Copied to
clipboard

## G = C14×SL2(𝔽3)  order 336 = 24·3·7

### Direct product of C14 and SL2(𝔽3)

Aliases: C14×SL2(𝔽3), Q8⋊C42, (C2×Q8)⋊C21, (C7×Q8)⋊7C6, (Q8×C14)⋊1C3, C2.2(A4×C14), (C2×C14).4A4, C14.11(C2×A4), C22.2(C7×A4), SmallGroup(336,169)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C14×SL2(𝔽3)
 Chief series C1 — C2 — Q8 — C7×Q8 — C7×SL2(𝔽3) — C14×SL2(𝔽3)
 Lower central Q8 — C14×SL2(𝔽3)
 Upper central C1 — C2×C14

Generators and relations for C14×SL2(𝔽3)
G = < a,b,c,d | a14=b4=d3=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=c, dcd-1=bc >

Smallest permutation representation of C14×SL2(𝔽3)
On 112 points
Generators in S112
(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)(99 100 101 102 103 104 105 106 107 108 109 110 111 112)
(1 85 66 72)(2 86 67 73)(3 87 68 74)(4 88 69 75)(5 89 70 76)(6 90 57 77)(7 91 58 78)(8 92 59 79)(9 93 60 80)(10 94 61 81)(11 95 62 82)(12 96 63 83)(13 97 64 84)(14 98 65 71)(15 37 100 48)(16 38 101 49)(17 39 102 50)(18 40 103 51)(19 41 104 52)(20 42 105 53)(21 29 106 54)(22 30 107 55)(23 31 108 56)(24 32 109 43)(25 33 110 44)(26 34 111 45)(27 35 112 46)(28 36 99 47)
(1 18 66 103)(2 19 67 104)(3 20 68 105)(4 21 69 106)(5 22 70 107)(6 23 57 108)(7 24 58 109)(8 25 59 110)(9 26 60 111)(10 27 61 112)(11 28 62 99)(12 15 63 100)(13 16 64 101)(14 17 65 102)(29 88 54 75)(30 89 55 76)(31 90 56 77)(32 91 43 78)(33 92 44 79)(34 93 45 80)(35 94 46 81)(36 95 47 82)(37 96 48 83)(38 97 49 84)(39 98 50 71)(40 85 51 72)(41 86 52 73)(42 87 53 74)
(15 96 48)(16 97 49)(17 98 50)(18 85 51)(19 86 52)(20 87 53)(21 88 54)(22 89 55)(23 90 56)(24 91 43)(25 92 44)(26 93 45)(27 94 46)(28 95 47)(29 106 75)(30 107 76)(31 108 77)(32 109 78)(33 110 79)(34 111 80)(35 112 81)(36 99 82)(37 100 83)(38 101 84)(39 102 71)(40 103 72)(41 104 73)(42 105 74)

G:=sub<Sym(112)| (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)(99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,85,66,72)(2,86,67,73)(3,87,68,74)(4,88,69,75)(5,89,70,76)(6,90,57,77)(7,91,58,78)(8,92,59,79)(9,93,60,80)(10,94,61,81)(11,95,62,82)(12,96,63,83)(13,97,64,84)(14,98,65,71)(15,37,100,48)(16,38,101,49)(17,39,102,50)(18,40,103,51)(19,41,104,52)(20,42,105,53)(21,29,106,54)(22,30,107,55)(23,31,108,56)(24,32,109,43)(25,33,110,44)(26,34,111,45)(27,35,112,46)(28,36,99,47), (1,18,66,103)(2,19,67,104)(3,20,68,105)(4,21,69,106)(5,22,70,107)(6,23,57,108)(7,24,58,109)(8,25,59,110)(9,26,60,111)(10,27,61,112)(11,28,62,99)(12,15,63,100)(13,16,64,101)(14,17,65,102)(29,88,54,75)(30,89,55,76)(31,90,56,77)(32,91,43,78)(33,92,44,79)(34,93,45,80)(35,94,46,81)(36,95,47,82)(37,96,48,83)(38,97,49,84)(39,98,50,71)(40,85,51,72)(41,86,52,73)(42,87,53,74), (15,96,48)(16,97,49)(17,98,50)(18,85,51)(19,86,52)(20,87,53)(21,88,54)(22,89,55)(23,90,56)(24,91,43)(25,92,44)(26,93,45)(27,94,46)(28,95,47)(29,106,75)(30,107,76)(31,108,77)(32,109,78)(33,110,79)(34,111,80)(35,112,81)(36,99,82)(37,100,83)(38,101,84)(39,102,71)(40,103,72)(41,104,73)(42,105,74)>;

