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G = D4.D17order 272 = 24·17

The non-split extension by D4 of D17 acting via D17/C17=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.D17, C34.8D4, C4.2D34, C172SD16, Dic342C2, C68.2C22, C173C82C2, (D4×C17).1C2, C2.5(C17⋊D4), SmallGroup(272,16)

Series: Derived Chief Lower central Upper central

C1C68 — D4.D17
C1C17C34C68Dic34 — D4.D17
C17C34C68 — D4.D17
C1C2C4D4

Generators and relations for D4.D17
 G = < a,b,c,d | a4=b2=c17=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >

4C2
2C22
34C4
4C34
17C8
17Q8
2Dic17
2C2×C34
17SD16

Smallest permutation representation of D4.D17
On 136 points
Generators in S136
(1 64 30 35)(2 65 31 36)(3 66 32 37)(4 67 33 38)(5 68 34 39)(6 52 18 40)(7 53 19 41)(8 54 20 42)(9 55 21 43)(10 56 22 44)(11 57 23 45)(12 58 24 46)(13 59 25 47)(14 60 26 48)(15 61 27 49)(16 62 28 50)(17 63 29 51)(69 103 86 120)(70 104 87 121)(71 105 88 122)(72 106 89 123)(73 107 90 124)(74 108 91 125)(75 109 92 126)(76 110 93 127)(77 111 94 128)(78 112 95 129)(79 113 96 130)(80 114 97 131)(81 115 98 132)(82 116 99 133)(83 117 100 134)(84 118 101 135)(85 119 102 136)
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 43)(10 44)(11 45)(12 46)(13 47)(14 48)(15 49)(16 50)(17 51)(18 52)(19 53)(20 54)(21 55)(22 56)(23 57)(24 58)(25 59)(26 60)(27 61)(28 62)(29 63)(30 64)(31 65)(32 66)(33 67)(34 68)(69 86)(70 87)(71 88)(72 89)(73 90)(74 91)(75 92)(76 93)(77 94)(78 95)(79 96)(80 97)(81 98)(82 99)(83 100)(84 101)(85 102)
(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)
(1 102 30 85)(2 101 31 84)(3 100 32 83)(4 99 33 82)(5 98 34 81)(6 97 18 80)(7 96 19 79)(8 95 20 78)(9 94 21 77)(10 93 22 76)(11 92 23 75)(12 91 24 74)(13 90 25 73)(14 89 26 72)(15 88 27 71)(16 87 28 70)(17 86 29 69)(35 136 64 119)(36 135 65 118)(37 134 66 117)(38 133 67 116)(39 132 68 115)(40 131 52 114)(41 130 53 113)(42 129 54 112)(43 128 55 111)(44 127 56 110)(45 126 57 109)(46 125 58 108)(47 124 59 107)(48 123 60 106)(49 122 61 105)(50 121 62 104)(51 120 63 103)

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

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

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

47 conjugacy classes

class 1 2A2B4A4B8A8B17A···17H34A···34H34I···34X68A···68H
order122448817···1734···3434···3468···68
size11426834342···22···24···44···4

47 irreducible representations

dim1111222224
type+++++++-
imageC1C2C2C2D4SD16D17D34C17⋊D4D4.D17
kernelD4.D17C173C8Dic34D4×C17C34C17D4C4C2C1
# reps11111288168

Matrix representation of D4.D17 in GL4(𝔽137) generated by

1000
0100
001119
0061136
,
1000
0100
001119
000136
,
103100
229200
0010
0001
,
717600
136600
00089
00200
G:=sub<GL(4,GF(137))| [1,0,0,0,0,1,0,0,0,0,1,61,0,0,119,136],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,119,136],[103,22,0,0,1,92,0,0,0,0,1,0,0,0,0,1],[71,13,0,0,76,66,0,0,0,0,0,20,0,0,89,0] >;

D4.D17 in GAP, Magma, Sage, TeX

D_4.D_{17}
% in TeX

G:=Group("D4.D17");
// GroupNames label

G:=SmallGroup(272,16);
// by ID

G=gap.SmallGroup(272,16);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-17,40,61,182,97,42,6404]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^17=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Subgroup lattice of D4.D17 in TeX

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