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G = Q84Q16order 128 = 27

3rd semidirect product of Q8 and Q16 acting via Q16/Q8=C2

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

Aliases: Q84Q16, C42.209C23, C4⋊C4.31D4, Q8⋊C8.6C2, (C2×Q8).51D4, C82Q8.3C2, C4.21(C2×Q16), C4.83(C4○D8), C4⋊C8.14C22, (C4×C8).46C22, Q8⋊Q8.4C2, Q83Q8.2C2, C42Q16.2C2, C4⋊Q8.29C22, C4.66(C8⋊C22), C4.10D8.5C2, (C4×Q8).37C22, C2.23(D4⋊D4), C4.39(C8.C22), C22.175C22≀C2, C2.16(C22⋊Q16), C2.13(D4.10D4), (C2×C4).966(C2×D4), SmallGroup(128,380)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — Q84Q16
C1C2C22C2×C4C42C4×Q8Q83Q8 — Q84Q16
C1C22C42 — Q84Q16
C1C22C42 — Q84Q16
C1C22C22C42 — Q84Q16

Generators and relations for Q84Q16
 G = < a,b,c,d | a4=c8=1, b2=a2, d2=c4, bab-1=cac-1=a-1, ad=da, cbc-1=ab, dbd-1=a2b, dcd-1=c-1 >

Subgroups: 192 in 97 conjugacy classes, 36 normal (32 characteristic)
C1, C2 [×3], C4 [×4], C4 [×9], C22, C8 [×4], C2×C4 [×3], C2×C4 [×7], Q8 [×2], Q8 [×7], C42, C42 [×4], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×11], C2×C8 [×3], Q16 [×2], C2×Q8 [×2], C2×Q8 [×2], C4×C8, Q8⋊C4 [×2], C4⋊C8 [×2], C4.Q8, C2.D8 [×2], C4×Q8 [×2], C4×Q8 [×2], C42.C2 [×3], C4⋊Q8 [×2], C4⋊Q8, C2×Q16, Q8⋊C8 [×2], C4.10D8, C42Q16, Q8⋊Q8, C82Q8, Q83Q8, Q84Q16
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, Q16 [×2], C2×D4 [×3], C22≀C2, C2×Q16, C4○D8, C8⋊C22, C8.C22, D4⋊D4, C22⋊Q16, D4.10D4, Q84Q16

