Q8:1Q16 - GroupNames

G = Q₈⋊₁Q₁₆ order 128 = 2⁷

The semidirect product of Q₈ and Q₁₆ acting via Q₁₆/C₈=C₂

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

Aliases: Q₈⋊₁Q₁₆, C₈.₃₄SD₁₆, C₄².₂₂₁C₂³, (C₈×Q₈).₄C₂, C₈⋊₁C₈.₉C₂, C₄⋊C₄.₁₉₆D₄, (C₂×C₈).₃₄₂D₄, C₄.₂₄(C₂×Q₁₆), C₄.₄₁(C₄○D₈), (C₄×C₈).₅₂C₂², (C₂×Q₈).₁₄₈D₄, Q₈⋊Q₈.₆C₂, C₄⋊Q₁₆.₄C₂, C₄.₈₀(C₂×SD₁₆), C₄⋊Q₈.₄₄C₂², C₄⋊C₈.₁₇₅C₂², C₂.₈(C₄⋊SD₁₆), C₄.₁₁₉(C₈⋊C₂²), C₂.₇(D₄.₅D₄), C₄.₁₀D₈.₁₀C₂, C₂.₇(C₈.₁₈D₄), (C₄×Q₈).₂₆₃C₂², C₂².₁₈₂(C₄⋊D₄), (C₂×C₄).₆(C₄○D₄), (C₂×C₄).₁₂₅₆(C₂×D₄), SmallGroup(128,402)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series

C₁ — C₄² — Q₈⋊₁Q₁₆

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄×Q₈ — C₈×Q₈ — Q₈⋊₁Q₁₆

Lower central

C₁ — C₂² — C₄² — Q₈⋊₁Q₁₆

Upper central

C₁ — C₂² — C₄² — Q₈⋊₁Q₁₆

Jennings

C₁ — C₂² — C₂² — C₄² — Q₈⋊₁Q₁₆

Generators and relations for Q₈⋊₁Q₁₆
G = < a,b,c,d | a⁴=c⁸=1, b²=a², d²=c⁴, bab^-1=dad^-1=a^-1, ac=ca, bc=cb, dbd^-1=a^-1b, dcd^-1=c^-1 >

Subgroups: 160 in 78 conjugacy classes, 36 normal (24 characteristic)
C₁, C₂, C₄, C₄, C₂², C₈, C₈, C₂×C₄, C₂×C₄, Q₈, Q₈, C₄², C₄², C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, Q₁₆, C₂×Q₈, C₂×Q₈, C₄×C₈, C₄×C₈, Q₈⋊C₄, C₄⋊C₈, C₄⋊C₈, C₄⋊C₈, C₄.Q₈, C₄×Q₈, C₄⋊Q₈, C₂×Q₁₆, C₄.₁₀D₈, C₈⋊₁C₈, C₈×Q₈, Q₈⋊Q₈, C₄⋊Q₁₆, Q₈⋊₁Q₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, SD₁₆, Q₁₆, C₂×D₄, C₄○D₄, C₄⋊D₄, C₂×SD₁₆, C₂×Q₁₆, C₄○D₈, C₈⋊C₂², C₄⋊SD₁₆, C₈.₁₈D₄, D₄.₅D₄, Q₈⋊₁Q₁₆

