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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

C1C2 — M4(2)×C17
C1C2C4C68C136 — M4(2)×C17
C1C2 — M4(2)×C17
C1C68 — M4(2)×C17

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

2C2
2C34

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 2A2B4A4B4C8A8B8C8D17A···17P34A···34P34Q···34AF68A···68AF68AG···68AV136A···136BL
order122444888817···1734···3434···3468···6868···68136···136
size11211222221···11···12···21···12···22···2

170 irreducible representations

dim111111111122
type+++
imageC1C2C2C4C4C17C34C34C68C68M4(2)M4(2)×C17
kernelM4(2)×C17C136C2×C68C68C2×C34M4(2)C8C2×C4C4C22C17C1
# reps121221632163232232

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

340
034
,
687
63131
,
152
0136
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

Export

Subgroup lattice of M4(2)×C17 in TeX

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