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G = Q165Q8order 128 = 27

5th semidirect product of Q16 and Q8 acting via Q8/C4=C2

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

Aliases: Q165Q8, C42.71C23, C4.1052- 1+4, C8⋊Q8.3C2, C8.10(C2×Q8), C2.48(D4×Q8), C4⋊C4.393D4, Q8.14(C2×Q8), C84Q8.7C2, Q8.Q8.4C2, (C4×Q16).18C2, Q83Q8.8C2, (C2×Q8).142D4, C2.69(Q8○D8), C8.5Q8.8C2, C4.48(C22×Q8), C4⋊C4.279C23, C4⋊C8.149C22, (C2×C8).217C23, (C2×C4).582C24, (C4×C8).206C22, Q16⋊C4.3C2, C4.Q16.12C2, C4⋊Q8.211C22, C2.D8.77C22, C4.Q8.80C22, C8⋊C4.75C22, (C2×Q8).415C23, (C4×Q8).209C22, (C2×Q16).165C22, Q8⋊C4.93C22, C22.842(C22×D4), C42.C2.80C22, C2.107(D8⋊C22), (C2×C4).652(C2×D4), SmallGroup(128,2122)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — Q165Q8
C1C2C4C2×C4C42C4×Q8Q83Q8 — Q165Q8
C1C2C2×C4 — Q165Q8
C1C22C4×Q8 — Q165Q8
C1C2C2C2×C4 — Q165Q8

Generators and relations for Q165Q8
 G = < a,b,c,d | a8=c4=1, b2=a4, d2=c2, bab-1=a-1, ac=ca, dad-1=a5, cbc-1=dbd-1=a4b, dcd-1=c-1 >

Subgroups: 248 in 160 conjugacy classes, 94 normal (26 characteristic)
C1, C2 [×3], C4 [×2], C4 [×16], C22, C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×8], Q8 [×4], Q8 [×7], C42, C42 [×2], C42 [×6], C4⋊C4, C4⋊C4 [×8], C4⋊C4 [×16], C2×C8 [×2], C2×C8 [×2], Q16 [×4], C2×Q8, C2×Q8 [×2], C2×Q8 [×2], C4×C8, C8⋊C4 [×2], Q8⋊C4 [×6], C4⋊C8, C4⋊C8 [×2], C4.Q8 [×4], C2.D8 [×3], C2.D8 [×2], C4×Q8, C4×Q8 [×6], C4×Q8 [×2], C42.C2 [×4], C42.C2 [×4], C4⋊Q8 [×2], C4⋊Q8 [×2], C2×Q16, C4×Q16, Q16⋊C4 [×2], C84Q8, C4.Q16 [×2], Q8.Q8 [×4], C8.5Q8, C8⋊Q8 [×2], Q83Q8 [×2], Q165Q8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C24, C22×D4, C22×Q8, 2- 1+4, D4×Q8, D8⋊C22, Q8○D8, Q165Q8

Character table of Q165Q8

 class 12A2B2C4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q4R4S8A8B8C8D8E8F
 size 11112222444444444888888444488
ρ111111111111111111111111111111    trivial
ρ21111-11-111-1-111-1-11-1111-1-1-1-1-111-11    linear of order 2
ρ31111-11-11-11-11-11-11-11-1-111-1-1-1111-1    linear of order 2
ρ411111111-1-111-1-11111-1-1-1-111111-1-1    linear of order 2
ρ511111111-1-11-1-1-1-1-1-1-11111-11111-1-1    linear of order 2
ρ61111-11-11-11-1-1-111-11-111-1-11-1-1111-1    linear of order 2
ρ71111-11-111-1-1-11-11-11-1-1-1111-1-111-11    linear of order 2
ρ811111111111-111-1-1-1-1-1-1-1-1-1111111    linear of order 2
ρ911111111-1-111-1-1-11-1-1-111-11-1-1-1-111    linear of order 2
ρ101111-11-11-11-11-11111-1-11-11-111-1-1-11    linear of order 2
ρ111111-11-111-1-111-1111-11-11-1-111-1-11-1    linear of order 2
ρ1211111111111111-11-1-11-1-111-1-1-1-1-1-1    linear of order 2
ρ1311111111111-1111-111-111-1-1-1-1-1-1-1-1    linear of order 2
ρ141111-11-111-1-1-11-1-1-1-11-11-11111-1-11-1    linear of order 2
ρ151111-11-11-11-1-1-11-1-1-111-11-1111-1-1-11    linear of order 2
ρ1611111111-1-11-1-1-11-1111-1-11-1-1-1-1-111    linear of order 2
ρ1722222-22-222-20-2-2000000000000000    orthogonal lifted from D4
ρ182222-2-2-2-22-220-22000000000000000    orthogonal lifted from D4
ρ1922222-22-2-2-2-2022000000000000000    orthogonal lifted from D4
ρ202222-2-2-2-2-22202-2000000000000000    orthogonal lifted from D4
ρ212-22-2020-2000-200-2220000002-20000    symplectic lifted from Q8, Schur index 2
ρ222-22-2020-20002002-2-20000002-20000    symplectic lifted from Q8, Schur index 2
ρ232-22-2020-2000-20022-2000000-220000    symplectic lifted from Q8, Schur index 2
ρ242-22-2020-2000200-2-22000000-220000    symplectic lifted from Q8, Schur index 2
ρ254-44-40-404000000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ2644-4-400000000000000000000022-2200    symplectic lifted from Q8○D8, Schur index 2
ρ2744-4-4000000000000000000000-222200    symplectic lifted from Q8○D8, Schur index 2
ρ284-4-444i0-4i0000000000000000000000    complex lifted from D8⋊C22
ρ294-4-44-4i04i0000000000000000000000    complex lifted from D8⋊C22

