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G = Q16.Q8order 128 = 27

1st non-split extension by Q16 of Q8 acting via Q8/C4=C2

p-group, metabelian, nilpotent (class 4), monomial

Aliases: Q16.1Q8, C42.150D4, (C2×C4).40D8, C4⋊C16.11C2, C8.34(C2×Q8), (C2×C8).177D4, C163C4.5C2, C164C4.5C2, C8.68(C4○D4), (C2×C16).9C22, (C4×Q16).13C2, C8.5Q8.2C2, C2.14(C4○D16), (C4×C8).106C22, (C2×C8).531C23, C2.Q32.5C2, C22.117(C2×D8), C4.50(C22⋊Q8), C2.D8.16C22, C2.17(D4⋊Q8), C2.16(Q32⋊C2), C4.13(C8.C22), (C2×Q16).112C22, (C2×C4).799(C2×D4), SmallGroup(128,961)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — Q16.Q8
C1C2C4C8C2×C8C2×Q16C4×Q16 — Q16.Q8
C1C2C4C2×C8 — Q16.Q8
C1C22C42C4×C8 — Q16.Q8
C1C2C2C2C2C4C4C2×C8 — Q16.Q8

Generators and relations for Q16.Q8
 G = < a,b,c,d | a8=c4=1, b2=a4, d2=a4c2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=a4c-1 >

Subgroups: 132 in 62 conjugacy classes, 32 normal (30 characteristic)
C1, C2 [×3], C4 [×2], C4 [×7], C22, C8 [×2], C8, C2×C4 [×3], C2×C4 [×4], Q8 [×3], C16 [×2], C42, C42, C4⋊C4 [×6], C2×C8 [×2], Q16 [×2], Q16, C2×Q8, C4×C8, Q8⋊C4, C4.Q8, C2.D8 [×3], C2×C16 [×2], C4×Q8, C42.C2, C2×Q16, C2.Q32 [×2], C4⋊C16, C163C4, C164C4, C4×Q16, C8.5Q8, Q16.Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, D8 [×2], C2×D4, C2×Q8, C4○D4, C22⋊Q8, C2×D8, C8.C22, D4⋊Q8, C4○D16, Q32⋊C2, Q16.Q8

Character table of Q16.Q8

 class 12A2B2C4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 1111222248888161622224444444444
ρ111111111111111111111111111111    trivial
ρ21111111111111-1-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ311111-1-11-1-1-111-111111-1-1-1111-1-1-11    linear of order 2
ρ411111-1-11-1-1-1111-11111-1-11-1-1-1111-1    linear of order 2
ρ511111-1-11-111-1-1-111111-1-11-1-1-1111-1    linear of order 2
ρ611111-1-11-111-1-11-11111-1-1-1111-1-1-11    linear of order 2
ρ7111111111-1-1-1-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ8111111111-1-1-1-1-1-111111111111111    linear of order 2
ρ92222-2-2-2-220000000000002-22-2-22-22    orthogonal lifted from D8
ρ102222-222-2-200000000000022-22-22-2-2    orthogonal lifted from D8
ρ11222222222000000-2-2-2-2-2-200000000    orthogonal lifted from D4
ρ1222222-2-22-2000000-2-2-2-22200000000    orthogonal lifted from D4
ρ132222-222-2-2000000000000-2-22-22-222    orthogonal lifted from D8
ρ142222-2-2-2-22000000000000-22-222-22-2    orthogonal lifted from D8
ρ152-2-22200-202-200002-22-20000000000    symplectic lifted from Q8, Schur index 2
ρ162-2-22200-20-2200002-22-20000000000    symplectic lifted from Q8, Schur index 2
ρ172-2-22200-2000-2i2i00-22-220000000000    complex lifted from C4○D4
ρ182-2-22200-20002i-2i00-22-220000000000    complex lifted from C4○D4
ρ1922-2-20-2i2i0000000022-2-2-2--2ζ165163ζ16716ζ16516316716ζ1615169ζ16131611ζ16716165163    complex lifted from C4○D16
ρ2022-2-202i-2i0000000022-2-2--2-2ζ16131611ζ16716ζ16516316716ζ16716ζ165163ζ1615169165163    complex lifted from C4○D16
ρ2122-2-20-2i2i00000000-2-222--2-2ζ16716ζ16516316716165163ζ165163ζ1615169ζ16131611ζ16716    complex lifted from C4○D16
ρ2222-2-202i-2i00000000-2-222-2--2ζ1615169ζ16516316716165163ζ16131611ζ16716ζ165163ζ16716    complex lifted from C4○D16
ρ2322-2-20-2i2i0000000022-2-2-2--2ζ1613161116716165163ζ16716ζ16716ζ165163ζ1615169ζ165163    complex lifted from C4○D16
ρ2422-2-20-2i2i00000000-2-222--2-2ζ1615169165163ζ16716ζ165163ζ16131611ζ16716ζ16516316716    complex lifted from C4○D16
ρ2522-2-202i-2i0000000022-2-2--2-2ζ16516316716165163ζ16716ζ1615169ζ16131611ζ16716ζ165163    complex lifted from C4○D16
ρ2622-2-202i-2i00000000-2-222-2--2ζ16716165163ζ16716ζ165163ζ165163ζ1615169ζ1613161116716    complex lifted from C4○D16
ρ274-4-44-4004000000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ284-44-40000000000022-22-22220000000000    symplectic lifted from Q32⋊C2, Schur index 2
ρ294-44-400000000000-222222-220000000000    symplectic lifted from Q32⋊C2, Schur index 2