Character table of Q84Q16

 class 12A2B2C4A4B4C4D4E4F4G4H4I4J4K4L4M4N8A8B8C8D8E8F8G8H
 size 111122224444488881644448888
ρ111111111111111111111111111    trivial
ρ211111111-111-111-1-1-11-1-1-1-111-1-1    linear of order 2
ρ311111111-111-111-1-1-1-11111-1-111    linear of order 2
ρ411111111111111111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ511111111-11-1-1-1-11-1111111-1-1-1-1    linear of order 2
ρ61111111111-11-1-1-11-11-1-1-1-1-1-111    linear of order 2
ρ71111111111-11-1-1-11-1-1111111-1-1    linear of order 2
ρ811111111-11-1-1-1-11-11-1-1-1-1-11111    linear of order 2
ρ92222-222-22-202000-20000000000    orthogonal lifted from D4
ρ102222-222-2-2-20-200020000000000    orthogonal lifted from D4
ρ112222-2-2-2-202000020-2000000000    orthogonal lifted from D4
ρ122222-2-2-2-2020000-202000000000    orthogonal lifted from D4
ρ1322222-2-220-2202-2000000000000    orthogonal lifted from D4
ρ1422222-2-220-2-20-22000000000000    orthogonal lifted from D4
ρ152-2-22200-20020-200000-2-222002-2    symplectic lifted from Q16, Schur index 2
ρ162-2-22200-20020-20000022-2-200-22    symplectic lifted from Q16, Schur index 2
ρ172-2-22200-200-20200000-2-22200-22    symplectic lifted from Q16, Schur index 2
ρ182-2-22200-200-2020000022-2-2002-2    symplectic lifted from Q16, Schur index 2
ρ1922-2-202-20-2i002i000000-222-2-2--200    complex lifted from C4○D8
ρ2022-2-202-202i00-2i0000002-2-22-2--200    complex lifted from C4○D8
ρ2122-2-202-20-2i002i0000002-2-22--2-200    complex lifted from C4○D8
ρ2222-2-202-202i00-2i000000-222-2--2-200    complex lifted from C4○D8
ρ2344-4-40-440000000000000000000    orthogonal lifted from C8⋊C22
ρ244-44-400000000000000-22-220000    symplectic lifted from D4.10D4, Schur index 2
ρ254-44-4000000000000002-22-20000    symplectic lifted from D4.10D4, Schur index 2
ρ264-4-44-4004000000000000000000    symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of Q84Q16
Regular action on 128 points
Generators in S128
(1 111 41 29)(2 30 42 112)(3 105 43 31)(4 32 44 106)(5 107 45 25)(6 26 46 108)(7 109 47 27)(8 28 48 110)(9 93 127 80)(10 73 128 94)(11 95 121 74)(12 75 122 96)(13 89 123 76)(14 77 124 90)(15 91 125 78)(16 79 126 92)(17 63 99 49)(18 50 100 64)(19 57 101 51)(20 52 102 58)(21 59 103 53)(22 54 104 60)(23 61 97 55)(24 56 98 62)(33 113 84 66)(34 67 85 114)(35 115 86 68)(36 69 87 116)(37 117 88 70)(38 71 81 118)(39 119 82 72)(40 65 83 120)
(1 39 41 82)(2 65 42 120)(3 33 43 84)(4 67 44 114)(5 35 45 86)(6 69 46 116)(7 37 47 88)(8 71 48 118)(9 99 127 17)(10 64 128 50)(11 101 121 19)(12 58 122 52)(13 103 123 21)(14 60 124 54)(15 97 125 23)(16 62 126 56)(18 73 100 94)(20 75 102 96)(22 77 104 90)(24 79 98 92)(25 115 107 68)(26 36 108 87)(27 117 109 70)(28 38 110 81)(29 119 111 72)(30 40 112 83)(31 113 105 66)(32 34 106 85)(49 93 63 80)(51 95 57 74)(53 89 59 76)(55 91 61 78)
(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 99 5 103)(2 98 6 102)(3 97 7 101)(4 104 8 100)(9 35 13 39)(10 34 14 38)(11 33 15 37)(12 40 16 36)(17 45 21 41)(18 44 22 48)(19 43 23 47)(20 42 24 46)(25 59 29 63)(26 58 30 62)(27 57 31 61)(28 64 32 60)(49 107 53 111)(50 106 54 110)(51 105 55 109)(52 112 56 108)(65 79 69 75)(66 78 70 74)(67 77 71 73)(68 76 72 80)(81 128 85 124)(82 127 86 123)(83 126 87 122)(84 125 88 121)(89 119 93 115)(90 118 94 114)(91 117 95 113)(92 116 96 120)

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

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

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

Matrix representation of Q84Q16 in GL4(𝔽17) generated by

1000
0100
00132
0004
,
1000
0100
00169
00131
,
2000
0900
00910
0028
,
0100
16000
0018
00016
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,13,0,0,0,2,4],[1,0,0,0,0,1,0,0,0,0,16,13,0,0,9,1],[2,0,0,0,0,9,0,0,0,0,9,2,0,0,10,8],[0,16,0,0,1,0,0,0,0,0,1,0,0,0,8,16] >;

Q84Q16 in GAP, Magma, Sage, TeX

Q_8\rtimes_4Q_{16}
% in TeX

G:=Group("Q8:4Q16");
// GroupNames label

G:=SmallGroup(128,380);
// by ID

G=gap.SmallGroup(128,380);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,456,422,184,1123,570,521,136,2804,1411,718,172]);
// Polycyclic

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

Export

Character table of Q84Q16 in TeX

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