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## G = Q8×C16order 128 = 27

### Direct product of C16 and Q8

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — Q8×C16
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C2×C16 — C4×C16 — Q8×C16
 Lower central C1 — C2 — Q8×C16
 Upper central C1 — C2×C16 — Q8×C16
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×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, C4, C4, C4, C22, C8, C8, C2×C4, C2×C4, Q8, C16, C16, C42, C4⋊C4, C2×C8, C2×C8, C2×Q8, C4×C8, C4⋊C8, C2×C16, C2×C16, C4×Q8, C4×C16, C4⋊C16, C8×Q8, Q8×C16
Quotients: C1, C2, C4, C22, C8, C2×C4, Q8, C23, C16, C2×C8, C22×C4, C2×Q8, C4○D4, C2×C16, 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)
(1 120 90 37)(2 121 91 38)(3 122 92 39)(4 123 93 40)(5 124 94 41)(6 125 95 42)(7 126 96 43)(8 127 81 44)(9 128 82 45)(10 113 83 46)(11 114 84 47)(12 115 85 48)(13 116 86 33)(14 117 87 34)(15 118 88 35)(16 119 89 36)(17 61 74 99)(18 62 75 100)(19 63 76 101)(20 64 77 102)(21 49 78 103)(22 50 79 104)(23 51 80 105)(24 52 65 106)(25 53 66 107)(26 54 67 108)(27 55 68 109)(28 56 69 110)(29 57 70 111)(30 58 71 112)(31 59 72 97)(32 60 73 98)
(1 108 90 54)(2 109 91 55)(3 110 92 56)(4 111 93 57)(5 112 94 58)(6 97 95 59)(7 98 96 60)(8 99 81 61)(9 100 82 62)(10 101 83 63)(11 102 84 64)(12 103 85 49)(13 104 86 50)(14 105 87 51)(15 106 88 52)(16 107 89 53)(17 127 74 44)(18 128 75 45)(19 113 76 46)(20 114 77 47)(21 115 78 48)(22 116 79 33)(23 117 80 34)(24 118 65 35)(25 119 66 36)(26 120 67 37)(27 121 68 38)(28 122 69 39)(29 123 70 40)(30 124 71 41)(31 125 72 42)(32 126 73 43)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E ··· 4P 8A ··· 8H 8I ··· 8T 16A ··· 16P 16Q ··· 16AN order 1 2 2 2 4 4 4 4 4 ··· 4 8 ··· 8 8 ··· 8 16 ··· 16 16 ··· 16 size 1 1 1 1 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + - image C1 C2 C2 C2 C4 C4 C8 C8 C16 Q8 C4○D4 C8○D4 D4○C16 kernel Q8×C16 C4×C16 C4⋊C16 C8×Q8 C4⋊C8 C4×Q8 C4⋊C4 C2×Q8 Q8 C16 C8 C4 C2 # reps 1 3 3 1 6 2 12 4 32 2 2 4 8

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

 10 0 0 0 2 0 0 0 2
,
 1 0 0 0 0 1 0 16 0
,
 1 0 0 0 13 0 0 0 4
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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