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G = Q8×C16order 128 = 27

Direct product of C16 and Q8

direct product, p-group, metabelian, nilpotent (class 2), monomial

Aliases: Q8×C16, C163(C4⋊C4), C162(C4⋊C8), C4⋊C8.28C4, C4⋊C4.14C8, C2.2(C8×Q8), (C4×C16).5C2, C4.4(C2×C16), C162(C4⋊C16), C4⋊C16.14C2, C8.43(C2×Q8), C4.47(C4×Q8), (C4×Q8).37C4, (C2×Q8).11C8, (C8×Q8).17C2, C4.60(C8○D4), C2.3(D4○C16), C2.5(C22×C16), C8.104(C4○D4), (C4×C8).377C22, C42.275(C2×C4), (C2×C8).635C23, (C2×C16).112C22, C22.30(C22×C8), (C2×C16)(C4×Q8), (C2×C16)(C8×Q8), (C2×C4).42(C2×C8), (C2×C8).164(C2×C4), (C2×C4).620(C22×C4), SmallGroup(128,914)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — Q8×C16
C1C2C4C8C2×C8C2×C16C4×C16 — Q8×C16
C1C2 — Q8×C16
C1C2×C16 — Q8×C16
C1C2C2C2C2C4C4C2×C8 — Q8×C16

Generators and relations for Q8×C16
 G = < a,b,c | a16=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 76 in 67 conjugacy classes, 58 normal (16 characteristic)
C1, C2 [×3], C4 [×2], C4 [×6], C4 [×3], C22, C8 [×2], C8 [×3], C2×C4, C2×C4 [×6], Q8 [×4], C16 [×2], C16 [×3], C42 [×3], C4⋊C4 [×3], C2×C8, C2×C8 [×3], C2×Q8, C4×C8 [×3], C4⋊C8 [×3], C2×C16, C2×C16 [×3], C4×Q8, C4×C16 [×3], C4⋊C16 [×3], C8×Q8, Q8×C16
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], Q8 [×2], C23, C16 [×4], C2×C8 [×6], C22×C4, C2×Q8, C4○D4, C2×C16 [×6], C4×Q8, C22×C8, C8○D4, C8×Q8, C22×C16, D4○C16, Q8×C16

Smallest permutation representation of Q8×C16
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 64 97 119)(2 49 98 120)(3 50 99 121)(4 51 100 122)(5 52 101 123)(6 53 102 124)(7 54 103 125)(8 55 104 126)(9 56 105 127)(10 57 106 128)(11 58 107 113)(12 59 108 114)(13 60 109 115)(14 61 110 116)(15 62 111 117)(16 63 112 118)(17 69 92 38)(18 70 93 39)(19 71 94 40)(20 72 95 41)(21 73 96 42)(22 74 81 43)(23 75 82 44)(24 76 83 45)(25 77 84 46)(26 78 85 47)(27 79 86 48)(28 80 87 33)(29 65 88 34)(30 66 89 35)(31 67 90 36)(32 68 91 37)
(1 76 97 45)(2 77 98 46)(3 78 99 47)(4 79 100 48)(5 80 101 33)(6 65 102 34)(7 66 103 35)(8 67 104 36)(9 68 105 37)(10 69 106 38)(11 70 107 39)(12 71 108 40)(13 72 109 41)(14 73 110 42)(15 74 111 43)(16 75 112 44)(17 128 92 57)(18 113 93 58)(19 114 94 59)(20 115 95 60)(21 116 96 61)(22 117 81 62)(23 118 82 63)(24 119 83 64)(25 120 84 49)(26 121 85 50)(27 122 86 51)(28 123 87 52)(29 124 88 53)(30 125 89 54)(31 126 90 55)(32 127 91 56)

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

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

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

80 conjugacy classes

class 1 2A2B2C4A4B4C4D4E···4P8A···8H8I···8T16A···16P16Q···16AN
order122244444···48···88···816···1616···16
size111111112···21···12···21···12···2

80 irreducible representations

dim1111111112222
type++++-
imageC1C2C2C2C4C4C8C8C16Q8C4○D4C8○D4D4○C16
kernelQ8×C16C4×C16C4⋊C16C8×Q8C4⋊C8C4×Q8C4⋊C4C2×Q8Q8C16C8C4C2
# reps133162124322248

Matrix representation of Q8×C16 in GL3(𝔽17) generated by

1000
020
002
,
100
001
0160
,
100
0130
004
G:=sub<GL(3,GF(17))| [10,0,0,0,2,0,0,0,2],[1,0,0,0,0,16,0,1,0],[1,0,0,0,13,0,0,0,4] >;

Q8×C16 in GAP, Magma, Sage, TeX

Q_8\times C_{16}
% in TeX

G:=Group("Q8xC16");
// GroupNames label

G:=SmallGroup(128,914);
// by ID

G=gap.SmallGroup(128,914);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,64,142,102,124]);
// Polycyclic

G:=Group<a,b,c|a^16=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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