Copied to
clipboard

G = C2×C52order 104 = 23·13

Abelian group of type [2,52]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C52, SmallGroup(104,9)

Series: Derived Chief Lower central Upper central

C1 — C2×C52
C1C2C26C52 — C2×C52
C1 — C2×C52
C1 — C2×C52

Generators and relations for C2×C52
 G = < a,b | a2=b52=1, ab=ba >


Smallest permutation representation of C2×C52
Regular action on 104 points
Generators in S104
(1 95)(2 96)(3 97)(4 98)(5 99)(6 100)(7 101)(8 102)(9 103)(10 104)(11 53)(12 54)(13 55)(14 56)(15 57)(16 58)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 65)(24 66)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 81)(40 82)(41 83)(42 84)(43 85)(44 86)(45 87)(46 88)(47 89)(48 90)(49 91)(50 92)(51 93)(52 94)
(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)

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

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

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

C2×C52 is a maximal subgroup of   C52.4C4  C26.D4  C523C4  D26⋊C4  D525C2

104 conjugacy classes

class 1 2A2B2C4A4B4C4D13A···13L26A···26AJ52A···52AV
order1222444413···1326···2652···52
size111111111···11···11···1

104 irreducible representations

dim11111111
type+++
imageC1C2C2C4C13C26C26C52
kernelC2×C52C52C2×C26C26C2×C4C4C22C2
# reps121412241248

Matrix representation of C2×C52 in GL2(𝔽53) generated by

520
01
,
360
020
G:=sub<GL(2,GF(53))| [52,0,0,1],[36,0,0,20] >;

C2×C52 in GAP, Magma, Sage, TeX

C_2\times C_{52}
% in TeX

G:=Group("C2xC52");
// GroupNames label

G:=SmallGroup(104,9);
// by ID

G=gap.SmallGroup(104,9);
# by ID

G:=PCGroup([4,-2,-2,-13,-2,208]);
// Polycyclic

G:=Group<a,b|a^2=b^52=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C52 in TeX

׿
×
𝔽