Smallest permutation representation of Q16.Q8
Regular action on 128 points
Generators in S128
(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 25 5 29)(2 32 6 28)(3 31 7 27)(4 30 8 26)(9 19 13 23)(10 18 14 22)(11 17 15 21)(12 24 16 20)(33 59 37 63)(34 58 38 62)(35 57 39 61)(36 64 40 60)(41 51 45 55)(42 50 46 54)(43 49 47 53)(44 56 48 52)(65 90 69 94)(66 89 70 93)(67 96 71 92)(68 95 72 91)(73 82 77 86)(74 81 78 85)(75 88 79 84)(76 87 80 83)(97 122 101 126)(98 121 102 125)(99 128 103 124)(100 127 104 123)(105 114 109 118)(106 113 110 117)(107 120 111 116)(108 119 112 115)
(1 47 15 35)(2 48 16 36)(3 41 9 37)(4 42 10 38)(5 43 11 39)(6 44 12 40)(7 45 13 33)(8 46 14 34)(17 61 29 49)(18 62 30 50)(19 63 31 51)(20 64 32 52)(21 57 25 53)(22 58 26 54)(23 59 27 55)(24 60 28 56)(65 101 77 105)(66 102 78 106)(67 103 79 107)(68 104 80 108)(69 97 73 109)(70 98 74 110)(71 99 75 111)(72 100 76 112)(81 117 93 121)(82 118 94 122)(83 119 95 123)(84 120 96 124)(85 113 89 125)(86 114 90 126)(87 115 91 127)(88 116 92 128)
(1 77 11 69)(2 76 12 68)(3 75 13 67)(4 74 14 66)(5 73 15 65)(6 80 16 72)(7 79 9 71)(8 78 10 70)(17 95 25 87)(18 94 26 86)(19 93 27 85)(20 92 28 84)(21 91 29 83)(22 90 30 82)(23 89 31 81)(24 96 32 88)(33 111 41 103)(34 110 42 102)(35 109 43 101)(36 108 44 100)(37 107 45 99)(38 106 46 98)(39 105 47 97)(40 112 48 104)(49 127 57 119)(50 126 58 118)(51 125 59 117)(52 124 60 116)(53 123 61 115)(54 122 62 114)(55 121 63 113)(56 128 64 120)

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

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

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

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

8000
21500
00160
00016
,
13300
0400
001515
00102
,
4000
0400
0088
00119
,
12800
14500
00130
0084
G:=sub<GL(4,GF(17))| [8,2,0,0,0,15,0,0,0,0,16,0,0,0,0,16],[13,0,0,0,3,4,0,0,0,0,15,10,0,0,15,2],[4,0,0,0,0,4,0,0,0,0,8,11,0,0,8,9],[12,14,0,0,8,5,0,0,0,0,13,8,0,0,0,4] >;

Q16.Q8 in GAP, Magma, Sage, TeX

Q_{16}.Q_8
% in TeX

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

G:=SmallGroup(128,961);
// by ID

G=gap.SmallGroup(128,961);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,56,141,512,422,352,1684,438,242,4037,1027,124]);
// Polycyclic

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

Export

Character table of Q16.Q8 in TeX

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