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## G = M4(2)×C17order 272 = 24·17

### Direct product of C17 and M4(2)

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: M4(2)×C17, C4.C68, C83C34, C1367C2, C68.7C4, C22.C68, C68.22C22, (C2×C68).8C2, C2.3(C2×C68), (C2×C4).2C34, (C2×C34).3C4, C4.6(C2×C34), C34.19(C2×C4), SmallGroup(272,24)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — M4(2)×C17
 Chief series C1 — C2 — C4 — C68 — C136 — M4(2)×C17
 Lower central C1 — C2 — M4(2)×C17
 Upper central C1 — C68 — M4(2)×C17

Generators and relations for M4(2)×C17
G = < a,b,c | a17=b8=c2=1, ab=ba, ac=ca, cbc=b5 >

Smallest permutation representation of M4(2)×C17
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 22 101 35 55 75 109 122)(2 23 102 36 56 76 110 123)(3 24 86 37 57 77 111 124)(4 25 87 38 58 78 112 125)(5 26 88 39 59 79 113 126)(6 27 89 40 60 80 114 127)(7 28 90 41 61 81 115 128)(8 29 91 42 62 82 116 129)(9 30 92 43 63 83 117 130)(10 31 93 44 64 84 118 131)(11 32 94 45 65 85 119 132)(12 33 95 46 66 69 103 133)(13 34 96 47 67 70 104 134)(14 18 97 48 68 71 105 135)(15 19 98 49 52 72 106 136)(16 20 99 50 53 73 107 120)(17 21 100 51 54 74 108 121)
(18 71)(19 72)(20 73)(21 74)(22 75)(23 76)(24 77)(25 78)(26 79)(27 80)(28 81)(29 82)(30 83)(31 84)(32 85)(33 69)(34 70)(35 122)(36 123)(37 124)(38 125)(39 126)(40 127)(41 128)(42 129)(43 130)(44 131)(45 132)(46 133)(47 134)(48 135)(49 136)(50 120)(51 121)

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

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

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

170 conjugacy classes

 class 1 2A 2B 4A 4B 4C 8A 8B 8C 8D 17A ··· 17P 34A ··· 34P 34Q ··· 34AF 68A ··· 68AF 68AG ··· 68AV 136A ··· 136BL order 1 2 2 4 4 4 8 8 8 8 17 ··· 17 34 ··· 34 34 ··· 34 68 ··· 68 68 ··· 68 136 ··· 136 size 1 1 2 1 1 2 2 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

170 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 type + + + image C1 C2 C2 C4 C4 C17 C34 C34 C68 C68 M4(2) M4(2)×C17 kernel M4(2)×C17 C136 C2×C68 C68 C2×C34 M4(2) C8 C2×C4 C4 C22 C17 C1 # reps 1 2 1 2 2 16 32 16 32 32 2 32

Matrix representation of M4(2)×C17 in GL2(𝔽137) generated by

 34 0 0 34
,
 6 87 63 131
,
 1 52 0 136
G:=sub<GL(2,GF(137))| [34,0,0,34],[6,63,87,131],[1,0,52,136] >;

M4(2)×C17 in GAP, Magma, Sage, TeX

M_4(2)\times C_{17}
% in TeX

G:=Group("M4(2)xC17");
// GroupNames label

G:=SmallGroup(272,24);
// by ID

G=gap.SmallGroup(272,24);
# by ID

G:=PCGroup([5,-2,-2,-17,-2,-2,340,1381,58]);
// Polycyclic

G:=Group<a,b,c|a^17=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;
// generators/relations

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