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## G = Q16.Q8order 128 = 27

### 1st non-split extension by Q16 of Q8 acting via Q8/C4=C2

p-group, metabelian, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C8 — Q16.Q8
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C2×Q16 — C4×Q16 — Q16.Q8
 Lower central C1 — C2 — C4 — C2×C8 — Q16.Q8
 Upper central C1 — C22 — C42 — C4×C8 — Q16.Q8
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — Q16.Q8

Generators and relations for Q16.Q8
G = < a,b,c,d | a8=c4=1, b2=a4, d2=a4c2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=a4c-1 >

Subgroups: 132 in 62 conjugacy classes, 32 normal (30 characteristic)
C1, C2 [×3], C4 [×2], C4 [×7], C22, C8 [×2], C8, C2×C4 [×3], C2×C4 [×4], Q8 [×3], C16 [×2], C42, C42, C4⋊C4 [×6], C2×C8 [×2], Q16 [×2], Q16, C2×Q8, C4×C8, Q8⋊C4, C4.Q8, C2.D8 [×3], C2×C16 [×2], C4×Q8, C42.C2, C2×Q16, C2.Q32 [×2], C4⋊C16, C163C4, C164C4, C4×Q16, C8.5Q8, Q16.Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, D8 [×2], C2×D4, C2×Q8, C4○D4, C22⋊Q8, C2×D8, C8.C22, D4⋊Q8, C4○D16, Q32⋊C2, Q16.Q8

Character table of Q16.Q8

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 8A 8B 8C 8D 8E 8F 16A 16B 16C 16D 16E 16F 16G 16H size 1 1 1 1 2 2 2 2 4 8 8 8 8 16 16 2 2 2 2 4 4 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 -1 -1 1 -1 -1 -1 1 1 -1 1 1 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 linear of order 2 ρ4 1 1 1 1 1 -1 -1 1 -1 -1 -1 1 1 1 -1 1 1 1 1 -1 -1 1 -1 -1 -1 1 1 1 -1 linear of order 2 ρ5 1 1 1 1 1 -1 -1 1 -1 1 1 -1 -1 -1 1 1 1 1 1 -1 -1 1 -1 -1 -1 1 1 1 -1 linear of order 2 ρ6 1 1 1 1 1 -1 -1 1 -1 1 1 -1 -1 1 -1 1 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 linear of order 2 ρ7 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ8 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ9 2 2 2 2 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 √2 -√2 √2 -√2 -√2 √2 -√2 √2 orthogonal lifted from D8 ρ10 2 2 2 2 -2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 √2 √2 -√2 √2 -√2 √2 -√2 -√2 orthogonal lifted from D8 ρ11 2 2 2 2 2 2 2 2 2 0 0 0 0 0 0 -2 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 2 -2 -2 2 -2 0 0 0 0 0 0 -2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 -2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 -√2 -√2 √2 -√2 √2 -√2 √2 √2 orthogonal lifted from D8 ρ14 2 2 2 2 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 -√2 √2 -√2 √2 √2 -√2 √2 -√2 orthogonal lifted from D8 ρ15 2 -2 -2 2 2 0 0 -2 0 2 -2 0 0 0 0 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ16 2 -2 -2 2 2 0 0 -2 0 -2 2 0 0 0 0 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ17 2 -2 -2 2 2 0 0 -2 0 0 0 -2i 2i 0 0 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ18 2 -2 -2 2 2 0 0 -2 0 0 0 2i -2i 0 0 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ19 2 2 -2 -2 0 -2i 2i 0 0 0 0 0 0 0 0 √2 √2 -√2 -√2 √-2 -√-2 ζ165+ζ163 ζ167-ζ16 ζ165-ζ163 -ζ167+ζ16 ζ1615+ζ169 ζ1613+ζ1611 ζ167+ζ16 -ζ165+ζ163 complex lifted from C4○D16 ρ20 2 2 -2 -2 0 2i -2i 0 0 0 0 0 0 0 0 √2 √2 -√2 -√2 -√-2 √-2 ζ1613+ζ1611 ζ167-ζ16 ζ165-ζ163 -ζ167+ζ16 ζ167+ζ16 ζ165+ζ163 ζ1615+ζ169 -ζ165+ζ163 complex lifted from C4○D16 ρ21 2 2 -2 -2 0 -2i 2i 0 0 0 0 0 0 0 0 -√2 -√2 √2 √2 -√-2 √-2 ζ167+ζ16 ζ165-ζ163 -ζ167+ζ16 -ζ165+ζ163 ζ165+ζ163 ζ1615+ζ169 ζ1613+ζ1611 ζ167-ζ16 complex lifted from C4○D16 ρ22 2 2 -2 -2 0 2i -2i 0 0 0 0 0 0 0 0 -√2 -√2 √2 √2 √-2 -√-2 ζ1615+ζ169 ζ165-ζ163 -ζ167+ζ16 -ζ165+ζ163 ζ1613+ζ1611 ζ167+ζ16 ζ165+ζ163 ζ167-ζ16 complex lifted from C4○D16 ρ23 2 2 -2 -2 0 -2i 2i 0 0 0 0 0 0 0 0 √2 √2 -√2 -√2 √-2 -√-2 ζ1613+ζ1611 -ζ167+ζ16 -ζ165+ζ163 ζ167-ζ16 ζ167+ζ16 ζ165+ζ163 ζ1615+ζ169 ζ165-ζ163 complex lifted from C4○D16 ρ24 2 2 -2 -2 0 -2i 2i 0 0 0 0 0 0 0 0 -√2 -√2 √2 √2 -√-2 √-2 ζ1615+ζ169 -ζ165+ζ163 ζ167-ζ16 ζ165-ζ163 ζ1613+ζ1611 ζ167+ζ16 ζ165+ζ163 -ζ167+ζ16 complex lifted from C4○D16 ρ25 2 2 -2 -2 0 2i -2i 0 0 0 0 0 0 0 0 √2 √2 -√2 -√2 -√-2 √-2 ζ165+ζ163 -ζ167+ζ16 -ζ165+ζ163 ζ167-ζ16 ζ1615+ζ169 ζ1613+ζ1611 ζ167+ζ16 ζ165-ζ163 complex lifted from C4○D16 ρ26 2 2 -2 -2 0 2i -2i 0 0 0 0 0 0 0 0 -√2 -√2 √2 √2 √-2 -√-2 ζ167+ζ16 -ζ165+ζ163 ζ167-ζ16 ζ165-ζ163 ζ165+ζ163 ζ1615+ζ169 ζ1613+ζ1611 -ζ167+ζ16 complex lifted from C4○D16 ρ27 4 -4 -4 4 -4 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2 ρ28 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 2√2 -2√2 -2√2 2√2 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q32⋊C2, Schur index 2 ρ29 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 -2√2 2√2 2√2 -2√2 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q32⋊C2, Schur index 2

