)(54,80)(56,82)(58,84)(60,86)(62,88)(64,90)(66,92)(68,94)(70,96)(72,98)(74,100)(76,102)(78,104), (1,57)(2,58)(3,59)(4,60)(5,61)(6,62)(7,63)(8,64)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,73)(18,74)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,53)(50,54)(51,55)(52,56), (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)(2,68,58,12)(3,67)(4,10,60,66)(5,9)(6,64,62,8)(7,63)(11,59)(14,56,70,52)(15,55)(16,50,72,54)(17,49)(18,104,74,48)(19,103)(20,46,76,102)(21,45)(22,100,78,44)(23,99)(24,42,80,98)(25,41)(26,96,82,40)(27,95)(28,38,84,94)(29,37)(30,92,86,36)(31,91)(32,34,88,90)(35,87)(39,83)(43,79)(47,75)(51,71)(53,73)(57,69)(61,65)(77,101)(81,97)(85,93) );
G=PermutationGroup([[(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,33),(8,34),(9,35),(10,36),(11,37),(12,38),(13,39),(14,40),(15,41),(16,42),(17,43),(18,44),(19,45),(20,46),(21,47),(22,48),(23,49),(24,50),(25,51),(26,52),(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)], [(1,83),(2,28),(3,85),(4,30),(5,87),(6,32),(7,89),(8,34),(9,91),(10,36),(11,93),(12,38),(13,95),(14,40),(15,97),(16,42),(17,99),(18,44),(19,101),(20,46),(21,103),(22,48),(23,53),(24,50),(25,55),(26,52),(27,57),(29,59),(31,61),(33,63),(35,65),(37,67),(39,69),(41,71),(43,73),(45,75),(47,77),(49,79),(51,81),(54,80),(56,82),(58,84),(60,86),(62,88),(64,90),(66,92),(68,94),(70,96),(72,98),(74,100),(76,102),(78,104)], [(1,57),(2,58),(3,59),(4,60),(5,61),(6,62),(7,63),(8,64),(9,65),(10,66),(11,67),(12,68),(13,69),(14,70),(15,71),(16,72),(17,73),(18,74),(19,75),(20,76),(21,77),(22,78),(23,79),(24,80),(25,81),(26,82),(27,83),(28,84),(29,85),(30,86),(31,87),(32,88),(33,89),(34,90),(35,91),(36,92),(37,93),(38,94),(39,95),(40,96),(41,97),(42,98),(43,99),(44,100),(45,101),(46,102),(47,103),(48,104),(49,53),(50,54),(51,55),(52,56)], [(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),(2,68,58,12),(3,67),(4,10,60,66),(5,9),(6,64,62,8),(7,63),(11,59),(14,56,70,52),(15,55),(16,50,72,54),(17,49),(18,104,74,48),(19,103),(20,46,76,102),(21,45),(22,100,78,44),(23,99),(24,42,80,98),(25,41),(26,96,82,40),(27,95),(28,38,84,94),(29,37),(30,92,86,36),(31,91),(32,34,88,90),(35,87),(39,83),(43,79),(47,75),(51,71),(53,73),(57,69),(61,65),(77,101),(81,97),(85,93)]])
71 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 13A | ··· | 13F | 26A | ··· | 26R | 26S | ··· | 26AD | 52A | ··· | 52X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 2 | 2 | 2 | 52 | 4 | 4 | 52 | 52 | 52 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
71 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D13 | D26 | C4×D13 | D52 | C13⋊D4 | C23⋊C4 | C22.2D52 |
| kernel | C22.2D52 | C23.D13 | C13×C22⋊C4 | C2×C13⋊D4 | C2×Dic13 | C22×D13 | C2×C26 | C22⋊C4 | C23 | C22 | C22 | C22 | C13 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 12 | 12 | 12 | 1 | 12 |
Matrix representation of C22.2D52 ►in GL4(𝔽53) generated by
| 8 | 13 | 35 | 11 |
| 40 | 45 | 0 | 18 |
| 0 | 0 | 40 | 17 |
| 0 | 0 | 40 | 13 |
| 45 | 40 | 44 | 31 |
| 13 | 8 | 8 | 45 |
| 0 | 0 | 40 | 17 |
| 0 | 0 | 40 | 13 |
| 52 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 |
| 0 | 0 | 52 | 0 |
| 0 | 0 | 0 | 52 |
| 2 | 32 | 38 | 14 |
| 31 | 26 | 0 | 38 |
| 3 | 25 | 11 | 16 |
| 22 | 19 | 22 | 14 |
| 14 | 42 | 12 | 0 |
| 37 | 39 | 31 | 27 |
| 0 | 0 | 13 | 44 |
| 0 | 0 | 13 | 40 |
G:=sub<GL(4,GF(53))| [8,40,0,0,13,45,0,0,35,0,40,40,11,18,17,13],[45,13,0,0,40,8,0,0,44,8,40,40,31,45,17,13],[52,0,0,0,0,52,0,0,0,0,52,0,0,0,0,52],[2,31,3,22,32,26,25,19,38,0,11,22,14,38,16,14],[14,37,0,0,42,39,0,0,12,31,13,13,0,27,44,40] >;
C22.2D52 in GAP, Magma, Sage, TeX
C_2^2._2D_{52} % in TeX
G:=Group("C2^2.2D52"); // GroupNames label
G:=SmallGroup(416,13);
// by ID
G=gap.SmallGroup(416,13);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,121,31,362,297,13829]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^26=a,e^2=a*b*c,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,d*b*d^-1=e*b*e^-1=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d^25>;
// generators/relations
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