Q8.1Q16 - GroupNames

G = Q₈.₁Q₁₆ order 128 = 2⁷

1^st non-split extension by Q₈ of Q₁₆ acting via Q₁₆/C₈=C₂

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

Aliases: Q₈.₁Q₁₆, C₈.₁₅Q₁₆, C₄².₂₂₇C₂³, (C₈×Q₈).₆C₂, C₈⋊₁C₈.₈C₂, C₄⋊C₄.₂₀₂D₄, (C₂×C₈).₃₄₆D₄, C₄.₄₁(C₂×Q₁₆), C₈⋊₂Q₈.₅C₂, C₄.₄₃(C₄○D₈), C₄⋊C₈.₂₃C₂², (C₄×C₈).₅₆C₂², C₄.Q₁₆.₄C₂, (C₂×Q₈).₁₅₁D₄, C₄⋊Q₈.₅₀C₂², C₂.₈(C₄⋊₂Q₁₆), C₄.₆Q₁₆.₃C₂, C₂.₉(D₄.₄D₄), C₂.₉(C₈.₁₈D₄), (C₄×Q₈).₂₆₆C₂², C₄.₁₁₅(C₈.C₂²), C₂².₁₈₈(C₄⋊D₄), (C₂×C₄).₁₂(C₄○D₄), (C₂×C₄).₁₂₆₂(C₂×D₄), SmallGroup(128,408)

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

Derived series

C₁ — C₄² — Q₈.₁Q₁₆

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄×Q₈ — C₈×Q₈ — Q₈.₁Q₁₆

Lower central

C₁ — C₂² — C₄² — Q₈.₁Q₁₆

Upper central

C₁ — C₂² — C₄² — Q₈.₁Q₁₆

Jennings

C₁ — C₂² — C₂² — C₄² — Q₈.₁Q₁₆

Generators and relations for Q₈.₁Q₁₆
G = < a,b,c,d | a⁴=c⁸=1, b²=a², d²=a²c⁴, bab^-1=dad^-1=a^-1, ac=ca, bc=cb, dbd^-1=a^-1b, dcd^-1=c^-1 >

Subgroups: 144 in 72 conjugacy classes, 36 normal (24 characteristic)
C₁, C₂, C₄, C₄, C₂², C₈, C₈, C₂×C₄, C₂×C₄, Q₈, Q₈, C₄², C₄², C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, C₂×Q₈, C₂×Q₈, C₄×C₈, C₄×C₈, Q₈⋊C₄, C₄⋊C₈, C₄⋊C₈, C₄⋊C₈, C₂.D₈, C₄×Q₈, C₄⋊Q₈, C₄.₆Q₁₆, C₈⋊₁C₈, C₈×Q₈, C₄.Q₁₆, C₈⋊₂Q₈, Q₈.₁Q₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, Q₁₆, C₂×D₄, C₄○D₄, C₄⋊D₄, C₂×Q₁₆, C₄○D₈, C₈.C₂², C₄⋊₂Q₁₆, C₈.₁₈D₄, D₄.₄D₄, Q₈.₁Q₁₆

