Q16.Q8 - GroupNames

G = Q₁₆.Q₈ order 128 = 2⁷

1^st non-split extension by Q₁₆ of Q₈ acting via Q₈/C₄=C₂

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

Aliases: Q₁₆.₁Q₈, C₄².₁₅₀D₄, (C₂×C₄).₄₀D₈, C₄⋊C₁₆.₁₁C₂, C₈.₃₄(C₂×Q₈), (C₂×C₈).₁₇₇D₄, C₁₆⋊₃C₄.₅C₂, C₁₆⋊₄C₄.₅C₂, C₈.₆₈(C₄○D₄), (C₂×C₁₆).₉C₂², (C₄×Q₁₆).₁₃C₂, C₈.₅Q₈.₂C₂, C₂.₁₄(C₄○D₁₆), (C₄×C₈).₁₀₆C₂², (C₂×C₈).₅₃₁C₂³, C₂.Q₃₂.₅C₂, C₂².₁₁₇(C₂×D₈), C₄.₅₀(C₂²⋊Q₈), C₂.D₈.₁₆C₂², C₂.₁₇(D₄⋊Q₈), C₂.₁₆(Q₃₂⋊C₂), C₄.₁₃(C₈.C₂²), (C₂×Q₁₆).₁₁₂C₂², (C₂×C₄).₇₉₉(C₂×D₄), SmallGroup(128,961)

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=a⁴c², bab^-1=dad^-1=a^-1, ac=ca, bc=cb, dbd^-1=a^-1b, dcd^-1=a⁴c^-1 >

Subgroups: 132 in 62 conjugacy classes, 32 normal (30 characteristic)
C₁, C₂, 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, C₄.Q₈, C₂.D₈, C₂×C₁₆, C₄×Q₈, C₄².C₂, C₂×Q₁₆, C₂.Q₃₂, C₄⋊C₁₆, C₁₆⋊₃C₄, C₁₆⋊₄C₄, C₄×Q₁₆, C₈.₅Q₈, Q₁₆.Q₈
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, D₈, C₂×D₄, C₂×Q₈, C₄○D₄, C₂²⋊Q₈, C₂×D₈, C₈.C₂², D₄⋊Q₈, C₄○D₁₆, 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	-2i	2i	0	0	0	0	0	0	0	0	√2	√2	-√2	-√2	√-2	-√-2	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷-ζ₁₆	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁷+ζ₁₆	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁷+ζ₁₆	-ζ₁₆⁵+ζ₁₆³	complex lifted from C₄○D₁₆
ρ₂₀	2	2	-2	-2	0	2i	-2i	0	0	0	0	0	0	0	0	√2	√2	-√2	-√2	-√-2	√-2	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁷-ζ₁₆	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁷+ζ₁₆	ζ₁₆⁷+ζ₁₆	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁹	-ζ₁₆⁵+ζ₁₆³	complex lifted from C₄○D₁₆
ρ₂₁	2	2	-2	-2	0	-2i	2i	0	0	0	0	0	0	0	0	-√2	-√2	√2	√2	-√-2	√-2	ζ₁₆⁷+ζ₁₆	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁷+ζ₁₆	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁷-ζ₁₆	complex lifted from C₄○D₁₆
ρ₂₂	2	2	-2	-2	0	2i	-2i	0	0	0	0	0	0	0	0	-√2	-√2	√2	√2	√-2	-√-2	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁷+ζ₁₆	-ζ₁₆⁵+ζ₁₆³	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁷+ζ₁₆	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷-ζ₁₆	complex lifted from C₄○D₁₆
ρ₂₃	2	2	-2	-2	0	-2i	2i	0	0	0	0	0	0	0	0	√2	√2	-√2	-√2	√-2	-√-2	ζ₁₆¹³+ζ₁₆¹¹	-ζ₁₆⁷+ζ₁₆	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷-ζ₁₆	ζ₁₆⁷+ζ₁₆	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵-ζ₁₆³	complex lifted from C₄○D₁₆
ρ₂₄	2	2	-2	-2	0	-2i	2i	0	0	0	0	0	0	0	0	-√2	-√2	√2	√2	-√-2	√-2	ζ₁₆¹⁵+ζ₁₆⁹	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷-ζ₁₆	ζ₁₆⁵-ζ₁₆³	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁷+ζ₁₆	ζ₁₆⁵+ζ₁₆³	-ζ₁₆⁷+ζ₁₆	complex lifted from C₄○D₁₆
ρ₂₅	2	2	-2	-2	0	2i	-2i	0	0	0	0	0	0	0	0	√2	√2	-√2	-√2	-√-2	√-2	ζ₁₆⁵+ζ₁₆³	-ζ₁₆⁷+ζ₁₆	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷-ζ₁₆	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁷+ζ₁₆	ζ₁₆⁵-ζ₁₆³	complex lifted from C₄○D₁₆
ρ₂₆	2	2	-2	-2	0	2i	-2i	0	0	0	0	0	0	0	0	-√2	-√2	√2	√2	√-2	-√-2	ζ₁₆⁷+ζ₁₆	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷-ζ₁₆	ζ₁₆⁵-ζ₁₆³	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹³+ζ₁₆¹¹	-ζ₁₆⁷+ζ₁₆	complex lifted from C₄○D₁₆
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 47 15 35)(2 48 16 36)(3 41 9 37)(4 42 10 38)(5 43 11 39)(6 44 12 40)(7 45 13 33)(8 46 14 34)(17 61 29 49)(18 62 30 50)(19 63 31 51)(20 64 32 52)(21 57 25 53)(22 58 26 54)(23 59 27 55)(24 60 28 56)(65 101 77 105)(66 102 78 106)(67 103 79 107)(68 104 80 108)(69 97 73 109)(70 98 74 110)(71 99 75 111)(72 100 76 112)(81 117 93 121)(82 118 94 122)(83 119 95 123)(84 120 96 124)(85 113 89 125)(86 114 90 126)(87 115 91 127)(88 116 92 128)

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

8	0	0	0
2	15	0	0
0	0	16	0
0	0	0	16

13	3	0	0
0	4	0	0
0	0	15	15
0	0	10	2

4	0	0	0
0	4	0	0
0	0	8	8
0	0	11	9

12	8	0	0
14	5	0	0
0	0	13	0
0	0	8	4

G:=sub<GL(4,GF(17))| [8,2,0,0,0,15,0,0,0,0,16,0,0,0,0,16],[13,0,0,0,3,4,0,0,0,0,15,10,0,0,15,2],[4,0,0,0,0,4,0,0,0,0,8,11,0,0,8,9],[12,14,0,0,8,5,0,0,0,0,13,8,0,0,0,4] >;

Q₁₆.Q₈ in GAP, Magma, Sage, TeX

Q_{16}.Q_8

% in TeX

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

// GroupNames label

G:=SmallGroup(128,961);

// by ID

G=gap.SmallGroup(128,961);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,56,141,512,422,352,1684,438,242,4037,1027,124]);

// Polycyclic

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

// generators/relations

Export

Character table of Q₁₆.Q₈ in TeX

G = Q16.Q8 order 128 = 27

1st non-split extension by Q16 of Q8 acting via Q8/C4=C2

G = Q₁₆.Q₈ order 128 = 2⁷

1^st non-split extension by Q₁₆ of Q₈ acting via Q₈/C₄=C₂