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G = C2×C26.C6order 312 = 23·3·13

Direct product of C2 and C26.C6

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C2×C26.C6, C262C12, Dic133C6, (C2×C26).C6, C133(C2×C12), (C2×Dic13)⋊C3, C26.4(C2×C6), C22.(C13⋊C6), C13⋊C33(C2×C4), (C2×C13⋊C3)⋊2C4, C2.2(C2×C13⋊C6), (C22×C13⋊C3).C2, (C2×C13⋊C3).4C22, SmallGroup(312,11)

Series: Derived Chief Lower central Upper central

C1C13 — C2×C26.C6
C1C13C26C2×C13⋊C3C26.C6 — C2×C26.C6
C13 — C2×C26.C6
C1C22

Generators and relations for C2×C26.C6
 G = < a,b,c | a2=b26=1, c6=b13, ab=ba, ac=ca, cbc-1=b23 >

13C3
13C4
13C4
13C6
13C6
13C6
13C2×C4
13C12
13C12
13C2×C6
13C2×C12

Smallest permutation representation of C2×C26.C6
On 104 points
Generators in S104
(1 44)(2 45)(3 46)(4 47)(5 48)(6 49)(7 50)(8 51)(9 52)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 33)(17 34)(18 35)(19 36)(20 37)(21 38)(22 39)(23 40)(24 41)(25 42)(26 43)(53 99)(54 100)(55 101)(56 102)(57 103)(58 104)(59 79)(60 80)(61 81)(62 82)(63 83)(64 84)(65 85)(66 86)(67 87)(68 88)(69 89)(70 90)(71 91)(72 92)(73 93)(74 94)(75 95)(76 96)(77 97)(78 98)
(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 58 14 71)(2 75 17 70 10 55 15 62 4 57 23 68)(3 66 20 69 19 78 16 53 7 56 6 65)(5 74 26 67 11 72 18 61 13 54 24 59)(8 73 9 64 12 63 21 60 22 77 25 76)(27 101 32 82 47 103 40 88 45 95 34 90)(28 92 35 81 30 100 41 79 48 94 43 87)(29 83 38 80 39 97 42 96 51 93 52 84)(31 91 44 104)(33 99 50 102 49 85 46 86 37 89 36 98)

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

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

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

32 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D6A···6F12A···12H13A13B26A···26F
order12223344446···612···12131326···26
size111113131313131313···1313···13666···6

32 irreducible representations

dim11111111666
type++++-+
imageC1C2C2C3C4C6C6C12C13⋊C6C26.C6C2×C13⋊C6
kernelC2×C26.C6C26.C6C22×C13⋊C3C2×Dic13C2×C13⋊C3Dic13C2×C26C26C22C2C2
# reps12124428242

Matrix representation of C2×C26.C6 in GL8(𝔽157)

1560000000
0156000000
00100000
00010000
00001000
00000100
00000010
00000001
,
1560000000
01000000
00156671556815567
0090929019191
0066646715513365
00922591912592
0065133155676466
0091911909290
,
1350000000
0144000000
00136467154134154
00817386062114
00120853010669136
00671541378783154
0031267020390
0043110103971467

G:=sub<GL(8,GF(157))| [156,0,0,0,0,0,0,0,0,156,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[156,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,156,90,66,92,65,91,0,0,67,92,64,25,133,91,0,0,155,90,67,91,155,1,0,0,68,1,155,91,67,90,0,0,155,91,133,25,64,92,0,0,67,91,65,92,66,90],[135,0,0,0,0,0,0,0,0,144,0,0,0,0,0,0,0,0,13,8,120,67,3,43,0,0,64,17,85,154,126,110,0,0,67,38,30,137,70,103,0,0,154,60,106,87,20,97,0,0,134,62,69,83,3,146,0,0,154,114,136,154,90,7] >;

C2×C26.C6 in GAP, Magma, Sage, TeX

C_2\times C_{26}.C_6
% in TeX

G:=Group("C2xC26.C6");
// GroupNames label

G:=SmallGroup(312,11);
// by ID

G=gap.SmallGroup(312,11);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-13,60,7204,464]);
// Polycyclic

G:=Group<a,b,c|a^2=b^26=1,c^6=b^13,a*b=b*a,a*c=c*a,c*b*c^-1=b^23>;
// generators/relations

Export

Subgroup lattice of C2×C26.C6 in TeX

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