Character table of Q₈.₁Q₁₆

class	1	2A	2B	2C	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	8A	8B	8C	8D	8E	8F	8G	8H	8I	8J	8K	8L	8M	8N
size	1	1	1	1	2	2	2	2	4	4	4	4	4	16	16	2	2	2	2	4	4	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	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	0	0	0	-2	0	0	0	-2	-2	-2	-2	0	0	0	2	2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	-2	2	-2	2	2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	-2	2	-2	2	-2	2	2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	2	-2	2	-2	0	0	0	-2	0	0	0	2	2	2	2	0	0	0	-2	-2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	-2	-2	0	2	0	-2	0	-2	2	0	0	0	0	-√2	√2	√2	-√2	√2	-√2	-√2	√2	-√2	√2	0	0	0	0	symplectic lifted from Q₁₆, Schur index 2
ρ₁₄	2	2	-2	-2	0	2	0	-2	0	2	-2	0	0	0	0	√2	-√2	-√2	√2	√2	-√2	-√2	-√2	√2	√2	0	0	0	0	symplectic lifted from Q₁₆, Schur index 2
ρ₁₅	2	-2	2	-2	-2	0	2	0	0	0	0	0	0	0	0	-2	2	-2	2	0	0	0	0	0	0	√2	-√2	-√2	√2	symplectic lifted from Q₁₆, Schur index 2
ρ₁₆	2	2	-2	-2	0	2	0	-2	0	2	-2	0	0	0	0	-√2	√2	√2	-√2	-√2	√2	√2	√2	-√2	-√2	0	0	0	0	symplectic lifted from Q₁₆, Schur index 2
ρ₁₇	2	-2	2	-2	-2	0	2	0	0	0	0	0	0	0	0	2	-2	2	-2	0	0	0	0	0	0	-√2	-√2	√2	√2	symplectic lifted from Q₁₆, Schur index 2
ρ₁₈	2	-2	2	-2	-2	0	2	0	0	0	0	0	0	0	0	-2	2	-2	2	0	0	0	0	0	0	-√2	√2	√2	-√2	symplectic lifted from Q₁₆, Schur index 2
ρ₁₉	2	-2	2	-2	-2	0	2	0	0	0	0	0	0	0	0	2	-2	2	-2	0	0	0	0	0	0	√2	√2	-√2	-√2	symplectic lifted from Q₁₆, Schur index 2
ρ₂₀	2	2	-2	-2	0	2	0	-2	0	-2	2	0	0	0	0	√2	-√2	-√2	√2	-√2	√2	√2	-√2	√2	-√2	0	0	0	0	symplectic lifted from Q₁₆, Schur index 2
ρ₂₁	2	2	-2	-2	0	-2	0	2	2i	0	0	0	-2i	0	0	√2	-√2	-√2	√2	√-2	-√-2	√-2	√2	-√2	-√-2	0	0	0	0	complex lifted from C₄○D₈
ρ₂₂	2	2	-2	-2	0	-2	0	2	-2i	0	0	0	2i	0	0	-√2	√2	√2	-√2	√-2	-√-2	√-2	-√2	√2	-√-2	0	0	0	0	complex lifted from C₄○D₈
ρ₂₃	2	2	-2	-2	0	-2	0	2	2i	0	0	0	-2i	0	0	-√2	√2	√2	-√2	-√-2	√-2	-√-2	-√2	√2	√-2	0	0	0	0	complex lifted from C₄○D₈
ρ₂₄	2	2	-2	-2	0	-2	0	2	-2i	0	0	0	2i	0	0	√2	-√2	-√2	√2	-√-2	√-2	-√-2	√2	-√2	√-2	0	0	0	0	complex lifted from C₄○D₈
ρ₂₅	2	2	2	2	-2	-2	-2	-2	0	0	0	2	0	0	0	0	0	0	0	2i	2i	-2i	0	0	-2i	0	0	0	0	complex lifted from C₄○D₄
ρ₂₆	2	2	2	2	-2	-2	-2	-2	0	0	0	2	0	0	0	0	0	0	0	-2i	-2i	2i	0	0	2i	0	0	0	0	complex lifted from C₄○D₄
ρ₂₇	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	orthogonal lifted from D₄.₄D₄
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	orthogonal lifted from D₄.₄D₄
ρ₂₉	4	-4	4	-4	4	0	-4	0	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

Smallest permutation representation of Q₈.₁Q₁₆
►Regular action on 128 points

Generators in S₁₂₈

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

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

(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)

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

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

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

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

Matrix representation of Q₈.₁Q₁₆ ►in GL₄(𝔽₁₇) generated by

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

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

14	3	0	0
14	14	0	0
0	0	1	0
0	0	0	1

16	10	0	0
10	1	0	0
0	0	0	8
0	0	2	0

G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,13,0,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,0,16,0,0,1,0],[14,14,0,0,3,14,0,0,0,0,1,0,0,0,0,1],[16,10,0,0,10,1,0,0,0,0,0,2,0,0,8,0] >;

Q₈.₁Q₁₆ in GAP, Magma, Sage, TeX

Q_8._1Q_{16}

% in TeX

G:=Group("Q8.1Q16");

// GroupNames label

G:=SmallGroup(128,408);

// by ID

G=gap.SmallGroup(128,408);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,672,141,288,422,352,1123,136,2804,718,172]);

// 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,b*c=c*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^-1>;

// generators/relations

Export

Character table of Q₈.₁Q₁₆ in TeX

G = Q8.1Q16 order 128 = 27

1st non-split extension by Q8 of Q16 acting via Q16/C8=C2

G = Q₈.₁Q₁₆ order 128 = 2⁷

1^st non-split extension by Q₈ of Q₁₆ acting via Q₁₆/C₈=C₂