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G = C17×SL2(𝔽3)  order 408 = 23·3·17

Direct product of C17 and SL2(𝔽3)

direct product, non-abelian, soluble

Aliases: C17×SL2(𝔽3), Q8⋊C51, C34.A4, (Q8×C17)⋊C3, C2.(A4×C17), SmallGroup(408,14)

Series: Derived Chief Lower central Upper central

C1C2Q8 — C17×SL2(𝔽3)
C1C2Q8Q8×C17 — C17×SL2(𝔽3)
Q8 — C17×SL2(𝔽3)
C1C34

Generators and relations for C17×SL2(𝔽3)
 G = < a,b,c,d | a17=b4=d3=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=c, dcd-1=bc >

4C3
3C4
4C6
4C51
3C68
4C102

Smallest permutation representation of C17×SL2(𝔽3)
On 136 points
Generators in S136
(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)(103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119)(120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136)
(1 119 67 94)(2 103 68 95)(3 104 52 96)(4 105 53 97)(5 106 54 98)(6 107 55 99)(7 108 56 100)(8 109 57 101)(9 110 58 102)(10 111 59 86)(11 112 60 87)(12 113 61 88)(13 114 62 89)(14 115 63 90)(15 116 64 91)(16 117 65 92)(17 118 66 93)(18 73 133 49)(19 74 134 50)(20 75 135 51)(21 76 136 35)(22 77 120 36)(23 78 121 37)(24 79 122 38)(25 80 123 39)(26 81 124 40)(27 82 125 41)(28 83 126 42)(29 84 127 43)(30 85 128 44)(31 69 129 45)(32 70 130 46)(33 71 131 47)(34 72 132 48)
(1 85 67 44)(2 69 68 45)(3 70 52 46)(4 71 53 47)(5 72 54 48)(6 73 55 49)(7 74 56 50)(8 75 57 51)(9 76 58 35)(10 77 59 36)(11 78 60 37)(12 79 61 38)(13 80 62 39)(14 81 63 40)(15 82 64 41)(16 83 65 42)(17 84 66 43)(18 99 133 107)(19 100 134 108)(20 101 135 109)(21 102 136 110)(22 86 120 111)(23 87 121 112)(24 88 122 113)(25 89 123 114)(26 90 124 115)(27 91 125 116)(28 92 126 117)(29 93 127 118)(30 94 128 119)(31 95 129 103)(32 96 130 104)(33 97 131 105)(34 98 132 106)
(18 73 107)(19 74 108)(20 75 109)(21 76 110)(22 77 111)(23 78 112)(24 79 113)(25 80 114)(26 81 115)(27 82 116)(28 83 117)(29 84 118)(30 85 119)(31 69 103)(32 70 104)(33 71 105)(34 72 106)(35 102 136)(36 86 120)(37 87 121)(38 88 122)(39 89 123)(40 90 124)(41 91 125)(42 92 126)(43 93 127)(44 94 128)(45 95 129)(46 96 130)(47 97 131)(48 98 132)(49 99 133)(50 100 134)(51 101 135)

G:=sub<Sym(136)| (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)(103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119)(120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136), (1,119,67,94)(2,103,68,95)(3,104,52,96)(4,105,53,97)(5,106,54,98)(6,107,55,99)(7,108,56,100)(8,109,57,101)(9,110,58,102)(10,111,59,86)(11,112,60,87)(12,113,61,88)(13,114,62,89)(14,115,63,90)(15,116,64,91)(16,117,65,92)(17,118,66,93)(18,73,133,49)(19,74,134,50)(20,75,135,51)(21,76,136,35)(22,77,120,36)(23,78,121,37)(24,79,122,38)(25,80,123,39)(26,81,124,40)(27,82,125,41)(28,83,126,42)(29,84,127,43)(30,85,128,44)(31,69,129,45)(32,70,130,46)(33,71,131,47)(34,72,132,48), (1,85,67,44)(2,69,68,45)(3,70,52,46)(4,71,53,47)(5,72,54,48)(6,73,55,49)(7,74,56,50)(8,75,57,51)(9,76,58,35)(10,77,59,36)(11,78,60,37)(12,79,61,38)(13,80,62,39)(14,81,63,40)(15,82,64,41)(16,83,65,42)(17,84,66,43)(18,99,133,107)(19,100,134,108)(20,101,135,109)(21,102,136,110)(22,86,120,111)(23,87,121,112)(24,88,122,113)(25,89,123,114)(26,90,124,115)(27,91,125,116)(28,92,126,117)(29,93,127,118)(30,94,128,119)(31,95,129,103)(32,96,130,104)(33,97,131,105)(34,98,132,106), (18,73,107)(19,74,108)(20,75,109)(21,76,110)(22,77,111)(23,78,112)(24,79,113)(25,80,114)(26,81,115)(27,82,116)(28,83,117)(29,84,118)(30,85,119)(31,69,103)(32,70,104)(33,71,105)(34,72,106)(35,102,136)(36,86,120)(37,87,121)(38,88,122)(39,89,123)(40,90,124)(41,91,125)(42,92,126)(43,93,127)(44,94,128)(45,95,129)(46,96,130)(47,97,131)(48,98,132)(49,99,133)(50,100,134)(51,101,135)>;

