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G = C8⋊Q16order 128 = 27

1st semidirect product of C8 and Q16 acting via Q16/C4=C22

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

Aliases: C81Q16, C42.243C23, C4⋊C4.65D4, (C2×C8).95D4, (C2×Q8).58D4, C4.42(C2×Q16), C83Q8.1C2, C84Q8.2C2, C81C8.11C2, C4⋊C8.31C22, C4.Q16.6C2, C42Q16.7C2, C4⋊Q8.64C22, C2.11(C8⋊D4), C4.43(C8⋊C22), (C4×C8).146C22, C2.9(C42Q16), C4.10D8.7C2, C4.6Q16.5C2, (C4×Q8).46C22, C4.73(C8.C22), C2.19(D4.3D4), C22.204(C4⋊D4), (C2×C4).28(C4○D4), (C2×C4).1278(C2×D4), SmallGroup(128,424)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C8⋊Q16
C1C2C22C2×C4C42C4×Q8C84Q8 — C8⋊Q16
C1C22C42 — C8⋊Q16
C1C22C42 — C8⋊Q16
C1C22C22C42 — C8⋊Q16

Generators and relations for C8⋊Q16
 G = < a,b,c | a8=b8=1, c2=b4, bab-1=a-1, cac-1=a5, cbc-1=b-1 >

Subgroups: 152 in 73 conjugacy classes, 34 normal (32 characteristic)
C1, C2 [×3], C4 [×4], C4 [×5], C22, C8 [×2], C8 [×4], C2×C4 [×3], C2×C4 [×4], Q8 [×5], C42, C42, C4⋊C4, C4⋊C4 [×5], C2×C8 [×2], C2×C8 [×3], Q16 [×2], C2×Q8, C2×Q8 [×2], C4×C8, C8⋊C4, Q8⋊C4 [×2], C4⋊C8 [×3], C4⋊C8, C4.Q8 [×2], C2.D8, C4×Q8, C4⋊Q8 [×2], C2×Q16, C4.10D8, C4.6Q16, C81C8, C84Q8, C42Q16, C4.Q16, C83Q8, C8⋊Q16
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, Q16 [×2], C2×D4 [×2], C4○D4, C4⋊D4, C2×Q16, C8⋊C22, C8.C22 [×2], C42Q16, C8⋊D4, D4.3D4, C8⋊Q16

Character table of C8⋊Q16

 class 12A2B2C4A4B4C4D4E4F4G4H4I8A8B8C8D8E8F8G8H8I8J
 size 1111222248816164444888888
ρ111111111111111111111111    trivial
ρ211111111111-11-1-1-1-11-11-1-1-1    linear of order 2
ρ3111111111-1-1-11-1-1-1-1-11-1111    linear of order 2
ρ4111111111-1-1111111-1-1-1-1-1-1    linear of order 2
ρ511111111111-1-11111-1-1-111-1    linear of order 2
ρ6111111111111-1-1-1-1-1-11-1-1-11    linear of order 2
ρ7111111111-1-11-1-1-1-1-11-1111-1    linear of order 2
ρ8111111111-1-1-1-11111111-1-11    linear of order 2
ρ92222-22-22-22-2000000000000    orthogonal lifted from D4
ρ102222-22-22-2-22000000000000    orthogonal lifted from D4
ρ1122222-22-2-200002-2-22000000    orthogonal lifted from D4
ρ1222222-22-2-20000-222-2000000    orthogonal lifted from D4
ρ132-2-22-20200000002-202-2-2002    symplectic lifted from Q16, Schur index 2
ρ142-2-22-2020000000-220-2-22002    symplectic lifted from Q16, Schur index 2
ρ152-2-22-20200000002-20-22200-2    symplectic lifted from Q16, Schur index 2
ρ162-2-22-2020000000-22022-200-2    symplectic lifted from Q16, Schur index 2
ρ172222-2-2-2-22000000000002i-2i0    complex lifted from C4○D4
ρ182222-2-2-2-2200000000000-2i2i0    complex lifted from C4○D4
ρ194-44-4040-4000000000000000    orthogonal lifted from C8⋊C22
ρ204-4-4440-40000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ214-44-40-404000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2244-4-40000000002-200-2-2000000    complex lifted from D4.3D4
ρ2344-4-4000000000-2-2002-2000000    complex lifted from D4.3D4

Smallest permutation representation of C8⋊Q16
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 41 76 106 49 12 57 96)(2 48 77 105 50 11 58 95)(3 47 78 112 51 10 59 94)(4 46 79 111 52 9 60 93)(5 45 80 110 53 16 61 92)(6 44 73 109 54 15 62 91)(7 43 74 108 55 14 63 90)(8 42 75 107 56 13 64 89)(17 121 72 86 32 37 99 115)(18 128 65 85 25 36 100 114)(19 127 66 84 26 35 101 113)(20 126 67 83 27 34 102 120)(21 125 68 82 28 33 103 119)(22 124 69 81 29 40 104 118)(23 123 70 88 30 39 97 117)(24 122 71 87 31 38 98 116)
(1 23 49 30)(2 20 50 27)(3 17 51 32)(4 22 52 29)(5 19 53 26)(6 24 54 31)(7 21 55 28)(8 18 56 25)(9 81 46 118)(10 86 47 115)(11 83 48 120)(12 88 41 117)(13 85 42 114)(14 82 43 119)(15 87 44 116)(16 84 45 113)(33 90 125 108)(34 95 126 105)(35 92 127 110)(36 89 128 107)(37 94 121 112)(38 91 122 109)(39 96 123 106)(40 93 124 111)(57 70 76 97)(58 67 77 102)(59 72 78 99)(60 69 79 104)(61 66 80 101)(62 71 73 98)(63 68 74 103)(64 65 75 100)

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

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

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

Matrix representation of C8⋊Q16 in GL8(𝔽17)

014890000
30990000
980140000
88300000
000000012
00000050
000001007
000070100
,
1111250000
6115150000
1226110000
25660000
000024126
00001321112
00001131513
00001411415
,
00100000
00010000
160000000
016000000
000014530
000053014
000011035
000006514

G:=sub<GL(8,GF(17))| [0,3,9,8,0,0,0,0,14,0,8,8,0,0,0,0,8,9,0,3,0,0,0,0,9,9,14,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,10,0,0,0,0,0,0,5,0,10,0,0,0,0,12,0,7,0],[11,6,12,2,0,0,0,0,11,11,2,5,0,0,0,0,2,5,6,6,0,0,0,0,5,15,11,6,0,0,0,0,0,0,0,0,2,13,11,14,0,0,0,0,4,2,3,11,0,0,0,0,12,11,15,4,0,0,0,0,6,12,13,15],[0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,14,5,11,0,0,0,0,0,5,3,0,6,0,0,0,0,3,0,3,5,0,0,0,0,0,14,5,14] >;

C8⋊Q16 in GAP, Magma, Sage, TeX

C_8\rtimes Q_{16}
% in TeX

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

G:=SmallGroup(128,424);
// by ID

G=gap.SmallGroup(128,424);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,64,422,387,352,1123,136,2804,718,172]);
// Polycyclic

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

Export

Character table of C8⋊Q16 in TeX

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