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G = D34⋊C4order 272 = 24·17

1st semidirect product of D34 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D341C4, C2.2D68, C34.6D4, C22.6D34, (C2×C68)⋊1C2, (C2×C4)⋊1D17, C2.5(C4×D17), C172(C22⋊C4), C34.12(C2×C4), (C2×Dic17)⋊1C2, C2.2(C17⋊D4), (C2×C34).6C22, (C22×D17).1C2, SmallGroup(272,14)

Series: Derived Chief Lower central Upper central

C1C34 — D34⋊C4
C1C17C34C2×C34C22×D17 — D34⋊C4
C17C34 — D34⋊C4
C1C22C2×C4

Generators and relations for D34⋊C4
 G = < a,b,c | a34=b2=c4=1, bab=a-1, ac=ca, cbc-1=a17b >

34C2
34C2
2C4
17C22
17C22
34C4
34C22
34C22
2D17
2D17
17C2×C4
17C23
2D34
2D34
2Dic17
2C68
17C22⋊C4

Smallest permutation representation of D34⋊C4
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 52)(2 51)(3 50)(4 49)(5 48)(6 47)(7 46)(8 45)(9 44)(10 43)(11 42)(12 41)(13 40)(14 39)(15 38)(16 37)(17 36)(18 35)(19 68)(20 67)(21 66)(22 65)(23 64)(24 63)(25 62)(26 61)(27 60)(28 59)(29 58)(30 57)(31 56)(32 55)(33 54)(34 53)(69 124)(70 123)(71 122)(72 121)(73 120)(74 119)(75 118)(76 117)(77 116)(78 115)(79 114)(80 113)(81 112)(82 111)(83 110)(84 109)(85 108)(86 107)(87 106)(88 105)(89 104)(90 103)(91 136)(92 135)(93 134)(94 133)(95 132)(96 131)(97 130)(98 129)(99 128)(100 127)(101 126)(102 125)
(1 136 53 75)(2 103 54 76)(3 104 55 77)(4 105 56 78)(5 106 57 79)(6 107 58 80)(7 108 59 81)(8 109 60 82)(9 110 61 83)(10 111 62 84)(11 112 63 85)(12 113 64 86)(13 114 65 87)(14 115 66 88)(15 116 67 89)(16 117 68 90)(17 118 35 91)(18 119 36 92)(19 120 37 93)(20 121 38 94)(21 122 39 95)(22 123 40 96)(23 124 41 97)(24 125 42 98)(25 126 43 99)(26 127 44 100)(27 128 45 101)(28 129 46 102)(29 130 47 69)(30 131 48 70)(31 132 49 71)(32 133 50 72)(33 134 51 73)(34 135 52 74)

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

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

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

74 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D17A···17H34A···34X68A···68AF
order122222444417···1734···3468···68
size111134342234342···22···22···2

74 irreducible representations

dim11111222222
type++++++++
imageC1C2C2C2C4D4D17D34C4×D17D68C17⋊D4
kernelD34⋊C4C2×Dic17C2×C68C22×D17D34C34C2×C4C22C2C2C2
# reps11114288161616

Matrix representation of D34⋊C4 in GL3(𝔽137) generated by

100
0116116
02134
,
100
0116116
03421
,
10000
0129101
0368
G:=sub<GL(3,GF(137))| [1,0,0,0,116,21,0,116,34],[1,0,0,0,116,34,0,116,21],[100,0,0,0,129,36,0,101,8] >;

D34⋊C4 in GAP, Magma, Sage, TeX

D_{34}\rtimes C_4
% in TeX

G:=Group("D34:C4");
// GroupNames label

G:=SmallGroup(272,14);
// by ID

G=gap.SmallGroup(272,14);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-17,101,26,6404]);
// Polycyclic

G:=Group<a,b,c|a^34=b^2=c^4=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^17*b>;
// generators/relations

Export

Subgroup lattice of D34⋊C4 in TeX

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