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)(99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,85,66,72)(2,86,67,73)(3,87,68,74)(4,88,69,75)(5,89,70,76)(6,90,57,77)(7,91,58,78)(8,92,59,79)(9,93,60,80)(10,94,61,81)(11,95,62,82)(12,96,63,83)(13,97,64,84)(14,98,65,71)(15,37,100,48)(16,38,101,49)(17,39,102,50)(18,40,103,51)(19,41,104,52)(20,42,105,53)(21,29,106,54)(22,30,107,55)(23,31,108,56)(24,32,109,43)(25,33,110,44)(26,34,111,45)(27,35,112,46)(28,36,99,47), (1,18,66,103)(2,19,67,104)(3,20,68,105)(4,21,69,106)(5,22,70,107)(6,23,57,108)(7,24,58,109)(8,25,59,110)(9,26,60,111)(10,27,61,112)(11,28,62,99)(12,15,63,100)(13,16,64,101)(14,17,65,102)(29,88,54,75)(30,89,55,76)(31,90,56,77)(32,91,43,78)(33,92,44,79)(34,93,45,80)(35,94,46,81)(36,95,47,82)(37,96,48,83)(38,97,49,84)(39,98,50,71)(40,85,51,72)(41,86,52,73)(42,87,53,74), (15,96,48)(16,97,49)(17,98,50)(18,85,51)(19,86,52)(20,87,53)(21,88,54)(22,89,55)(23,90,56)(24,91,43)(25,92,44)(26,93,45)(27,94,46)(28,95,47)(29,106,75)(30,107,76)(31,108,77)(32,109,78)(33,110,79)(34,111,80)(35,112,81)(36,99,82)(37,100,83)(38,101,84)(39,102,71)(40,103,72)(41,104,73)(42,105,74) );

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),(99,100,101,102,103,104,105,106,107,108,109,110,111,112)], [(1,85,66,72),(2,86,67,73),(3,87,68,74),(4,88,69,75),(5,89,70,76),(6,90,57,77),(7,91,58,78),(8,92,59,79),(9,93,60,80),(10,94,61,81),(11,95,62,82),(12,96,63,83),(13,97,64,84),(14,98,65,71),(15,37,100,48),(16,38,101,49),(17,39,102,50),(18,40,103,51),(19,41,104,52),(20,42,105,53),(21,29,106,54),(22,30,107,55),(23,31,108,56),(24,32,109,43),(25,33,110,44),(26,34,111,45),(27,35,112,46),(28,36,99,47)], [(1,18,66,103),(2,19,67,104),(3,20,68,105),(4,21,69,106),(5,22,70,107),(6,23,57,108),(7,24,58,109),(8,25,59,110),(9,26,60,111),(10,27,61,112),(11,28,62,99),(12,15,63,100),(13,16,64,101),(14,17,65,102),(29,88,54,75),(30,89,55,76),(31,90,56,77),(32,91,43,78),(33,92,44,79),(34,93,45,80),(35,94,46,81),(36,95,47,82),(37,96,48,83),(38,97,49,84),(39,98,50,71),(40,85,51,72),(41,86,52,73),(42,87,53,74)], [(15,96,48),(16,97,49),(17,98,50),(18,85,51),(19,86,52),(20,87,53),(21,88,54),(22,89,55),(23,90,56),(24,91,43),(25,92,44),(26,93,45),(27,94,46),(28,95,47),(29,106,75),(30,107,76),(31,108,77),(32,109,78),(33,110,79),(34,111,80),(35,112,81),(36,99,82),(37,100,83),(38,101,84),(39,102,71),(40,103,72),(41,104,73),(42,105,74)]])

98 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 6A ··· 6F 7A ··· 7F 14A ··· 14R 21A ··· 21L 28A ··· 28L 42A ··· 42AJ order 1 2 2 2 3 3 4 4 6 ··· 6 7 ··· 7 14 ··· 14 21 ··· 21 28 ··· 28 42 ··· 42 size 1 1 1 1 4 4 6 6 4 ··· 4 1 ··· 1 1 ··· 1 4 ··· 4 6 ··· 6 4 ··· 4

98 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 3 3 3 3 type + + - + + image C1 C2 C3 C6 C7 C14 C21 C42 SL2(𝔽3) SL2(𝔽3) C7×SL2(𝔽3) A4 C2×A4 C7×A4 A4×C14 kernel C14×SL2(𝔽3) C7×SL2(𝔽3) Q8×C14 C7×Q8 C2×SL2(𝔽3) SL2(𝔽3) C2×Q8 Q8 C14 C14 C2 C2×C14 C14 C22 C2 # reps 1 1 2 2 6 6 12 12 2 4 36 1 1 6 6

Matrix representation of C14×SL2(𝔽3) in GL3(𝔽337) generated by

 336 0 0 0 285 0 0 0 285
,
 1 0 0 0 0 336 0 1 0
,
 1 0 0 0 209 208 0 208 128
,
 128 0 0 0 1 208 0 0 128
G:=sub<GL(3,GF(337))| [336,0,0,0,285,0,0,0,285],[1,0,0,0,0,1,0,336,0],[1,0,0,0,209,208,0,208,128],[128,0,0,0,1,0,0,208,128] >;

C14×SL2(𝔽3) in GAP, Magma, Sage, TeX

C_{14}\times {\rm SL}_2({\mathbb F}_3)
% in TeX

G:=Group("C14xSL(2,3)");
// GroupNames label

G:=SmallGroup(336,169);
// by ID

G=gap.SmallGroup(336,169);
# by ID

G:=PCGroup([6,-2,-3,-7,-2,2,-2,1017,117,1900,202,88]);
// Polycyclic

G:=Group<a,b,c,d|a^14=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

׿
×
𝔽