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G = Q8.1Q16order 128 = 27

1st 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, C81C8.8C2, C4⋊C4.202D4, (C2×C8).346D4, C4.41(C2×Q16), C82Q8.5C2, C4.43(C4○D8), C4⋊C8.23C22, (C4×C8).56C22, C4.Q16.4C2, (C2×Q8).151D4, C4⋊Q8.50C22, C2.8(C42Q16), 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

C1C42 — Q8.1Q16
C1C2C22C2×C4C42C4×Q8C8×Q8 — Q8.1Q16
C1C22C42 — Q8.1Q16
C1C22C42 — Q8.1Q16
C1C22C22C42 — 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 [×3], C4 [×4], C4 [×6], C22, C8 [×2], C8 [×4], C2×C4 [×3], C2×C4 [×4], Q8 [×2], Q8 [×3], C42, C42, C4⋊C4, C4⋊C4 [×5], C2×C8 [×2], C2×C8 [×3], C2×Q8, C2×Q8 [×2], C4×C8, C4×C8, Q8⋊C4 [×2], C4⋊C8, C4⋊C8 [×2], C4⋊C8, C2.D8 [×4], C4×Q8, C4⋊Q8 [×2], C4.6Q16 [×2], C81C8, C8×Q8, C4.Q16 [×2], C82Q8, Q8.1Q16
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, Q16 [×4], C2×D4 [×2], C4○D4, C4⋊D4, C2×Q16 [×2], C4○D8, C8.C22, C42Q16, C8.18D4, D4.4D4, Q8.1Q16

Character table of Q8.1Q16

 class 12A2B2C4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D8E8F8G8H8I8J8K8L8M8N
 size 1111222244444161622224444448888
ρ111111111111111111111111111111    trivial
ρ211111111-1-1-11-1-11-1-1-1-1111-1-11-11-11    linear of order 2
ρ31111111111111-11-1-1-1-1-1-1-1-1-1-11-11-1    linear of order 2
ρ411111111-1-1-11-1111111-1-1-111-1-1-1-1-1    linear of order 2
ρ511111111111111-1-1-1-1-1-1-1-1-1-1-1-11-11    linear of order 2
ρ611111111-1-1-11-1-1-11111-1-1-111-11111    linear of order 2
ρ71111111111111-1-11111111111-1-1-1-1    linear of order 2
ρ811111111-1-1-11-11-1-1-1-1-1111-1-111-11-1    linear of order 2
ρ922222-22-2000-2000-2-2-2-20002200000    orthogonal lifted from D4
ρ102222-22-222-2-2-220000000000000000    orthogonal lifted from D4
ρ112222-22-22-222-2-20000000000000000    orthogonal lifted from D4
ρ1222222-22-2000-20002222000-2-200000    orthogonal lifted from D4
ρ1322-2-2020-20-220000-222-22-2-22-220000    symplectic lifted from Q16, Schur index 2
ρ1422-2-2020-202-200002-2-222-2-2-2220000    symplectic lifted from Q16, Schur index 2
ρ152-22-2-20200000000-22-220000002-2-22    symplectic lifted from Q16, Schur index 2
ρ1622-2-2020-202-20000-222-2-2222-2-20000    symplectic lifted from Q16, Schur index 2
ρ172-22-2-202000000002-22-2000000-2-222    symplectic lifted from Q16, Schur index 2
ρ182-22-2-20200000000-22-22000000-222-2    symplectic lifted from Q16, Schur index 2
ρ192-22-2-202000000002-22-200000022-2-2    symplectic lifted from Q16, Schur index 2
ρ2022-2-2020-20-2200002-2-22-222-22-20000    symplectic lifted from Q16, Schur index 2
ρ2122-2-20-2022i000-2i002-2-22-2--2-22-2--20000    complex lifted from C4○D8
ρ2222-2-20-202-2i0002i00-222-2-2--2-2-22--20000    complex lifted from C4○D8
ρ2322-2-20-2022i000-2i00-222-2--2-2--2-22-20000    complex lifted from C4○D8
ρ2422-2-20-202-2i0002i002-2-22--2-2--22-2-20000    complex lifted from C4○D8
ρ252222-2-2-2-2000200000002i2i-2i00-2i0000    complex lifted from C4○D4
ρ262222-2-2-2-200020000000-2i-2i2i002i0000    complex lifted from C4○D4
ρ274-4-44000000000002222-22-220000000000    orthogonal lifted from D4.4D4
ρ284-4-4400000000000-22-2222220000000000    orthogonal lifted from D4.4D4
ρ294-44-440-40000000000000000000000    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

1000
0100
00130
0004
,
1000
0100
0001
00160
,
14300
141400
0010
0001
,
161000
10100
0008
0020
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

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