Q8:Q16 - GroupNames

G = Q₈⋊Q₁₆ order 128 = 2⁷

1^st semidirect product of Q₈ and Q₁₆ acting via Q₁₆/Q₈=C₂

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

Aliases: Q₈⋊₂Q₁₆, C₄².₁₉₄C₂³, Q₈².₁C₂, C₄⋊C₄.₅₃D₄, Q₈⋊C₈.₃C₂, C₄⋊C₈.₇C₂², C₄.₁₆(C₂×Q₁₆), (C₄×C₈).₄₃C₂², C₄.Q₁₆.₁C₂, (C₂×Q₈).₁₉₇D₄, C₄⋊Q₁₆.₃C₂, C₄⋊Q₈.₁₅C₂², C₄.₆Q₁₆.₁C₂, (C₄×Q₈).₂₇C₂², C₂.₂₂(D₄⋊₄D₄), C₄.₃₂(C₈.C₂²), C₂².₁₆₀C₂²≀C₂, C₂.₁₁(C₂²⋊Q₁₆), (C₂×C₄).₉₅₁(C₂×D₄), SmallGroup(128,365)

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

Derived series

C₁ — C₄² — Q₈⋊Q₁₆

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄×Q₈ — Q₈² — Q₈⋊Q₁₆

Lower central

C₁ — C₂² — C₄² — Q₈⋊Q₁₆

Upper central

C₁ — C₂² — C₄² — Q₈⋊Q₁₆

Jennings

C₁ — C₂² — C₂² — C₄² — Q₈⋊Q₁₆

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

Subgroups: 216 in 103 conjugacy classes, 38 normal (12 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, Q₈, Q₈, C₄², C₄², C₄⋊C₄, C₄⋊C₄, C₂×C₈, Q₁₆, C₂×Q₈, C₂×Q₈, C₄×C₈, Q₈⋊C₄, C₄⋊C₈, C₂.D₈, C₄×Q₈, C₄×Q₈, C₄⋊Q₈, C₄⋊Q₈, C₂×Q₁₆, Q₈⋊C₈, C₄.₆Q₁₆, C₄.Q₁₆, C₄⋊Q₁₆, Q₈², Q₈⋊Q₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, Q₁₆, C₂×D₄, C₂²≀C₂, C₂×Q₁₆, C₈.C₂², C₂²⋊Q₁₆, D₄⋊₄D₄, Q₈⋊Q₁₆

Character table of Q₈⋊Q₁₆

class	1	2A	2B	2C	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L	4M	4N	8A	8B	8C	8D	8E	8F	8G	8H
size	1	1	1	1	2	2	2	2	4	4	4	4	4	8	8	8	8	16	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	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	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	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	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	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	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	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	linear of order 2
ρ₉	2	2	2	2	-2	2	2	-2	2	-2	0	2	0	0	0	-2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	-2	2	2	-2	-2	-2	0	-2	0	0	0	2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	-2	-2	-2	-2	0	2	0	0	0	0	2	0	-2	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	-2	-2	-2	-2	0	2	0	0	0	0	-2	0	2	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	2	2	2	-2	-2	2	0	-2	2	0	2	-2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₄	2	2	2	2	2	-2	-2	2	0	-2	-2	0	-2	2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₅	2	2	-2	-2	0	2	-2	0	2	0	0	-2	0	0	0	0	0	0	-√2	√2	√2	-√2	-√2	√2	0	0	symplectic lifted from Q₁₆, Schur index 2
ρ₁₆	2	2	-2	-2	0	2	-2	0	2	0	0	-2	0	0	0	0	0	0	√2	-√2	-√2	√2	√2	-√2	0	0	symplectic lifted from Q₁₆, Schur index 2
ρ₁₇	2	2	-2	-2	0	2	-2	0	-2	0	0	2	0	0	0	0	0	0	√2	-√2	-√2	√2	-√2	√2	0	0	symplectic lifted from Q₁₆, Schur index 2
ρ₁₈	2	-2	-2	2	2	0	0	-2	0	0	2	0	-2	0	0	0	0	0	√2	√2	-√2	-√2	0	0	-√2	√2	symplectic lifted from Q₁₆, Schur index 2
ρ₁₉	2	-2	-2	2	2	0	0	-2	0	0	-2	0	2	0	0	0	0	0	-√2	-√2	√2	√2	0	0	-√2	√2	symplectic lifted from Q₁₆, Schur index 2
ρ₂₀	2	2	-2	-2	0	2	-2	0	-2	0	0	2	0	0	0	0	0	0	-√2	√2	√2	-√2	√2	-√2	0	0	symplectic lifted from Q₁₆, Schur index 2
ρ₂₁	2	-2	-2	2	2	0	0	-2	0	0	2	0	-2	0	0	0	0	0	-√2	-√2	√2	√2	0	0	√2	-√2	symplectic lifted from Q₁₆, Schur index 2
ρ₂₂	2	-2	-2	2	2	0	0	-2	0	0	-2	0	2	0	0	0	0	0	√2	√2	-√2	-√2	0	0	√2	-√2	symplectic lifted from Q₁₆, Schur index 2
ρ₂₃	4	-4	4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	-2	2	-2	2	0	0	0	0	orthogonal lifted from D₄⋊₄D₄
ρ₂₄	4	-4	4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	2	-2	2	-2	0	0	0	0	orthogonal lifted from D₄⋊₄D₄
ρ₂₅	4	4	-4	-4	0	-4	4	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
ρ₂₆	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	symplectic lifted from C₈.C₂², Schur index 2

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

Generators in S₁₂₈

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

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

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

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

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

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

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

1	0	0	0
0	1	0	0
0	0	16	15
0	0	1	1

16	0	0	0
0	16	0	0
0	0	9	3
0	0	1	8

11	11	0	0
3	0	0	0
0	0	11	11
0	0	3	0

6	14	0	0
1	11	0	0
0	0	8	14
0	0	16	9

G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,16,1,0,0,15,1],[16,0,0,0,0,16,0,0,0,0,9,1,0,0,3,8],[11,3,0,0,11,0,0,0,0,0,11,3,0,0,11,0],[6,1,0,0,14,11,0,0,0,0,8,16,0,0,14,9] >;

Q₈⋊Q₁₆ in GAP, Magma, Sage, TeX

Q_8\rtimes Q_{16}

% in TeX

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

// GroupNames label

G:=SmallGroup(128,365);

// by ID

G=gap.SmallGroup(128,365);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,232,422,352,1123,570,521,136,2804,1411,718,172]);

// 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,c*b*c^-1=a^-1*b,b*d=d*b,d*c*d^-1=c^-1>;

// generators/relations

Export

Character table of Q₈⋊Q₁₆ in TeX

G = Q8⋊Q16 order 128 = 27

1st semidirect product of Q8 and Q16 acting via Q16/Q8=C2

G = Q₈⋊Q₁₆ order 128 = 2⁷

1^st semidirect product of Q₈ and Q₁₆ acting via Q₁₆/Q₈=C₂