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)(103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119)(120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136), (1,119,67,94)(2,103,68,95)(3,104,52,96)(4,105,53,97)(5,106,54,98)(6,107,55,99)(7,108,56,100)(8,109,57,101)(9,110,58,102)(10,111,59,86)(11,112,60,87)(12,113,61,88)(13,114,62,89)(14,115,63,90)(15,116,64,91)(16,117,65,92)(17,118,66,93)(18,73,133,49)(19,74,134,50)(20,75,135,51)(21,76,136,35)(22,77,120,36)(23,78,121,37)(24,79,122,38)(25,80,123,39)(26,81,124,40)(27,82,125,41)(28,83,126,42)(29,84,127,43)(30,85,128,44)(31,69,129,45)(32,70,130,46)(33,71,131,47)(34,72,132,48), (1,85,67,44)(2,69,68,45)(3,70,52,46)(4,71,53,47)(5,72,54,48)(6,73,55,49)(7,74,56,50)(8,75,57,51)(9,76,58,35)(10,77,59,36)(11,78,60,37)(12,79,61,38)(13,80,62,39)(14,81,63,40)(15,82,64,41)(16,83,65,42)(17,84,66,43)(18,99,133,107)(19,100,134,108)(20,101,135,109)(21,102,136,110)(22,86,120,111)(23,87,121,112)(24,88,122,113)(25,89,123,114)(26,90,124,115)(27,91,125,116)(28,92,126,117)(29,93,127,118)(30,94,128,119)(31,95,129,103)(32,96,130,104)(33,97,131,105)(34,98,132,106), (18,73,107)(19,74,108)(20,75,109)(21,76,110)(22,77,111)(23,78,112)(24,79,113)(25,80,114)(26,81,115)(27,82,116)(28,83,117)(29,84,118)(30,85,119)(31,69,103)(32,70,104)(33,71,105)(34,72,106)(35,102,136)(36,86,120)(37,87,121)(38,88,122)(39,89,123)(40,90,124)(41,91,125)(42,92,126)(43,93,127)(44,94,128)(45,95,129)(46,96,130)(47,97,131)(48,98,132)(49,99,133)(50,100,134)(51,101,135) );

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),(103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119),(120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136)], [(1,119,67,94),(2,103,68,95),(3,104,52,96),(4,105,53,97),(5,106,54,98),(6,107,55,99),(7,108,56,100),(8,109,57,101),(9,110,58,102),(10,111,59,86),(11,112,60,87),(12,113,61,88),(13,114,62,89),(14,115,63,90),(15,116,64,91),(16,117,65,92),(17,118,66,93),(18,73,133,49),(19,74,134,50),(20,75,135,51),(21,76,136,35),(22,77,120,36),(23,78,121,37),(24,79,122,38),(25,80,123,39),(26,81,124,40),(27,82,125,41),(28,83,126,42),(29,84,127,43),(30,85,128,44),(31,69,129,45),(32,70,130,46),(33,71,131,47),(34,72,132,48)], [(1,85,67,44),(2,69,68,45),(3,70,52,46),(4,71,53,47),(5,72,54,48),(6,73,55,49),(7,74,56,50),(8,75,57,51),(9,76,58,35),(10,77,59,36),(11,78,60,37),(12,79,61,38),(13,80,62,39),(14,81,63,40),(15,82,64,41),(16,83,65,42),(17,84,66,43),(18,99,133,107),(19,100,134,108),(20,101,135,109),(21,102,136,110),(22,86,120,111),(23,87,121,112),(24,88,122,113),(25,89,123,114),(26,90,124,115),(27,91,125,116),(28,92,126,117),(29,93,127,118),(30,94,128,119),(31,95,129,103),(32,96,130,104),(33,97,131,105),(34,98,132,106)], [(18,73,107),(19,74,108),(20,75,109),(21,76,110),(22,77,111),(23,78,112),(24,79,113),(25,80,114),(26,81,115),(27,82,116),(28,83,117),(29,84,118),(30,85,119),(31,69,103),(32,70,104),(33,71,105),(34,72,106),(35,102,136),(36,86,120),(37,87,121),(38,88,122),(39,89,123),(40,90,124),(41,91,125),(42,92,126),(43,93,127),(44,94,128),(45,95,129),(46,96,130),(47,97,131),(48,98,132),(49,99,133),(50,100,134),(51,101,135)])

119 conjugacy classes

class 1  2 3A3B 4 6A6B17A···17P34A···34P51A···51AF68A···68P102A···102AF
order123346617···1734···3451···5168···68102···102
size11446441···11···14···46···64···4

119 irreducible representations

dim111122233
type+-+
imageC1C3C17C51SL2(𝔽3)SL2(𝔽3)C17×SL2(𝔽3)A4A4×C17
kernelC17×SL2(𝔽3)Q8×C17SL2(𝔽3)Q8C17C17C1C34C2
# reps1216321248116

Matrix representation of C17×SL2(𝔽3) in GL2(𝔽409) generated by

360
036
,
53355
355356
,
0408
10
,
153
0355
G:=sub<GL(2,GF(409))| [36,0,0,36],[53,355,355,356],[0,1,408,0],[1,0,53,355] >;

C17×SL2(𝔽3) in GAP, Magma, Sage, TeX

C_{17}\times {\rm SL}_2({\mathbb F}_3)
% in TeX

G:=Group("C17xSL(2,3)");
// GroupNames label

G:=SmallGroup(408,14);
// by ID

G=gap.SmallGroup(408,14);
# by ID

G:=PCGroup([5,-3,-17,-2,2,-2,1532,72,3063,133,58]);
// Polycyclic

G:=Group<a,b,c,d|a^17=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 C17×SL2(𝔽3) in TeX

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