Q16:Q8 - GroupNames

G = Q₁₆⋊Q₈ order 128 = 2⁷

1^st semidirect product of Q₁₆ and Q₈ acting via Q₈/C₄=C₂

p-group, metabelian, nilpotent (class 4), monomial

Aliases: Q₁₆⋊₁Q₈, C₄.₁₃SD₃₂, C₄².₁₄₆D₄, (C₂×C₈).₇₀D₄, C₄⋊C₁₆.₁₂C₂, C₈.₃₀(C₂×Q₈), (C₂×C₄).₁₅₂D₈, (C₄×Q₁₆).₆C₂, C₁₆⋊₄C₄.₄C₂, C₈.₆₄(C₄○D₄), (C₄×C₈).₆₆C₂², C₈⋊₂Q₈.₁₁C₂, C₂.₁₀(C₂×SD₃₂), (C₂×C₁₆).₄₂C₂², (C₂×C₈).₅₂₇C₂³, C₂.Q₃₂.₄C₂, C₂².₁₁₃(C₂×D₈), C₄.₉(C₈.C₂²), C₄.₄₆(C₂²⋊Q₈), C₂.D₈.₁₂C₂², C₂.₁₃(D₄⋊Q₈), C₂.₁₅(Q₃₂⋊C₂), (C₂×Q₁₆).₁₁₀C₂², (C₂×C₄).₇₉₅(C₂×D₄), SmallGroup(128,957)

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

Derived series

C₁ — C₂×C₈ — Q₁₆⋊Q₈

Chief series

C₁ — C₂ — C₄ — C₈ — C₂×C₈ — C₂×Q₁₆ — C₄×Q₁₆ — Q₁₆⋊Q₈

Lower central

C₁ — C₂ — C₄ — C₂×C₈ — Q₁₆⋊Q₈

Upper central

C₁ — C₂² — C₄² — C₄×C₈ — Q₁₆⋊Q₈

Jennings

C₁ — C₂ — C₂ — C₂ — C₂ — C₄ — C₄ — C₂×C₈ — Q₁₆⋊Q₈

Generators and relations for Q₁₆⋊Q₈
G = < a,b,c,d | a⁸=c⁴=1, b²=a⁴, d²=c², bab^-1=dad^-1=a^-1, ac=ca, bc=cb, dbd^-1=a^-1b, dcd^-1=c^-1 >

Subgroups: 148 in 65 conjugacy classes, 34 normal (22 characteristic)
C₁, C₂, C₄, C₄, C₄, C₂², C₈, C₈, C₂×C₄, C₂×C₄, Q₈, C₁₆, C₄², C₄², C₄⋊C₄, C₂×C₈, Q₁₆, Q₁₆, C₂×Q₈, C₄×C₈, Q₈⋊C₄, C₂.D₈, C₂.D₈, C₂.D₈, C₂×C₁₆, C₄×Q₈, C₄⋊Q₈, C₂×Q₁₆, C₂.Q₃₂, C₄⋊C₁₆, C₁₆⋊₄C₄, C₄×Q₁₆, C₈⋊₂Q₈, Q₁₆⋊Q₈
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, D₈, C₂×D₄, C₂×Q₈, C₄○D₄, SD₃₂, C₂²⋊Q₈, C₂×D₈, C₈.C₂², D₄⋊Q₈, C₂×SD₃₂, Q₃₂⋊C₂, Q₁₆⋊Q₈

