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G = C13×C22≀C2order 416 = 25·13

Direct product of C13 and C22≀C2

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C13×C22≀C2, C241C26, (C2×C26)⋊7D4, (C2×D4)⋊1C26, C2.4(D4×C26), (D4×C26)⋊10C2, C22⋊C42C26, (C2×C52)⋊8C22, (C23×C26)⋊1C2, C231(C2×C26), C26.67(C2×D4), C222(D4×C13), (C2×C26).75C23, (C22×C26)⋊1C22, C22.10(C22×C26), (C2×C4)⋊1(C2×C26), (C13×C22⋊C4)⋊10C2, SmallGroup(416,181)

Series: Derived Chief Lower central Upper central

C1C22 — C13×C22≀C2
C1C2C22C2×C26C22×C26D4×C26 — C13×C22≀C2
C1C22 — C13×C22≀C2
C1C2×C26 — C13×C22≀C2

Generators and relations for C13×C22≀C2
 G = < a,b,c,d,e,f | a13=b2=c2=d2=e2=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf=bd=db, be=eb, cd=dc, fcf=ce=ec, de=ed, df=fd, ef=fe >

Subgroups: 212 in 130 conjugacy classes, 52 normal (10 characteristic)
C1, C2 [×3], C2 [×7], C4 [×3], C22, C22 [×6], C22 [×17], C2×C4 [×3], D4 [×6], C23, C23 [×3], C23 [×6], C13, C22⋊C4 [×3], C2×D4 [×3], C24, C26 [×3], C26 [×7], C22≀C2, C52 [×3], C2×C26, C2×C26 [×6], C2×C26 [×17], C2×C52 [×3], D4×C13 [×6], C22×C26, C22×C26 [×3], C22×C26 [×6], C13×C22⋊C4 [×3], D4×C26 [×3], C23×C26, C13×C22≀C2
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C13, C2×D4 [×3], C26 [×7], C22≀C2, C2×C26 [×7], D4×C13 [×6], C22×C26, D4×C26 [×3], C13×C22≀C2

Smallest permutation representation of C13×C22≀C2
On 104 points
Generators in S104
(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 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 14)(9 15)(10 16)(11 17)(12 18)(13 19)(27 66)(28 67)(29 68)(30 69)(31 70)(32 71)(33 72)(34 73)(35 74)(36 75)(37 76)(38 77)(39 78)(40 91)(41 79)(42 80)(43 81)(44 82)(45 83)(46 84)(47 85)(48 86)(49 87)(50 88)(51 89)(52 90)(53 104)(54 92)(55 93)(56 94)(57 95)(58 96)(59 97)(60 98)(61 99)(62 100)(63 101)(64 102)(65 103)
(27 94)(28 95)(29 96)(30 97)(31 98)(32 99)(33 100)(34 101)(35 102)(36 103)(37 104)(38 92)(39 93)(53 76)(54 77)(55 78)(56 66)(57 67)(58 68)(59 69)(60 70)(61 71)(62 72)(63 73)(64 74)(65 75)
(1 52)(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 84)(15 85)(16 86)(17 87)(18 88)(19 89)(20 90)(21 91)(22 79)(23 80)(24 81)(25 82)(26 83)(27 56)(28 57)(29 58)(30 59)(31 60)(32 61)(33 62)(34 63)(35 64)(36 65)(37 53)(38 54)(39 55)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 100)(73 101)(74 102)(75 103)(76 104)(77 92)(78 93)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 14)(9 15)(10 16)(11 17)(12 18)(13 19)(27 94)(28 95)(29 96)(30 97)(31 98)(32 99)(33 100)(34 101)(35 102)(36 103)(37 104)(38 92)(39 93)(40 91)(41 79)(42 80)(43 81)(44 82)(45 83)(46 84)(47 85)(48 86)(49 87)(50 88)(51 89)(52 90)(53 76)(54 77)(55 78)(56 66)(57 67)(58 68)(59 69)(60 70)(61 71)(62 72)(63 73)(64 74)(65 75)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 65)(8 53)(9 54)(10 55)(11 56)(12 57)(13 58)(14 76)(15 77)(16 78)(17 66)(18 67)(19 68)(20 69)(21 70)(22 71)(23 72)(24 73)(25 74)(26 75)(27 49)(28 50)(29 51)(30 52)(31 40)(32 41)(33 42)(34 43)(35 44)(36 45)(37 46)(38 47)(39 48)(79 99)(80 100)(81 101)(82 102)(83 103)(84 104)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)

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

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

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

182 conjugacy classes

class 1 2A2B2C2D···2I2J4A4B4C13A···13L26A···26AJ26AK···26DD26DE···26DP52A···52AJ
order12222···2244413···1326···2626···2626···2652···52
size11112···244441···11···12···24···44···4

182 irreducible representations

dim1111111122
type+++++
imageC1C2C2C2C13C26C26C26D4D4×C13
kernelC13×C22≀C2C13×C22⋊C4D4×C26C23×C26C22≀C2C22⋊C4C2×D4C24C2×C26C22
# reps133112363612672

Matrix representation of C13×C22≀C2 in GL4(𝔽53) generated by

16000
01600
0010
0001
,
52000
05200
00520
00301
,
12400
05200
0010
002352
,
1000
0100
00520
00052
,
52000
05200
00520
00052
,
244900
512900
002351
005230
G:=sub<GL(4,GF(53))| [16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[52,0,0,0,0,52,0,0,0,0,52,30,0,0,0,1],[1,0,0,0,24,52,0,0,0,0,1,23,0,0,0,52],[1,0,0,0,0,1,0,0,0,0,52,0,0,0,0,52],[52,0,0,0,0,52,0,0,0,0,52,0,0,0,0,52],[24,51,0,0,49,29,0,0,0,0,23,52,0,0,51,30] >;

C13×C22≀C2 in GAP, Magma, Sage, TeX

C_{13}\times C_2^2\wr C_2
% in TeX

G:=Group("C13xC2^2wrC2");
// GroupNames label

G:=SmallGroup(416,181);
// by ID

G=gap.SmallGroup(416,181);
# by ID

G:=PCGroup([6,-2,-2,-2,-13,-2,-2,1273,3818]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^13=b^2=c^2=d^2=e^2=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f=b*d=d*b,b*e=e*b,c*d=d*c,f*c*f=c*e=e*c,d*e=e*d,d*f=f*d,e*f=f*e>;
// generators/relations

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