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G = Q8×C17order 136 = 23·17

Direct product of C17 and Q8

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

Aliases: Q8×C17, C4.C34, C68.3C2, C34.7C22, C2.2(C2×C34), SmallGroup(136,11)

Series: Derived Chief Lower central Upper central

C1C2 — Q8×C17
C1C2C34C68 — Q8×C17
C1C2 — Q8×C17
C1C34 — Q8×C17

Generators and relations for Q8×C17
 G = < a,b,c | a17=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >


Smallest permutation representation of Q8×C17
Regular action 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 47 126 33)(2 48 127 34)(3 49 128 18)(4 50 129 19)(5 51 130 20)(6 35 131 21)(7 36 132 22)(8 37 133 23)(9 38 134 24)(10 39 135 25)(11 40 136 26)(12 41 120 27)(13 42 121 28)(14 43 122 29)(15 44 123 30)(16 45 124 31)(17 46 125 32)(52 101 73 106)(53 102 74 107)(54 86 75 108)(55 87 76 109)(56 88 77 110)(57 89 78 111)(58 90 79 112)(59 91 80 113)(60 92 81 114)(61 93 82 115)(62 94 83 116)(63 95 84 117)(64 96 85 118)(65 97 69 119)(66 98 70 103)(67 99 71 104)(68 100 72 105)
(1 79 126 58)(2 80 127 59)(3 81 128 60)(4 82 129 61)(5 83 130 62)(6 84 131 63)(7 85 132 64)(8 69 133 65)(9 70 134 66)(10 71 135 67)(11 72 136 68)(12 73 120 52)(13 74 121 53)(14 75 122 54)(15 76 123 55)(16 77 124 56)(17 78 125 57)(18 114 49 92)(19 115 50 93)(20 116 51 94)(21 117 35 95)(22 118 36 96)(23 119 37 97)(24 103 38 98)(25 104 39 99)(26 105 40 100)(27 106 41 101)(28 107 42 102)(29 108 43 86)(30 109 44 87)(31 110 45 88)(32 111 46 89)(33 112 47 90)(34 113 48 91)

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

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

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

Q8×C17 is a maximal subgroup of   Q8⋊D17  C17⋊Q16  D68⋊C2

85 conjugacy classes

class 1  2 4A4B4C17A···17P34A···34P68A···68AV
order1244417···1734···3468···68
size112221···11···12···2

85 irreducible representations

dim111122
type++-
imageC1C2C17C34Q8Q8×C17
kernelQ8×C17C68Q8C4C17C1
# reps131648116

Matrix representation of Q8×C17 in GL2(𝔽137) generated by

380
038
,
01
1360
,
3956
5698
G:=sub<GL(2,GF(137))| [38,0,0,38],[0,136,1,0],[39,56,56,98] >;

Q8×C17 in GAP, Magma, Sage, TeX

Q_8\times C_{17}
% in TeX

G:=Group("Q8xC17");
// GroupNames label

G:=SmallGroup(136,11);
// by ID

G=gap.SmallGroup(136,11);
# by ID

G:=PCGroup([4,-2,-2,-17,-2,272,561,277]);
// Polycyclic

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

Export

Subgroup lattice of Q8×C17 in TeX

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