Character table of Q₁₆⋊Q₈

class	1	2A	2B	2C	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	8A	8B	8C	8D	8E	8F	16A	16B	16C	16D	16E	16F	16G	16H
size	1	1	1	1	2	2	2	2	4	8	8	8	8	16	16	2	2	2	2	4	4	4	4	4	4	4	4	4	4
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	2	0	0	0	0	0	0	0	0	0	0	0	0	√2	-√2	√2	-√2	-√2	√2	-√2	√2	orthogonal lifted from D₈
ρ₁₀	2	2	2	2	-2	2	2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	0	√2	√2	-√2	√2	-√2	√2	-√2	-√2	orthogonal lifted from D₈
ρ₁₁	2	2	2	2	2	2	2	2	2	0	0	0	0	0	0	-2	-2	-2	-2	-2	-2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	2	-2	-2	2	-2	0	0	0	0	0	0	-2	-2	-2	-2	2	2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	2	2	-2	2	2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	0	-√2	-√2	√2	-√2	√2	-√2	√2	√2	orthogonal lifted from D₈
ρ₁₄	2	2	2	2	-2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	0	-√2	√2	-√2	√2	√2	-√2	√2	-√2	orthogonal lifted from D₈
ρ₁₅	2	-2	-2	2	2	0	0	-2	0	2	-2	0	0	0	0	2	-2	2	-2	0	0	0	0	0	0	0	0	0	0	symplectic lifted from Q₈, Schur index 2
ρ₁₆	2	-2	-2	2	2	0	0	-2	0	-2	2	0	0	0	0	2	-2	2	-2	0	0	0	0	0	0	0	0	0	0	symplectic lifted from Q₈, Schur index 2
ρ₁₇	2	-2	-2	2	2	0	0	-2	0	0	0	-2i	2i	0	0	-2	2	-2	2	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₁₈	2	-2	-2	2	2	0	0	-2	0	0	0	2i	-2i	0	0	-2	2	-2	2	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₁₉	2	2	-2	-2	0	-2	2	0	0	0	0	0	0	0	0	-√2	-√2	√2	√2	-√2	√2	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷+ζ₁₆	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁷+ζ₁₆	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹³+ζ₁₆¹¹	complex lifted from SD₃₂
ρ₂₀	2	2	-2	-2	0	-2	2	0	0	0	0	0	0	0	0	-√2	-√2	√2	√2	-√2	√2	ζ₁₆⁷+ζ₁₆	ζ₁₆⁷+ζ₁₆	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁵+ζ₁₆³	complex lifted from SD₃₂
ρ₂₁	2	2	-2	-2	0	2	-2	0	0	0	0	0	0	0	0	√2	√2	-√2	-√2	-√2	√2	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷+ζ₁₆	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁷+ζ₁₆	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹⁵+ζ₁₆⁹	complex lifted from SD₃₂
ρ₂₂	2	2	-2	-2	0	2	-2	0	0	0	0	0	0	0	0	√2	√2	-√2	-√2	-√2	√2	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁷+ζ₁₆	ζ₁₆⁷+ζ₁₆	complex lifted from SD₃₂
ρ₂₃	2	2	-2	-2	0	-2	2	0	0	0	0	0	0	0	0	√2	√2	-√2	-√2	√2	-√2	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷+ζ₁₆	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁷+ζ₁₆	ζ₁₆¹⁵+ζ₁₆⁹	complex lifted from SD₃₂
ρ₂₄	2	2	-2	-2	0	-2	2	0	0	0	0	0	0	0	0	√2	√2	-√2	-√2	√2	-√2	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷+ζ₁₆	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁷+ζ₁₆	complex lifted from SD₃₂
ρ₂₅	2	2	-2	-2	0	2	-2	0	0	0	0	0	0	0	0	-√2	-√2	√2	√2	√2	-√2	ζ₁₆⁷+ζ₁₆	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷+ζ₁₆	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹³+ζ₁₆¹¹	complex lifted from SD₃₂
ρ₂₆	2	2	-2	-2	0	2	-2	0	0	0	0	0	0	0	0	-√2	-√2	√2	√2	√2	-√2	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁷+ζ₁₆	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁷+ζ₁₆	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁵+ζ₁₆³	complex lifted from SD₃₂
ρ₂₇	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	0	0	0	symplectic lifted from C₈.C₂², Schur index 2
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 Q₃₂⋊C₂, 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 Q₃₂⋊C₂, Schur index 2

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

Generators in S₁₂₈

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

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

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

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

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

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

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

1	0	0	0
0	1	0	0
0	0	9	0
0	0	0	2

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

1	2	0	0
16	16	0	0
0	0	16	0
0	0	0	16

6	15	0	0
10	11	0	0
0	0	0	12
0	0	10	0

G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,2],[16,0,0,0,0,16,0,0,0,0,0,16,0,0,1,0],[1,16,0,0,2,16,0,0,0,0,16,0,0,0,0,16],[6,10,0,0,15,11,0,0,0,0,0,10,0,0,12,0] >;

Q₁₆⋊Q₈ in GAP, Magma, Sage, TeX

Q_{16}\rtimes Q_8

% in TeX

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

// GroupNames label

G:=SmallGroup(128,957);

// by ID

G=gap.SmallGroup(128,957);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,56,141,64,422,352,1684,438,242,4037,1027,124]);

// Polycyclic

G:=Group<a,b,c,d|a^8=c^4=1,b^2=a^4,d^2=c^2,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 Q₁₆⋊Q₈ in TeX

G = Q16⋊Q8 order 128 = 27

1st semidirect product of Q16 and Q8 acting via Q8/C4=C2

G = Q₁₆⋊Q₈ order 128 = 2⁷

1^st semidirect product of Q₁₆ and Q₈ acting via Q₈/C₄=C₂