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G = C8.14SD16order 128 = 27

14th non-split extension by C8 of SD16 acting via SD16/C4=C22

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

Aliases: C8.14SD16, C42.155D4, (C2×C4).47D8, (C2×C8).137D4, C8.55(C4○D4), C165C4.10C2, C4.19(C2×SD16), C8.5Q8.6C2, (C2×C16).60C22, (C4×C8).169C22, (C2×C8).543C23, C2.Q32.7C2, C4⋊Q16.15C2, C22.129(C2×D8), C2.D8.28C22, C4.12(C4.4D4), C2.21(Q32⋊C2), C2.14(C4.4D8), (C2×Q16).16C22, (C2×C4).811(C2×D4), SmallGroup(128,977)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — C8.14SD16
C1C2C4C8C2×C8C2×C16C165C4 — C8.14SD16
C1C2C4C2×C8 — C8.14SD16
C1C22C42C4×C8 — C8.14SD16
C1C2C2C2C2C4C4C2×C8 — C8.14SD16

Generators and relations for C8.14SD16
 G = < a,b,c | a8=1, b8=c2=a4, bab-1=a5, cac-1=a-1, cbc-1=a6b3 >

Subgroups: 152 in 66 conjugacy classes, 32 normal (14 characteristic)
C1, C2, C2 [×2], C4 [×2], C4 [×6], C22, C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×4], C16 [×2], C42, C4⋊C4 [×6], C2×C8 [×2], Q16 [×6], C2×Q8 [×2], C4×C8, C4.Q8, C2.D8 [×2], C2×C16 [×2], C42.C2, C4⋊Q8, C2×Q16 [×2], C2×Q16, C165C4, C2.Q32 [×4], C4⋊Q16, C8.5Q8, C8.14SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D8 [×2], SD16 [×2], C2×D4, C4○D4 [×2], C4.4D4, C2×D8, C2×SD16, C4.4D8, Q32⋊C2 [×2], C8.14SD16

Character table of C8.14SD16

 class 12A2B2C4A4B4C4D4E4F4G4H8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 111122441616161622224444444444
ρ111111111111111111111111111    trivial
ρ2111111-1-1-111-11111-1-1-11-1-1-1111    linear of order 2
ρ3111111-1-1-1-1111111-1-11-1111-1-1-1    linear of order 2
ρ4111111111-11-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ511111111-11-11111111-1-1-1-1-1-1-1-1    linear of order 2
ρ6111111-1-111-1-11111-1-11-1111-1-1-1    linear of order 2
ρ711111111-1-1-1-111111111111111    linear of order 2
ρ8111111-1-11-1-111111-1-1-11-1-1-1111    linear of order 2
ρ9222222220000-2-2-2-2-2-200000000    orthogonal lifted from D4
ρ10222222-2-20000-2-2-2-22200000000    orthogonal lifted from D4
ρ112222-2-22-20000000000-2-22-222-22    orthogonal lifted from D8
ρ122222-2-2-2200000000002-2-22-22-22    orthogonal lifted from D8
ρ132222-2-2-220000000000-222-22-22-2    orthogonal lifted from D8
ρ142222-2-22-2000000000022-22-2-22-2    orthogonal lifted from D8
ρ152-22-22-20000002-22-200-2i0-2i2i2i000    complex lifted from C4○D4
ρ162-22-22-2000000-22-220002i000-2i-2i2i    complex lifted from C4○D4
ρ172-22-2-220000000000-22--2--2-2-2--2--2-2-2    complex lifted from SD16
ρ182-22-22-2000000-22-22000-2i0002i2i-2i    complex lifted from C4○D4
ρ192-22-2-220000000000-22-2-2--2--2-2-2--2--2    complex lifted from SD16
ρ202-22-22-20000002-22-2002i02i-2i-2i000    complex lifted from C4○D4
ρ212-22-2-2200000000002-2--2-2-2-2--2-2--2--2    complex lifted from SD16
ρ222-22-2-2200000000002-2-2--2--2--2-2--2-2-2    complex lifted from SD16
ρ234-4-4400000000-222222-220000000000    symplectic lifted from Q32⋊C2, Schur index 2
ρ2444-4-400000000-22-2222220000000000    symplectic lifted from Q32⋊C2, Schur index 2
ρ254-4-440000000022-22-22220000000000    symplectic lifted from Q32⋊C2, Schur index 2
ρ2644-4-4000000002222-22-220000000000    symplectic lifted from Q32⋊C2, Schur index 2

Smallest permutation representation of C8.14SD16
Regular action on 128 points
Generators in S128
(1 19 47 101 9 27 39 109)(2 28 48 110 10 20 40 102)(3 21 33 103 11 29 41 111)(4 30 34 112 12 22 42 104)(5 23 35 105 13 31 43 97)(6 32 36 98 14 24 44 106)(7 25 37 107 15 17 45 99)(8 18 38 100 16 26 46 108)(49 65 85 119 57 73 93 127)(50 74 86 128 58 66 94 120)(51 67 87 121 59 75 95 113)(52 76 88 114 60 68 96 122)(53 69 89 123 61 77 81 115)(54 78 90 116 62 70 82 124)(55 71 91 125 63 79 83 117)(56 80 92 118 64 72 84 126)
(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 9 55)(2 94 10 86)(3 61 11 53)(4 92 12 84)(5 59 13 51)(6 90 14 82)(7 57 15 49)(8 88 16 96)(17 127 25 119)(18 76 26 68)(19 125 27 117)(20 74 28 66)(21 123 29 115)(22 72 30 80)(23 121 31 113)(24 70 32 78)(33 89 41 81)(34 56 42 64)(35 87 43 95)(36 54 44 62)(37 85 45 93)(38 52 46 60)(39 83 47 91)(40 50 48 58)(65 99 73 107)(67 97 75 105)(69 111 77 103)(71 109 79 101)(98 124 106 116)(100 122 108 114)(102 120 110 128)(104 118 112 126)

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

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

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

Matrix representation of C8.14SD16 in GL6(𝔽17)

690000
11110000
00111393
0010671
0000149
001583
,
720000
10100000
00124015
001115105
00815911
0090315
,
11140000
660000
0020216
0091477
00261511
0091203

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

C8.14SD16 in GAP, Magma, Sage, TeX

C_8._{14}{\rm SD}_{16}
% in TeX

G:=Group("C8.14SD16");
// GroupNames label

G:=SmallGroup(128,977);
// by ID

G=gap.SmallGroup(128,977);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,568,422,723,58,1123,360,3924,102,4037,124]);
// Polycyclic

G:=Group<a,b,c|a^8=1,b^8=c^2=a^4,b*a*b^-1=a^5,c*a*c^-1=a^-1,c*b*c^-1=a^6*b^3>;
// generators/relations

Export

Character table of C8.14SD16 in TeX

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