Smallest permutation representation of Q16.Q8
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 25 5 29)(2 32 6 28)(3 31 7 27)(4 30 8 26)(9 19 13 23)(10 18 14 22)(11 17 15 21)(12 24 16 20)(33 59 37 63)(34 58 38 62)(35 57 39 61)(36 64 40 60)(41 51 45 55)(42 50 46 54)(43 49 47 53)(44 56 48 52)(65 90 69 94)(66 89 70 93)(67 96 71 92)(68 95 72 91)(73 82 77 86)(74 81 78 85)(75 88 79 84)(76 87 80 83)(97 122 101 126)(98 121 102 125)(99 128 103 124)(100 127 104 123)(105 114 109 118)(106 113 110 117)(107 120 111 116)(108 119 112 115)
(1 47 15 35)(2 48 16 36)(3 41 9 37)(4 42 10 38)(5 43 11 39)(6 44 12 40)(7 45 13 33)(8 46 14 34)(17 61 29 49)(18 62 30 50)(19 63 31 51)(20 64 32 52)(21 57 25 53)(22 58 26 54)(23 59 27 55)(24 60 28 56)(65 101 77 105)(66 102 78 106)(67 103 79 107)(68 104 80 108)(69 97 73 109)(70 98 74 110)(71 99 75 111)(72 100 76 112)(81 117 93 121)(82 118 94 122)(83 119 95 123)(84 120 96 124)(85 113 89 125)(86 114 90 126)(87 115 91 127)(88 116 92 128)
(1 77 11 69)(2 76 12 68)(3 75 13 67)(4 74 14 66)(5 73 15 65)(6 80 16 72)(7 79 9 71)(8 78 10 70)(17 95 25 87)(18 94 26 86)(19 93 27 85)(20 92 28 84)(21 91 29 83)(22 90 30 82)(23 89 31 81)(24 96 32 88)(33 111 41 103)(34 110 42 102)(35 109 43 101)(36 108 44 100)(37 107 45 99)(38 106 46 98)(39 105 47 97)(40 112 48 104)(49 127 57 119)(50 126 58 118)(51 125 59 117)(52 124 60 116)(53 123 61 115)(54 122 62 114)(55 121 63 113)(56 128 64 120)```

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

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

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

Matrix representation of Q16.Q8 in GL4(𝔽17) generated by

 8 0 0 0 2 15 0 0 0 0 16 0 0 0 0 16
,
 13 3 0 0 0 4 0 0 0 0 15 15 0 0 10 2
,
 4 0 0 0 0 4 0 0 0 0 8 8 0 0 11 9
,
 12 8 0 0 14 5 0 0 0 0 13 0 0 0 8 4
`G:=sub<GL(4,GF(17))| [8,2,0,0,0,15,0,0,0,0,16,0,0,0,0,16],[13,0,0,0,3,4,0,0,0,0,15,10,0,0,15,2],[4,0,0,0,0,4,0,0,0,0,8,11,0,0,8,9],[12,14,0,0,8,5,0,0,0,0,13,8,0,0,0,4] >;`

Q16.Q8 in GAP, Magma, Sage, TeX

`Q_{16}.Q_8`
`% in TeX`

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

`G:=SmallGroup(128,961);`
`// by ID`

`G=gap.SmallGroup(128,961);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,-2,56,141,512,422,352,1684,438,242,4037,1027,124]);`
`// Polycyclic`

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

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