Q8.3SD16 - GroupNames

G = Q₈.₃SD₁₆ order 128 = 2⁷

3^rd non-split extension by Q₈ of SD₁₆ acting via SD₁₆/C₈=C₂

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

Aliases: Q₈.₃SD₁₆, C₄².₂₃₁C₂³, C₈⋊₂C₈.₃C₂, C₄⋊C₄.₂₀₆D₄, (C₂×C₈).₃₁₄D₄, (C₈×Q₈).₁₃C₂, C₄.₇₁(C₄○D₈), C₄⋊C₈.₂₅C₂², Q₈⋊Q₈.₇C₂, (C₂×Q₈).₁₅₃D₄, C₄⋊₂Q₁₆.₆C₂, C₄.₄₃(C₂×SD₁₆), C₄⋊Q₈.₅₄C₂², C₂.₁₅(C₈⋊₈D₄), (C₄×C₈).₂₅₆C₂², C₄.₁₀D₈.₁₁C₂, C₄.₉₂(C₈.C₂²), C₂.₉(Q₈.D₄), C₄.SD₁₆.₁₁C₂, (C₄×Q₈).₂₆₈C₂², C₂.₁₁(D₄.₅D₄), C₂².₁₉₂(C₄⋊D₄), (C₂×C₄).₁₆(C₄○D₄), (C₂×C₄).₁₂₆₆(C₂×D₄), SmallGroup(128,412)

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

Derived series

C₁ — C₄² — Q₈.₃SD₁₆

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄×Q₈ — C₈×Q₈ — Q₈.₃SD₁₆

Lower central

C₁ — C₂² — C₄² — Q₈.₃SD₁₆

Upper central

C₁ — C₂² — C₄² — Q₈.₃SD₁₆

Jennings

C₁ — C₂² — C₂² — C₄² — Q₈.₃SD₁₆

Generators and relations for Q₈.₃SD₁₆
G = < a,b,c,d | a⁴=c⁸=1, b²=d²=a², bab^-1=dad^-1=a^-1, ac=ca, bc=cb, dbd^-1=a^-1b, dcd^-1=a²c³ >

Subgroups: 152 in 75 conjugacy classes, 34 normal (32 characteristic)
C₁, C₂, C₄, C₄, C₂², C₈, C₂×C₄, C₂×C₄, Q₈, Q₈, C₄², C₄², C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, Q₁₆, C₂×Q₈, C₂×Q₈, C₄×C₈, C₄×C₈, Q₈⋊C₄, C₄⋊C₈, C₄⋊C₈, C₄.Q₈, C₄×Q₈, C₄⋊Q₈, C₂×Q₁₆, C₄.₁₀D₈, C₈⋊₂C₈, C₈×Q₈, C₄⋊₂Q₁₆, Q₈⋊Q₈, C₄.SD₁₆, Q₈.₃SD₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, SD₁₆, C₂×D₄, C₄○D₄, C₄⋊D₄, C₂×SD₁₆, C₄○D₈, C₈.C₂², Q₈.D₄, C₈⋊₈D₄, D₄.₅D₄, Q₈.₃SD₁₆

Character table of Q₈.₃SD₁₆

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	-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
ρ₁₄	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
ρ₁₅	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	complex lifted from SD₁₆
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	complex lifted from SD₁₆
ρ₁₇	2	-2	2	-2	-2	0	2	0	0	0	0	0	0	0	0	2i	-2i	2i	-2i	0	0	0	0	0	0	√2	-√-2	-√2	√-2	complex lifted from C₄○D₈
ρ₁₈	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
ρ₁₉	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
ρ₂₀	2	-2	2	-2	-2	0	2	0	0	0	0	0	0	0	0	-2i	2i	-2i	2i	0	0	0	0	0	0	-√2	-√-2	√2	√-2	complex lifted from C₄○D₈
ρ₂₁	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	complex lifted from SD₁₆
ρ₂₂	2	-2	2	-2	-2	0	2	0	0	0	0	0	0	0	0	2i	-2i	2i	-2i	0	0	0	0	0	0	-√2	√-2	√2	-√-2	complex lifted from C₄○D₈
ρ₂₃	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	complex lifted from SD₁₆
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	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
ρ₂₆	2	-2	2	-2	-2	0	2	0	0	0	0	0	0	0	0	-2i	2i	-2i	2i	0	0	0	0	0	0	√2	√-2	-√2	-√-2	complex lifted from C₄○D₈
ρ₂₇	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
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 D₄.₅D₄, 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 D₄.₅D₄, Schur index 2

Smallest permutation representation of Q₈.₃SD₁₆
►Regular action on 128 points

Generators in S₁₂₈

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

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

(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)

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

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

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

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

Matrix representation of Q₈.₃SD₁₆ ►in GL₄(𝔽₁₇) generated by

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

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

0	5	0	0
7	7	0	0
0	0	4	0
0	0	0	4

0	14	0	0
11	0	0	0
0	0	0	9
0	0	15	0

G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,13,0,0,0,0,4],[16,0,0,0,0,16,0,0,0,0,0,16,0,0,1,0],[0,7,0,0,5,7,0,0,0,0,4,0,0,0,0,4],[0,11,0,0,14,0,0,0,0,0,0,15,0,0,9,0] >;

Q₈.₃SD₁₆ in GAP, Magma, Sage, TeX

Q_8._3{\rm SD}_{16}

% in TeX

G:=Group("Q8.3SD16");

// GroupNames label

G:=SmallGroup(128,412);

// by ID

G=gap.SmallGroup(128,412);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,512,422,352,1123,136,2804,718,172]);

// Polycyclic

G:=Group<a,b,c,d|a^4=c^8=1,b^2=d^2=a^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^2*c^3>;

// generators/relations

Export

Character table of Q₈.₃SD₁₆ in TeX

G = Q8.3SD16 order 128 = 27

3rd non-split extension by Q8 of SD16 acting via SD16/C8=C2

G = Q₈.₃SD₁₆ order 128 = 2⁷

3^rd non-split extension by Q₈ of SD₁₆ acting via SD₁₆/C₈=C₂