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G = Q166Q8order 128 = 27

The semidirect product of Q16 and Q8 acting through Inn(Q16)

p-group, metabelian, nilpotent (class 3), monomial

Aliases: Q166Q8, C42.524C23, C4.982- 1+4, C4⋊C43Q16, C2.41(D4×Q8), C8.37(C2×Q8), C4⋊C4.417D4, (C8×Q8).11C2, Q8.10(C2×Q8), Q8.Q8.1C2, C4.93(C4○D8), Q83Q8.6C2, (C4×Q16).10C2, (C2×Q8).188D4, C2.67(Q8○D8), C82Q8.17C2, C8.5Q8.4C2, C4.41(C22×Q8), C4⋊C8.329C22, C4⋊C4.272C23, (C2×C8).213C23, (C2×C4).575C24, (C4×C8).125C22, Q8⋊Q8.12C2, C4⋊Q8.204C22, C2.D8.74C22, (C2×Q8).410C23, (C4×Q8).313C22, C4.Q8.117C22, (C2×Q16).173C22, C22.835(C22×D4), C42.C2.73C22, Q8⋊C4.187C22, C2.78(C2×C4○D8), (C2×C4).180(C2×D4), SmallGroup(128,2115)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — Q166Q8
C1C2C4C2×C4C42C4×Q8Q83Q8 — Q166Q8
C1C2C2×C4 — Q166Q8
C1C22C4×Q8 — Q166Q8
C1C2C2C2×C4 — Q166Q8

Generators and relations for Q166Q8
 G = < a,b,c,d | a8=c4=1, b2=a4, d2=c2, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=a4b, dcd-1=c-1 >

Subgroups: 248 in 162 conjugacy classes, 96 normal (24 characteristic)
C1, C2 [×3], C4 [×2], C4 [×2], C4 [×15], C22, C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×8], Q8 [×4], Q8 [×8], C42, C42 [×2], C42 [×6], C4⋊C4, C4⋊C4 [×8], C4⋊C4 [×16], C2×C8 [×2], C2×C8 [×2], Q16 [×4], C2×Q8, C2×Q8 [×2], C2×Q8 [×2], C4×C8, C4×C8 [×2], Q8⋊C4 [×6], C4⋊C8, C4⋊C8 [×2], C4.Q8 [×4], C2.D8, C2.D8 [×4], C4×Q8, C4×Q8 [×6], C4×Q8 [×2], C42.C2 [×4], C42.C2 [×4], C4⋊Q8 [×2], C4⋊Q8 [×2], C2×Q16, C4×Q16, C4×Q16 [×2], C8×Q8, Q8⋊Q8 [×2], Q8.Q8 [×4], C8.5Q8 [×2], C82Q8, Q83Q8 [×2], Q166Q8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C24, C4○D8 [×2], C22×D4, C22×Q8, 2- 1+4, D4×Q8, C2×C4○D8, Q8○D8, Q166Q8

Smallest permutation representation of Q166Q8
Regular action on 128 points
Generators in S128
(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)
(1 54 5 50)(2 53 6 49)(3 52 7 56)(4 51 8 55)(9 107 13 111)(10 106 14 110)(11 105 15 109)(12 112 16 108)(17 46 21 42)(18 45 22 41)(19 44 23 48)(20 43 24 47)(25 103 29 99)(26 102 30 98)(27 101 31 97)(28 100 32 104)(33 60 37 64)(34 59 38 63)(35 58 39 62)(36 57 40 61)(65 124 69 128)(66 123 70 127)(67 122 71 126)(68 121 72 125)(73 120 77 116)(74 119 78 115)(75 118 79 114)(76 117 80 113)(81 95 85 91)(82 94 86 90)(83 93 87 89)(84 92 88 96)
(1 29 43 107)(2 30 44 108)(3 31 45 109)(4 32 46 110)(5 25 47 111)(6 26 48 112)(7 27 41 105)(8 28 42 106)(9 50 103 20)(10 51 104 21)(11 52 97 22)(12 53 98 23)(13 54 99 24)(14 55 100 17)(15 56 101 18)(16 49 102 19)(33 72 88 117)(34 65 81 118)(35 66 82 119)(36 67 83 120)(37 68 84 113)(38 69 85 114)(39 70 86 115)(40 71 87 116)(57 122 93 77)(58 123 94 78)(59 124 95 79)(60 125 96 80)(61 126 89 73)(62 127 90 74)(63 128 91 75)(64 121 92 76)
(1 60 43 96)(2 61 44 89)(3 62 45 90)(4 63 46 91)(5 64 47 92)(6 57 48 93)(7 58 41 94)(8 59 42 95)(9 68 103 113)(10 69 104 114)(11 70 97 115)(12 71 98 116)(13 72 99 117)(14 65 100 118)(15 66 101 119)(16 67 102 120)(17 81 55 34)(18 82 56 35)(19 83 49 36)(20 84 50 37)(21 85 51 38)(22 86 52 39)(23 87 53 40)(24 88 54 33)(25 76 111 121)(26 77 112 122)(27 78 105 123)(28 79 106 124)(29 80 107 125)(30 73 108 126)(31 74 109 127)(32 75 110 128)

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

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

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

35 conjugacy classes

class 1 2A2B2C4A···4H4I···4O4P···4U8A8B8C8D8E···8J
order12224···44···44···488888···8
size11112···24···48···822224···4

35 irreducible representations

dim11111111222244
type+++++++++-+--
imageC1C2C2C2C2C2C2C2D4Q8D4C4○D82- 1+4Q8○D8
kernelQ166Q8C4×Q16C8×Q8Q8⋊Q8Q8.Q8C8.5Q8C82Q8Q83Q8C4⋊C4Q16C2×Q8C4C4C2
# reps13124212341812

Matrix representation of Q166Q8 in GL4(𝔽17) generated by

14300
141400
00160
00016
,
11000
101600
00160
00016
,
1000
0100
0012
001616
,
01300
4000
00112
0076
G:=sub<GL(4,GF(17))| [14,14,0,0,3,14,0,0,0,0,16,0,0,0,0,16],[1,10,0,0,10,16,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,1,16,0,0,2,16],[0,4,0,0,13,0,0,0,0,0,11,7,0,0,2,6] >;

Q166Q8 in GAP, Magma, Sage, TeX

Q_{16}\rtimes_6Q_8
% in TeX

G:=Group("Q16:6Q8");
// GroupNames label

G:=SmallGroup(128,2115);
// by ID

G=gap.SmallGroup(128,2115);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,568,758,352,346,80,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^8=c^4=1,b^2=a^4,d^2=c^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=c^-1>;
// generators/relations

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