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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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C8 — C8.14SD16
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C2×C16 — C16⋊5C4 — C8.14SD16
 Lower central C1 — C2 — C4 — C2×C8 — C8.14SD16
 Upper central C1 — C22 — C42 — C4×C8 — C8.14SD16
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×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 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 8A 8B 8C 8D 8E 8F 16A 16B 16C 16D 16E 16F 16G 16H size 1 1 1 1 2 2 4 4 16 16 16 16 2 2 2 2 4 4 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 -1 -1 -1 1 1 -1 1 1 1 1 -1 -1 -1 1 -1 -1 -1 1 1 1 linear of order 2 ρ3 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 1 -1 1 1 1 -1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 1 -1 1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ5 1 1 1 1 1 1 1 1 -1 1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ6 1 1 1 1 1 1 -1 -1 1 1 -1 -1 1 1 1 1 -1 -1 1 -1 1 1 1 -1 -1 -1 linear of order 2 ρ7 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ8 1 1 1 1 1 1 -1 -1 1 -1 -1 1 1 1 1 1 -1 -1 -1 1 -1 -1 -1 1 1 1 linear of order 2 ρ9 2 2 2 2 2 2 2 2 0 0 0 0 -2 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 2 2 -2 -2 0 0 0 0 -2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 -√2 -√2 √2 -√2 √2 √2 -√2 √2 orthogonal lifted from D8 ρ12 2 2 2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 √2 -√2 -√2 √2 -√2 √2 -√2 √2 orthogonal lifted from D8 ρ13 2 2 2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 -√2 √2 √2 -√2 √2 -√2 √2 -√2 orthogonal lifted from D8 ρ14 2 2 2 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 √2 √2 -√2 √2 -√2 -√2 √2 -√2 orthogonal lifted from D8 ρ15 2 -2 2 -2 2 -2 0 0 0 0 0 0 2 -2 2 -2 0 0 -2i 0 -2i 2i 2i 0 0 0 complex lifted from C4○D4 ρ16 2 -2 2 -2 2 -2 0 0 0 0 0 0 -2 2 -2 2 0 0 0 2i 0 0 0 -2i -2i 2i complex lifted from C4○D4 ρ17 2 -2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 -2 2 -√-2 -√-2 √-2 √-2 -√-2 -√-2 √-2 √-2 complex lifted from SD16 ρ18 2 -2 2 -2 2 -2 0 0 0 0 0 0 -2 2 -2 2 0 0 0 -2i 0 0 0 2i 2i -2i complex lifted from C4○D4 ρ19 2 -2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 -2 2 √-2 √-2 -√-2 -√-2 √-2 √-2 -√-2 -√-2 complex lifted from SD16 ρ20 2 -2 2 -2 2 -2 0 0 0 0 0 0 2 -2 2 -2 0 0 2i 0 2i -2i -2i 0 0 0 complex lifted from C4○D4 ρ21 2 -2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 2 -2 -√-2 √-2 √-2 √-2 -√-2 √-2 -√-2 -√-2 complex lifted from SD16 ρ22 2 -2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 2 -2 √-2 -√-2 -√-2 -√-2 √-2 -√-2 √-2 √-2 complex lifted from SD16 ρ23 4 -4 -4 4 0 0 0 0 0 0 0 0 -2√2 2√2 2√2 -2√2 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q32⋊C2, Schur index 2 ρ24 4 4 -4 -4 0 0 0 0 0 0 0 0 -2√2 -2√2 2√2 2√2 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q32⋊C2, Schur index 2 ρ25 4 -4 -4 4 0 0 0 0 0 0 0 0 2√2 -2√2 -2√2 2√2 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q32⋊C2, Schur index 2 ρ26 4 4 -4 -4 0 0 0 0 0 0 0 0 2√2 2√2 -2√2 -2√2 0 0 0 0 0 0 0 0 0 0 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)

 6 9 0 0 0 0 11 11 0 0 0 0 0 0 11 13 9 3 0 0 10 6 7 1 0 0 0 0 14 9 0 0 1 5 8 3
,
 7 2 0 0 0 0 10 10 0 0 0 0 0 0 12 4 0 15 0 0 11 15 10 5 0 0 8 15 9 11 0 0 9 0 3 15
,
 11 14 0 0 0 0 6 6 0 0 0 0 0 0 2 0 2 16 0 0 9 14 7 7 0 0 2 6 15 11 0 0 9 12 0 3

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

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