Q8.2Q16 - GroupNames

G = Q₈.₂Q₁₆ order 128 = 2⁷

2^nd non-split extension by Q₈ of Q₁₆ acting via Q₁₆/C₈=C₂

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

Aliases: Q₈.₂Q₁₆, C₄².₂₃₅C₂³, (C₈×Q₈).₇C₂, C₄⋊C₄.₂₁₀D₄, (C₂×C₈).₃₁₈D₄, C₄.₂₇(C₂×Q₁₆), C₈⋊₁C₈.₁₀C₂, C₄.₄₅(C₄○D₈), C₄⋊C₈.₂₇C₂², (C₄×C₈).₆₀C₂², C₄.Q₁₆.₅C₂, (C₂×Q₈).₁₅₅D₄, Q₈⋊Q₈.₈C₂, C₄⋊Q₈.₅₈C₂², C₄.₁₀D₈.₄C₂, C₄.₆Q₁₆.₇C₂, C₄.SD₁₆.₅C₂, C₄.₉₄(C₈.C₂²), (C₄×Q₈).₂₇₀C₂², C₂.₁₅(D₄.₃D₄), C₂.₁₁(C₈.₁₈D₄), C₂.₁₁(Q₈.D₄), C₂².₁₉₆(C₄⋊D₄), (C₂×C₄).₂₀(C₄○D₄), (C₂×C₄).₁₂₇₀(C₂×D₄), SmallGroup(128,416)

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

Derived series

C₁ — C₄² — Q₈.₂Q₁₆

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄×Q₈ — C₈×Q₈ — Q₈.₂Q₁₆

Lower central

C₁ — C₂² — C₄² — Q₈.₂Q₁₆

Upper central

C₁ — C₂² — C₄² — Q₈.₂Q₁₆

Jennings

C₁ — C₂² — C₂² — C₄² — Q₈.₂Q₁₆

Generators and relations for Q₈.₂Q₁₆
G = < a,b,c,d | a⁴=c⁸=1, b²=a², d²=c⁴, bab^-1=dad^-1=a^-1, ac=ca, bc=cb, dbd^-1=a^-1b, dcd^-1=a²c^-1 >

Subgroups: 144 in 72 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, C₂×Q₈, C₂×Q₈, C₄×C₈, C₄×C₈, Q₈⋊C₄, C₄⋊C₈, C₄⋊C₈, C₄.Q₈, C₂.D₈, C₄×Q₈, C₄⋊Q₈, C₄.₁₀D₈, C₄.₆Q₁₆, C₈⋊₁C₈, C₈×Q₈, Q₈⋊Q₈, C₄.Q₁₆, C₄.SD₁₆, Q₈.₂Q₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, Q₁₆, C₂×D₄, C₄○D₄, C₄⋊D₄, C₂×Q₁₆, C₄○D₈, C₈.C₂², Q₈.D₄, C₈.₁₈D₄, D₄.₃D₄, Q₈.₂Q₁₆

Character table of Q₈.₂Q₁₆

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	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
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	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
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	-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
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	-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
ρ₂₂	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
ρ₂₃	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
ρ₂₄	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
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	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
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	complex lifted from D₄.₃D₄
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	complex lifted from D₄.₃D₄

Smallest permutation representation of Q₈.₂Q₁₆
►Regular action on 128 points

Generators in S₁₂₈

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

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

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

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

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

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

Matrix representation of Q₈.₂Q₁₆ ►in GL₄(𝔽₁₇) generated by

16	0	0	0
0	16	0	0
0	0	13	0
0	0	0	4

0	13	0	0
4	0	0	0
0	0	0	15
0	0	9	0

3	14	0	0
3	3	0	0
0	0	4	0
0	0	0	4

5	5	0	0
5	12	0	0
0	0	0	4
0	0	13	0

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

Q₈.₂Q₁₆ in GAP, Magma, Sage, TeX

Q_8._2Q_{16}

% in TeX

G:=Group("Q8.2Q16");

// GroupNames label

G:=SmallGroup(128,416);

// by ID

G=gap.SmallGroup(128,416);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,736,422,352,1123,136,2804,718,172]);

// Polycyclic

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

// generators/relations

Export

Character table of Q₈.₂Q₁₆ in TeX

G = Q8.2Q16 order 128 = 27

2nd non-split extension by Q8 of Q16 acting via Q16/C8=C2

G = Q₈.₂Q₁₆ order 128 = 2⁷

2^nd non-split extension by Q₈ of Q₁₆ acting via Q₁₆/C₈=C₂