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G = C13×SL2(𝔽3)  order 312 = 23·3·13

Direct product of C13 and SL2(𝔽3)

direct product, non-abelian, soluble

Aliases: C13×SL2(𝔽3), Q8⋊C39, C26.2A4, C2.(A4×C13), (Q8×C13)⋊1C3, SmallGroup(312,25)

Series: Derived Chief Lower central Upper central

C1C2Q8 — C13×SL2(𝔽3)
C1C2Q8Q8×C13 — C13×SL2(𝔽3)
Q8 — C13×SL2(𝔽3)
C1C26

Generators and relations for C13×SL2(𝔽3)
 G = < a,b,c,d | a13=b4=d3=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=c, dcd-1=bc >

4C3
3C4
4C6
4C39
3C52
4C78

Smallest permutation representation of C13×SL2(𝔽3)
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 40 70 14)(2 41 71 15)(3 42 72 16)(4 43 73 17)(5 44 74 18)(6 45 75 19)(7 46 76 20)(8 47 77 21)(9 48 78 22)(10 49 66 23)(11 50 67 24)(12 51 68 25)(13 52 69 26)(27 85 56 92)(28 86 57 93)(29 87 58 94)(30 88 59 95)(31 89 60 96)(32 90 61 97)(33 91 62 98)(34 79 63 99)(35 80 64 100)(36 81 65 101)(37 82 53 102)(38 83 54 103)(39 84 55 104)
(1 82 70 102)(2 83 71 103)(3 84 72 104)(4 85 73 92)(5 86 74 93)(6 87 75 94)(7 88 76 95)(8 89 77 96)(9 90 78 97)(10 91 66 98)(11 79 67 99)(12 80 68 100)(13 81 69 101)(14 53 40 37)(15 54 41 38)(16 55 42 39)(17 56 43 27)(18 57 44 28)(19 58 45 29)(20 59 46 30)(21 60 47 31)(22 61 48 32)(23 62 49 33)(24 63 50 34)(25 64 51 35)(26 65 52 36)
(14 53 102)(15 54 103)(16 55 104)(17 56 92)(18 57 93)(19 58 94)(20 59 95)(21 60 96)(22 61 97)(23 62 98)(24 63 99)(25 64 100)(26 65 101)(27 85 43)(28 86 44)(29 87 45)(30 88 46)(31 89 47)(32 90 48)(33 91 49)(34 79 50)(35 80 51)(36 81 52)(37 82 40)(38 83 41)(39 84 42)

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

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

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

91 conjugacy classes

class 1  2 3A3B 4 6A6B13A···13L26A···26L39A···39X52A···52L78A···78X
order123346613···1326···2639···3952···5278···78
size11446441···11···14···46···64···4

91 irreducible representations

dim111122233
type+-+
imageC1C3C13C39SL2(𝔽3)SL2(𝔽3)C13×SL2(𝔽3)A4A4×C13
kernelC13×SL2(𝔽3)Q8×C13SL2(𝔽3)Q8C13C13C1C26C2
# reps1212241236112

Matrix representation of C13×SL2(𝔽3) in GL2(𝔽157) generated by

160
016
,
13145
145144
,
0156
10
,
10
145144
G:=sub<GL(2,GF(157))| [16,0,0,16],[13,145,145,144],[0,1,156,0],[1,145,0,144] >;

C13×SL2(𝔽3) in GAP, Magma, Sage, TeX

C_{13}\times {\rm SL}_2({\mathbb F}_3)
% in TeX

G:=Group("C13xSL(2,3)");
// GroupNames label

G:=SmallGroup(312,25);
// by ID

G=gap.SmallGroup(312,25);
# by ID

G:=PCGroup([5,-3,-13,-2,2,-2,1172,72,2343,133,58]);
// Polycyclic

G:=Group<a,b,c,d|a^13=b^4=d^3=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=c,d*c*d^-1=b*c>;
// generators/relations

Export

Subgroup lattice of C13×SL2(𝔽3) in TeX

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