Q8:3Q16 - GroupNames

G = Q₈⋊₃Q₁₆ order 128 = 2⁷

2^nd semidirect product of Q₈ and Q₁₆ acting via Q₁₆/Q₈=C₂

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

Aliases: Q₈⋊₃Q₁₆, Q₈.₅SD₁₆, C₄².₂₀₆C₂³, Q₈².₂C₂, C₄⋊C₄.₅₇D₄, Q₈⋊C₈.₅C₂, C₄.₂₀(C₂×Q₁₆), C₄⋊C₈.₁₃C₂², (C₄×C₈).₂₇C₂², Q₈⋊Q₈.₃C₂, (C₂×Q₈).₂₀₁D₄, C₄⋊₂Q₁₆.₁C₂, C₄.₃₄(C₂×SD₁₆), C₄⋊Q₈.₂₆C₂², C₄.₆Q₁₆.₄C₂, C₄.SD₁₆.₂C₂, (C₄×Q₈).₃₆C₂², C₂.₁₅(Q₈⋊D₄), C₄.₆₆(C₈.C₂²), C₂.₁₈(D₄.₉D₄), C₂².₁₇₂C₂²≀C₂, C₂.₁₅(C₂²⋊Q₁₆), (C₂×C₄).₉₆₃(C₂×D₄), SmallGroup(128,377)

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

Derived series

C₁ — C₄² — Q₈⋊₃Q₁₆

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄×Q₈ — Q₈² — Q₈⋊₃Q₁₆

Lower central

C₁ — C₂² — C₄² — Q₈⋊₃Q₁₆

Upper central

C₁ — C₂² — C₄² — Q₈⋊₃Q₁₆

Jennings

C₁ — C₂² — C₂² — C₄² — Q₈⋊₃Q₁₆

Generators and relations for Q₈⋊₃Q₁₆
G = < a,b,c,d | a⁴=c⁸=1, b²=a², d²=c⁴, bab^-1=cac^-1=a^-1, ad=da, cbc^-1=a^-1b, bd=db, dcd^-1=c^-1 >

Subgroups: 208 in 100 conjugacy classes, 38 normal (32 characteristic)
C₁, C₂, C₄, C₄, C₂², C₈, C₂×C₄, C₂×C₄, Q₈, Q₈, C₄², C₄², C₄⋊C₄, C₄⋊C₄, C₂×C₈, Q₁₆, C₂×Q₈, C₂×Q₈, C₂×Q₈, C₄×C₈, Q₈⋊C₄, C₄⋊C₈, C₄.Q₈, C₄×Q₈, C₄×Q₈, C₄⋊Q₈, C₄⋊Q₈, C₂×Q₁₆, Q₈⋊C₈, C₄.₆Q₁₆, C₄⋊₂Q₁₆, Q₈⋊Q₈, C₄.SD₁₆, Q₈², Q₈⋊₃Q₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, SD₁₆, Q₁₆, C₂×D₄, C₂²≀C₂, C₂×SD₁₆, C₂×Q₁₆, C₈.C₂², Q₈⋊D₄, C₂²⋊Q₁₆, D₄.₉D₄, Q₈⋊₃Q₁₆

Character table of Q₈⋊₃Q₁₆

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	-2	0	0	2	0	0	0	0	0	0	-√-2	√-2	√-2	-√-2	√-2	-√-2	0	0	complex lifted from SD₁₆
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	complex lifted from SD₁₆
ρ₂₁	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	complex lifted from SD₁₆
ρ₂₂	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	complex lifted from SD₁₆
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 D₄.₉D₄
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 D₄.₉D₄

Smallest permutation representation of Q₈⋊₃Q₁₆
►Regular action on 128 points

Generators in S₁₂₈

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

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

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

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

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

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

Matrix representation of Q₈⋊₃Q₁₆ ►in GL₄(𝔽₁₇) generated by

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

0	4	0	0
4	0	0	0
0	0	16	0
0	0	0	16

14	14	0	0
14	3	0	0
0	0	0	6
0	0	14	6

1	0	0	0
0	1	0	0
0	0	11	15
0	0	10	6

G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,4,0,0,4,0,0,0,0,0,16,0,0,0,0,16],[14,14,0,0,14,3,0,0,0,0,0,14,0,0,6,6],[1,0,0,0,0,1,0,0,0,0,11,10,0,0,15,6] >;

Q₈⋊₃Q₁₆ in GAP, Magma, Sage, TeX

Q_8\rtimes_3Q_{16}

% in TeX

G:=Group("Q8:3Q16");

// GroupNames label

G:=SmallGroup(128,377);

// by ID

G=gap.SmallGroup(128,377);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,232,422,352,1123,570,521,136,2804,1411,718,172]);

// Polycyclic

G:=Group<a,b,c,d|a^4=c^8=1,b^2=a^2,d^2=c^4,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*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⋊₃Q₁₆ in TeX

G = Q8⋊3Q16 order 128 = 27

2nd semidirect product of Q8 and Q16 acting via Q16/Q8=C2

G = Q₈⋊₃Q₁₆ order 128 = 2⁷

2^nd semidirect product of Q₈ and Q₁₆ acting via Q₁₆/Q₈=C₂