Character table of Q₈⋊₁Q₁₆

class	1	2A	2B	2C	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	8A	8B	8C	8D	8E	8F	8G	8H	8I	8J	8K	8L	8M	8N
size	1	1	1	1	2	2	2	2	4	4	4	4	4	16	16	2	2	2	2	4	4	4	4	4	4	8	8	8	8
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	0	0	0	-2	0	0	0	-2	-2	-2	-2	0	0	0	2	2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	-2	2	-2	2	2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	-2	2	-2	2	-2	2	2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	2	-2	2	-2	0	0	0	-2	0	0	0	2	2	2	2	0	0	0	-2	-2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	-2	-2	0	2	0	-2	0	-2	2	0	0	0	0	-√2	√2	√2	-√2	√2	-√2	-√2	√2	-√2	√2	0	0	0	0	symplectic lifted from Q₁₆, Schur index 2
ρ₁₄	2	2	-2	-2	0	2	0	-2	0	2	-2	0	0	0	0	√2	-√2	-√2	√2	√2	-√2	-√2	-√2	√2	√2	0	0	0	0	symplectic lifted from Q₁₆, Schur index 2
ρ₁₅	2	2	-2	-2	0	2	0	-2	0	2	-2	0	0	0	0	-√2	√2	√2	-√2	-√2	√2	√2	√2	-√2	-√2	0	0	0	0	symplectic lifted from Q₁₆, Schur index 2
ρ₁₆	2	2	-2	-2	0	2	0	-2	0	-2	2	0	0	0	0	√2	-√2	-√2	√2	-√2	√2	√2	-√2	√2	-√2	0	0	0	0	symplectic lifted from Q₁₆, Schur index 2
ρ₁₇	2	2	-2	-2	0	-2	0	2	2i	0	0	0	-2i	0	0	√2	-√2	-√2	√2	√-2	-√-2	√-2	√2	-√2	-√-2	0	0	0	0	complex lifted from C₄○D₈
ρ₁₈	2	2	-2	-2	0	-2	0	2	-2i	0	0	0	2i	0	0	-√2	√2	√2	-√2	√-2	-√-2	√-2	-√2	√2	-√-2	0	0	0	0	complex lifted from C₄○D₈
ρ₁₉	2	-2	2	-2	-2	0	2	0	0	0	0	0	0	0	0	-2	2	-2	2	0	0	0	0	0	0	√-2	-√-2	-√-2	√-2	complex lifted from SD₁₆
ρ₂₀	2	-2	2	-2	-2	0	2	0	0	0	0	0	0	0	0	2	-2	2	-2	0	0	0	0	0	0	-√-2	-√-2	√-2	√-2	complex lifted from SD₁₆
ρ₂₁	2	2	2	2	-2	-2	-2	-2	0	0	0	2	0	0	0	0	0	0	0	2i	2i	-2i	0	0	-2i	0	0	0	0	complex lifted from C₄○D₄
ρ₂₂	2	2	2	2	-2	-2	-2	-2	0	0	0	2	0	0	0	0	0	0	0	-2i	-2i	2i	0	0	2i	0	0	0	0	complex lifted from C₄○D₄
ρ₂₃	2	-2	2	-2	-2	0	2	0	0	0	0	0	0	0	0	-2	2	-2	2	0	0	0	0	0	0	-√-2	√-2	√-2	-√-2	complex lifted from SD₁₆
ρ₂₄	2	-2	2	-2	-2	0	2	0	0	0	0	0	0	0	0	2	-2	2	-2	0	0	0	0	0	0	√-2	√-2	-√-2	-√-2	complex lifted from SD₁₆
ρ₂₅	2	2	-2	-2	0	-2	0	2	2i	0	0	0	-2i	0	0	-√2	√2	√2	-√2	-√-2	√-2	-√-2	-√2	√2	√-2	0	0	0	0	complex lifted from C₄○D₈
ρ₂₆	2	2	-2	-2	0	-2	0	2	-2i	0	0	0	2i	0	0	√2	-√2	-√2	√2	-√-2	√-2	-√-2	√2	-√2	√-2	0	0	0	0	complex lifted from C₄○D₈
ρ₂₇	4	-4	4	-4	4	0	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₈⋊C₂²
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 D₄.₅D₄, 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 D₄.₅D₄, Schur index 2

Smallest permutation representation of Q₈⋊₁Q₁₆
►Regular action on 128 points

Generators in S₁₂₈

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

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

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

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

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

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

Matrix representation of Q₈⋊₁Q₁₆ ►in GL₄(𝔽₁₇) generated by

16	0	0	0
0	16	0	0
0	0	0	1
0	0	16	0

13	9	0	0
4	4	0	0
0	0	1	10
0	0	10	16

6	6	0	0
14	0	0	0
0	0	16	0
0	0	0	16

7	7	0	0
5	10	0	0
0	0	13	6
0	0	6	4

G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,0,16,0,0,1,0],[13,4,0,0,9,4,0,0,0,0,1,10,0,0,10,16],[6,14,0,0,6,0,0,0,0,0,16,0,0,0,0,16],[7,5,0,0,7,10,0,0,0,0,13,6,0,0,6,4] >;

Q₈⋊₁Q₁₆ in GAP, Magma, Sage, TeX

Q_8\rtimes_1Q_{16}

% in TeX

G:=Group("Q8:1Q16");

// GroupNames label

G:=SmallGroup(128,402);

// by ID

G=gap.SmallGroup(128,402);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,288,422,352,1123,136,2804,718,172]);

// Polycyclic

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

// generators/relations

Export

Character table of Q₈⋊₁Q₁₆ in TeX

G = Q8⋊1Q16 order 128 = 27

The semidirect product of Q8 and Q16 acting via Q16/C8=C2

G = Q₈⋊₁Q₁₆ order 128 = 2⁷

The semidirect product of Q₈ and Q₁₆ acting via Q₁₆/C₈=C₂