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G = C22.2D52order 416 = 25·13

1st non-split extension by C22 of D52 acting via D52/D26=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C22.2D52, C23.1D26, C13:3(C23:C4), C22:C4:1D13, (C2xC26).27D4, (C2xDic13):1C4, (C22xD13):1C4, C23.D13:1C2, C22.3(C4xD13), C2.4(D26:C4), C26.13(C22:C4), C22.8(C13:D4), (C22xC26).5C22, (C13xC22:C4):1C2, (C2xC26).21(C2xC4), (C2xC13:D4).1C2, SmallGroup(416,13)

Series: Derived Chief Lower central Upper central

C1C2xC26 — C22.2D52
C1C13C26C2xC26C22xC26C2xC13:D4 — C22.2D52
C13C26C2xC26 — C22.2D52
C1C2C23C22:C4

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 >

Subgroups: 424 in 52 conjugacy classes, 19 normal (all characteristic)
Quotients: C1, C2, C4, C22, C2xC4, D4, C22:C4, D13, C23:C4, D26, C4xD13, D52, C13:D4, D26:C4, C22.2D52
2C2
2C2
2C2
52C2
4C4
4C22
26C22
26C4
52C4
52C22
2C26
2C26
2C26
4D13
2C2xC4
13C23
13C2xC4
26C2xC4
26D4
26D4
2Dic13
2D26
4Dic13
4D26
4C2xC26
4C52
13C22:C4
13C2xD4
2C2xC52
2C13:D4
2C13:D4
2C2xDic13
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 2A2B2C2D2E4A4B4C4D4E13A···13F26A···26R26S···26AD52A···52X
order1222224444413···1326···2626···2652···52
size1122252445252522···22···24···44···4

71 irreducible representations

dim11111122222244
type+++++++++
imageC1C2C2C2C4C4D4D13D26C4xD13D52C13:D4C23:C4C22.2D52
kernelC22.2D52C23.D13C13xC22:C4C2xC13:D4C2xDic13C22xD13C2xC26C22:C4C23C22C22C22C13C1
# reps111122266121212112

Matrix representation of C22.2D52 in GL4(F53) generated by

8133511
4045018
004017
004013
,
45404431
138845
004017
004013
,
52000
05200
00520
00052
,
2323814
3126038
3251116
22192214
,
1442120
37393127
001344
001340
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

Export

Subgroup lattice of C22.2D52 in TeX

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