Smallest permutation representation of Q165Q8
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 51 5 55)(2 50 6 54)(3 49 7 53)(4 56 8 52)(9 124 13 128)(10 123 14 127)(11 122 15 126)(12 121 16 125)(17 107 21 111)(18 106 22 110)(19 105 23 109)(20 112 24 108)(25 104 29 100)(26 103 30 99)(27 102 31 98)(28 101 32 97)(33 79 37 75)(34 78 38 74)(35 77 39 73)(36 76 40 80)(41 72 45 68)(42 71 46 67)(43 70 47 66)(44 69 48 65)(57 116 61 120)(58 115 62 119)(59 114 63 118)(60 113 64 117)(81 91 85 95)(82 90 86 94)(83 89 87 93)(84 96 88 92)
(1 32 43 105)(2 25 44 106)(3 26 45 107)(4 27 46 108)(5 28 47 109)(6 29 48 110)(7 30 41 111)(8 31 42 112)(9 87 118 35)(10 88 119 36)(11 81 120 37)(12 82 113 38)(13 83 114 39)(14 84 115 40)(15 85 116 33)(16 86 117 34)(17 49 99 68)(18 50 100 69)(19 51 101 70)(20 52 102 71)(21 53 103 72)(22 54 104 65)(23 55 97 66)(24 56 98 67)(57 79 122 95)(58 80 123 96)(59 73 124 89)(60 74 125 90)(61 75 126 91)(62 76 127 92)(63 77 128 93)(64 78 121 94)
(1 74 43 90)(2 79 44 95)(3 76 45 92)(4 73 46 89)(5 78 47 94)(6 75 48 91)(7 80 41 96)(8 77 42 93)(9 102 118 20)(10 99 119 17)(11 104 120 22)(12 101 113 19)(13 98 114 24)(14 103 115 21)(15 100 116 18)(16 97 117 23)(25 57 106 122)(26 62 107 127)(27 59 108 124)(28 64 109 121)(29 61 110 126)(30 58 111 123)(31 63 112 128)(32 60 105 125)(33 69 85 50)(34 66 86 55)(35 71 87 52)(36 68 88 49)(37 65 81 54)(38 70 82 51)(39 67 83 56)(40 72 84 53)

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

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

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

Matrix representation of Q165Q8 in GL6(𝔽17)

1600000
0160000
000020
000009
0015000
000800
,
100000
010000
000100
0016000
0000016
000010
,
420000
0130000
001000
0001600
000010
0000016
,
15100000
820000
000010
000001
001000
000100

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0,8,0,0,2,0,0,0,0,0,0,9,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[4,0,0,0,0,0,2,13,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[15,8,0,0,0,0,10,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

Q165Q8 in GAP, Magma, Sage, TeX

Q_{16}\rtimes_5Q_8
% in TeX

G:=Group("Q16:5Q8");
// GroupNames label

G:=SmallGroup(128,2122);
// by ID

G=gap.SmallGroup(128,2122);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,723,352,346,304,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^8=c^4=1,b^2=a^4,d^2=c^2,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^5,c*b*c^-1=d*b*d^-1=a^4*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Character table of Q165Q8 in TeX

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