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G = C83Q16order 128 = 27

3rd semidirect product of C8 and Q16 acting via Q16/C4=C22

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

Aliases: C83Q16, C42.670C23, (C2×C8).39D4, C4.4(C2×Q16), C8⋊C8.6C2, C83Q8.2C2, C4.7(C8⋊C22), C4⋊Q16.7C2, C82Q8.10C2, C4⋊Q8.94C22, C2.12(C83D4), (C4×C8).156C22, C2.7(C4⋊Q16), C4.7(C8.C22), C4.SD16.7C2, C2.12(C8.2D4), C22.71(C41D4), (C2×C4).727(C2×D4), SmallGroup(128,455)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C83Q16
C1C2C22C2×C4C42C4×C8C8⋊C8 — C83Q16
C1C22C42 — C83Q16
C1C22C42 — C83Q16
C1C22C22C42 — C83Q16

Generators and relations for C83Q16
 G = < a,b,c | a8=b8=1, c2=b4, bab-1=a5, cac-1=a3, cbc-1=b-1 >

Subgroups: 192 in 88 conjugacy classes, 40 normal (24 characteristic)
C1, C2 [×3], C4 [×6], C4 [×4], C22, C8 [×4], C8 [×4], C2×C4 [×3], C2×C4 [×4], Q8 [×8], C42, C4⋊C4 [×8], C2×C8 [×6], Q16 [×4], C2×Q8 [×4], C4×C8 [×3], Q8⋊C4 [×4], C4.Q8 [×2], C2.D8 [×4], C4⋊Q8 [×4], C2×Q16 [×2], C8⋊C8, C4.SD16 [×2], C4⋊Q16, C83Q8, C82Q8 [×2], C83Q16
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, Q16 [×4], C2×D4 [×3], C41D4, C2×Q16 [×2], C8⋊C22 [×2], C8.C22 [×2], C4⋊Q16, C83D4, C8.2D4, C83Q16

Character table of C83Q16

 class 12A2B2C4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E8F8G8H8I8J8K8L
 size 111122222216161616444444444444
ρ111111111111111111111111111    trivial
ρ211111111111-11-1-1-1-1-11-1-11-1-111    linear of order 2
ρ31111111111-1-11111-1-1-11-1-11-1-1-1    linear of order 2
ρ41111111111-111-1-1-111-1-11-1-11-1-1    linear of order 2
ρ51111111111-1-1-1-1111111111111    linear of order 2
ρ61111111111-11-11-1-1-1-11-1-11-1-111    linear of order 2
ρ7111111111111-1-111-1-1-11-1-11-1-1-1    linear of order 2
ρ811111111111-1-11-1-111-1-11-1-11-1-1    linear of order 2
ρ92222-22-2-22-200000000200-200-22    orthogonal lifted from D4
ρ1022222-22-2-2-200002-2000-2002000    orthogonal lifted from D4
ρ112222-2-2-22-22000000-2-200200200    orthogonal lifted from D4
ρ122222-2-2-22-220000002200-200-200    orthogonal lifted from D4
ρ132222-22-2-22-200000000-2002002-2    orthogonal lifted from D4
ρ1422222-22-2-2-20000-22000200-2000    orthogonal lifted from D4
ρ152-22-20200-200000-22-222-2-20220-2    symplectic lifted from Q16, Schur index 2
ρ162-22-20200-2000002-2-22-22-20-2202    symplectic lifted from Q16, Schur index 2
ρ172-22-20-200200000-2-2-22022-22-220    symplectic lifted from Q16, Schur index 2
ρ182-22-20-200200000-2-22-202-2222-20    symplectic lifted from Q16, Schur index 2
ρ192-22-20200-200000-222-2-2-2202-202    symplectic lifted from Q16, Schur index 2
ρ202-22-20200-2000002-22-22220-2-20-2    symplectic lifted from Q16, Schur index 2
ρ212-22-20-20020000022-220-222-2-2-20    symplectic lifted from Q16, Schur index 2
ρ222-22-20-200200000222-20-2-2-2-2220    symplectic lifted from Q16, Schur index 2
ρ2344-4-4000-4040000000000000000    orthogonal lifted from C8⋊C22
ρ2444-4-400040-40000000000000000    orthogonal lifted from C8⋊C22
ρ254-4-4440-40000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ264-4-44-4040000000000000000000    symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of C83Q16
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 63 18 67 74 25 54 98)(2 60 19 72 75 30 55 103)(3 57 20 69 76 27 56 100)(4 62 21 66 77 32 49 97)(5 59 22 71 78 29 50 102)(6 64 23 68 79 26 51 99)(7 61 24 65 80 31 52 104)(8 58 17 70 73 28 53 101)(9 106 119 35 42 91 84 125)(10 111 120 40 43 96 85 122)(11 108 113 37 44 93 86 127)(12 105 114 34 45 90 87 124)(13 110 115 39 46 95 88 121)(14 107 116 36 47 92 81 126)(15 112 117 33 48 89 82 123)(16 109 118 38 41 94 83 128)
(1 37 74 127)(2 40 75 122)(3 35 76 125)(4 38 77 128)(5 33 78 123)(6 36 79 126)(7 39 80 121)(8 34 73 124)(9 100 42 69)(10 103 43 72)(11 98 44 67)(12 101 45 70)(13 104 46 65)(14 99 47 68)(15 102 48 71)(16 97 41 66)(17 105 53 90)(18 108 54 93)(19 111 55 96)(20 106 56 91)(21 109 49 94)(22 112 50 89)(23 107 51 92)(24 110 52 95)(25 86 63 113)(26 81 64 116)(27 84 57 119)(28 87 58 114)(29 82 59 117)(30 85 60 120)(31 88 61 115)(32 83 62 118)

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

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

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

Matrix representation of C83Q16 in GL6(𝔽17)

010000
1600000
0069414
0086112
005121
0031583
,
1430000
14140000
0014310
001414115
000006
0000146
,
170000
7160000
00110100
00101673
0000914
0000168

G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,6,8,5,3,0,0,9,6,1,15,0,0,4,1,2,8,0,0,14,12,1,3],[14,14,0,0,0,0,3,14,0,0,0,0,0,0,14,14,0,0,0,0,3,14,0,0,0,0,1,1,0,14,0,0,0,15,6,6],[1,7,0,0,0,0,7,16,0,0,0,0,0,0,1,10,0,0,0,0,10,16,0,0,0,0,10,7,9,16,0,0,0,3,14,8] >;

C83Q16 in GAP, Magma, Sage, TeX

C_8\rtimes_3Q_{16}
% in TeX

G:=Group("C8:3Q16");
// GroupNames label

G:=SmallGroup(128,455);
// by ID

G=gap.SmallGroup(128,455);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,64,422,387,436,1123,136,2804,172]);
// Polycyclic

G:=Group<a,b,c|a^8=b^8=1,c^2=b^4,b*a*b^-1=a^5,c*a*c^-1=a^3,c*b*c^-1=b^-1>;
// generators/relations

Export

Character table of C83Q16 in TeX

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