Q8.Q16 - GroupNames

G = Q₈.Q₁₆ order 128 = 2⁷

The non-split extension by Q₈ of Q₁₆ acting via Q₁₆/Q₈=C₂

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

Aliases: Q₈.₃Q₁₆, C₄².₁₉₇C₂³, C₄⋊C₄.₂₇D₄, Q₈⋊C₈.₄C₂, C₄⋊C₈.₈C₂², (C₂×Q₈).₄₇D₄, C₄.₁₇(C₂×Q₁₆), C₄.₅₇(C₄○D₈), (C₄×C₈).₁₉C₂², Q₈⋊₃Q₈.₁C₂, C₄.Q₁₆.₂C₂, C₄⋊Q₈.₁₈C₂², C₄.₁₀D₈.₁C₂, C₄.SD₁₆.₁C₂, (C₄×Q₈).₂₈C₂², C₄.₆₁(C₈.C₂²), C₂.₁₇(D₄.₈D₄), C₂.₁₇(D₄.₇D₄), C₂².₁₆₃C₂²≀C₂, C₂.₁₂(C₂²⋊Q₁₆), (C₂×C₄).₉₅₄(C₂×D₄), SmallGroup(128,368)

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

Derived series

C₁ — C₄² — Q₈.Q₁₆

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄×Q₈ — Q₈⋊₃Q₈ — 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=a²c⁴, bab^-1=dad^-1=a^-1, ac=ca, cbc^-1=a^-1b, bd=db, dcd^-1=a²c^-1 >

Subgroups: 184 in 94 conjugacy classes, 36 normal (32 characteristic)
C₁, C₂, C₄, C₄, C₂², C₈, C₂×C₄, C₂×C₄, Q₈, Q₈, C₄², C₄², C₄⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×Q₈, C₂×Q₈, C₄×C₈, Q₈⋊C₄, C₄⋊C₈, C₂.D₈, C₄×Q₈, C₄×Q₈, C₄².C₂, C₄⋊Q₈, C₄⋊Q₈, Q₈⋊C₈, C₄.₁₀D₈, C₄.Q₁₆, C₄.SD₁₆, Q₈⋊₃Q₈, Q₈.Q₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, Q₁₆, C₂×D₄, C₂²≀C₂, C₂×Q₁₆, C₄○D₈, C₈.C₂², C₂²⋊Q₁₆, D₄.₇D₄, D₄.₈D₄, 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	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
ρ₁₆	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
ρ₁₇	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
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	0	2	-2	0	-2i	0	0	2i	0	0	0	0	0	0	√-2	-√-2	-√-2	√-2	√2	-√2	0	0	complex lifted from C₄○D₈
ρ₂₀	2	2	-2	-2	0	2	-2	0	2i	0	0	-2i	0	0	0	0	0	0	-√-2	√-2	√-2	-√-2	√2	-√2	0	0	complex lifted from C₄○D₈
ρ₂₁	2	2	-2	-2	0	2	-2	0	-2i	0	0	2i	0	0	0	0	0	0	-√-2	√-2	√-2	-√-2	-√2	√2	0	0	complex lifted from C₄○D₈
ρ₂₂	2	2	-2	-2	0	2	-2	0	2i	0	0	-2i	0	0	0	0	0	0	√-2	-√-2	-√-2	√-2	-√2	√2	0	0	complex lifted from C₄○D₈
ρ₂₃	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
ρ₂₄	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
ρ₂₅	4	-4	4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	-2i	2i	-2i	2i	0	0	0	0	complex lifted from D₄.₈D₄
ρ₂₆	4	-4	4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	2i	-2i	2i	-2i	0	0	0	0	complex lifted from D₄.₈D₄

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

Generators in S₁₂₈

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

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

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

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

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

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

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

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

16	0	0	0
0	16	0	0
0	0	12	12
0	0	12	5

11	11	0	0
3	0	0	0
0	0	5	12
0	0	5	5

10	10	0	0
12	7	0	0
0	0	14	14
0	0	14	3

G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,0,16,0,0,1,0],[16,0,0,0,0,16,0,0,0,0,12,12,0,0,12,5],[11,3,0,0,11,0,0,0,0,0,5,5,0,0,12,5],[10,12,0,0,10,7,0,0,0,0,14,14,0,0,14,3] >;

Q₈.Q₁₆ in GAP, Magma, Sage, TeX

Q_8.Q_{16}

% in TeX

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

// GroupNames label

G:=SmallGroup(128,368);

// by ID

G=gap.SmallGroup(128,368);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,456,422,184,1123,570,521,136,2804,1411,718,172]);

// Polycyclic

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

// generators/relations

Export

Character table of Q₈.Q₁₆ in TeX

G = Q8.Q16 order 128 = 27

The non-split extension by Q8 of Q16 acting via Q16/Q8=C2

G = Q₈.Q₁₆ order 128 = 2⁷

The non-split extension by Q₈ of Q₁₆ acting via Q₁₆/Q₈=C₂