Q8:4Q16 - GroupNames

G = Q₈⋊₄Q₁₆ order 128 = 2⁷

3^rd semidirect product of Q₈ and Q₁₆ acting via Q₁₆/Q₈=C₂

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

Aliases: Q₈⋊₄Q₁₆, C₄².₂₀₉C₂³, C₄⋊C₄.₃₁D₄, Q₈⋊C₈.₆C₂, (C₂×Q₈).₅₁D₄, C₈⋊₂Q₈.₃C₂, C₄.₂₁(C₂×Q₁₆), C₄.₈₃(C₄○D₈), C₄⋊C₈.₁₄C₂², (C₄×C₈).₄₆C₂², Q₈⋊Q₈.₄C₂, Q₈⋊₃Q₈.₂C₂, C₄⋊₂Q₁₆.₂C₂, C₄⋊Q₈.₂₉C₂², C₄.₆₆(C₈⋊C₂²), C₄.₁₀D₈.₅C₂, (C₄×Q₈).₃₇C₂², C₂.₂₃(D₄⋊D₄), C₄.₃₉(C₈.C₂²), C₂².₁₇₅C₂²≀C₂, C₂.₁₆(C₂²⋊Q₁₆), C₂.₁₃(D₄.₁₀D₄), (C₂×C₄).₉₆₆(C₂×D₄), SmallGroup(128,380)

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

Derived series

C₁ — C₄² — Q₈⋊₄Q₁₆

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄×Q₈ — Q₈⋊₃Q₈ — Q₈⋊₄Q₁₆

Lower central

C₁ — C₂² — C₄² — Q₈⋊₄Q₁₆

Upper central

C₁ — C₂² — C₄² — Q₈⋊₄Q₁₆

Jennings

C₁ — C₂² — C₂² — C₄² — Q₈⋊₄Q₁₆

Generators and relations for Q₈⋊₄Q₁₆
G = < a,b,c,d | a⁴=c⁸=1, b²=a², d²=c⁴, bab^-1=cac^-1=a^-1, ad=da, cbc^-1=ab, dbd^-1=a²b, dcd^-1=c^-1 >

Subgroups: 192 in 97 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, Q₁₆, C₂×Q₈, C₂×Q₈, C₄×C₈, Q₈⋊C₄, C₄⋊C₈, C₄.Q₈, C₂.D₈, C₄×Q₈, C₄×Q₈, C₄².C₂, C₄⋊Q₈, C₄⋊Q₈, C₂×Q₁₆, Q₈⋊C₈, C₄.₁₀D₈, C₄⋊₂Q₁₆, Q₈⋊Q₈, C₈⋊₂Q₈, Q₈⋊₃Q₈, Q₈⋊₄Q₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, Q₁₆, C₂×D₄, C₂²≀C₂, C₂×Q₁₆, C₄○D₈, C₈⋊C₂², C₈.C₂², D₄⋊D₄, C₂²⋊Q₁₆, D₄.₁₀D₄, Q₈⋊₄Q₁₆

Character table of Q₈⋊₄Q₁₆

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	orthogonal lifted from C₈⋊C₂²
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	symplectic lifted from D₄.₁₀D₄, Schur index 2
ρ₂₅	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	symplectic lifted from D₄.₁₀D₄, 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⋊₄Q₁₆
►Regular action on 128 points

Generators in S₁₂₈

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

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

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

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

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

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

Matrix representation of Q₈⋊₄Q₁₆ ►in GL₄(𝔽₁₇) generated by

1	0	0	0
0	1	0	0
0	0	13	2
0	0	0	4

1	0	0	0
0	1	0	0
0	0	16	9
0	0	13	1

2	0	0	0
0	9	0	0
0	0	9	10
0	0	2	8

0	1	0	0
16	0	0	0
0	0	1	8
0	0	0	16

G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,13,0,0,0,2,4],[1,0,0,0,0,1,0,0,0,0,16,13,0,0,9,1],[2,0,0,0,0,9,0,0,0,0,9,2,0,0,10,8],[0,16,0,0,1,0,0,0,0,0,1,0,0,0,8,16] >;

Q₈⋊₄Q₁₆ in GAP, Magma, Sage, TeX

Q_8\rtimes_4Q_{16}

% in TeX

G:=Group("Q8:4Q16");

// GroupNames label

G:=SmallGroup(128,380);

// by ID

G=gap.SmallGroup(128,380);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,456,422,184,1123,570,521,136,2804,1411,718,172]);

// Polycyclic

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

// generators/relations

Export

Character table of Q₈⋊₄Q₁₆ in TeX

G = Q8⋊4Q16 order 128 = 27

3rd semidirect product of Q8 and Q16 acting via Q16/Q8=C2

G = Q₈⋊₄Q₁₆ order 128 = 2⁷

3^rd semidirect product of Q₈ and Q₁₆ acting via Q₁₆/Q₈=C₂