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G = C8.27D8order 128 = 27

4th non-split extension by C8 of D8 acting via D8/D4=C2

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

Aliases: C8.27D8, C4.5Q32, C4.6SD32, C8.33SD16, C42.37D4, C4⋊C16.3C2, C81C8.2C2, (C2×C4).120D8, (C2×C8).334D4, (C2×Q16).3C4, (C4×C8).34C22, C4⋊Q16.1C2, (C2×C4).17SD16, C4.6(D4⋊C4), C2.8(C4.D8), C4.3(C4.D4), C2.4(C2.Q32), C2.4(C8.17D4), C22.61(D4⋊C4), (C2×C8).23(C2×C4), (C2×C4).223(C22⋊C4), SmallGroup(128,94)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — C8.27D8
C1C2C4C2×C4C42C4×C8C4⋊Q16 — C8.27D8
C1C2C2×C4C2×C8 — C8.27D8
C1C22C42C4×C8 — C8.27D8
C1C2C2C2C2C2×C4C2×C4C4×C8 — C8.27D8

Generators and relations for C8.27D8
 G = < a,b,c | a8=b8=1, c2=a5, bab-1=a-1, ac=ca, cbc-1=ab-1 >

2C4
8C4
8C4
2C8
4C2×C4
4Q8
4Q8
4C2×C4
4Q8
4Q8
8C8
2C2×Q8
2C2×Q8
4C4⋊C4
4Q16
4Q16
4C16
4Q16
4C4⋊C4
4C2×C8
4Q16
2C4⋊Q8
2C2×C16
2C4⋊C8
2C2×Q16

Character table of C8.27D8

 class 12A2B2C4A4B4C4D4E4F4G8A8B8C8D8E8F8G8H8I8J16A16B16C16D16E16F16G16H
 size 1111222241616222244888844444444
ρ111111111111111111111111111111    trivial
ρ2111111111-1-11111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ311111111111111111-1-1-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ4111111111-1-1111111-1-1-1-111111111    linear of order 2
ρ51111-111-1-1-11-1-1-1-111ii-i-ii-iii-i-i-ii    linear of order 4
ρ61111-111-1-11-1-1-1-1-111ii-i-i-ii-i-iiii-i    linear of order 4
ρ71111-111-1-11-1-1-1-1-111-i-iiii-iii-i-i-ii    linear of order 4
ρ81111-111-1-1-11-1-1-1-111-i-iii-ii-i-iiii-i    linear of order 4
ρ922222222200-2-2-2-2-2-2000000000000    orthogonal lifted from D4
ρ102222-222-2-2002222-2-2000000000000    orthogonal lifted from D4
ρ112-22-20-2200002-2-22002-2-2200000000    orthogonal lifted from D8
ρ1222222-2-22-2000000000000222-2-22-2-2    orthogonal lifted from D8
ρ132-22-20-2200002-2-2200-222-200000000    orthogonal lifted from D8
ρ1422222-2-22-2000000000000-2-2-222-222    orthogonal lifted from D8
ρ152-2-22-2002000-22-22-220000ζ16716ζ1671616716ζ165163ζ16516316716165163165163    symplectic lifted from Q32, Schur index 2
ρ162-2-22-2002000-22-22-2200001671616716ζ16716165163165163ζ16716ζ165163ζ165163    symplectic lifted from Q32, Schur index 2
ρ172-2-22-20020002-22-22-20000165163165163ζ165163ζ16716ζ16716ζ1651631671616716    symplectic lifted from Q32, Schur index 2
ρ182-2-22-20020002-22-22-20000ζ165163ζ1651631651631671616716165163ζ16716ζ16716    symplectic lifted from Q32, Schur index 2
ρ192-2-22200-2000-22-222-20000ζ16716ζ1615169ζ1615169ζ16131611ζ165163ζ16716ζ16131611ζ165163    complex lifted from SD32
ρ202-22-20-220000-222-200--2-2--2-200000000    complex lifted from SD16
ρ212-22-20-220000-222-200-2--2-2--200000000    complex lifted from SD16
ρ222222-2-2-2-22000000000000-2--2-2--2-2--2-2--2    complex lifted from SD16
ρ232-2-22200-20002-22-2-220000ζ165163ζ16131611ζ16131611ζ16716ζ1615169ζ165163ζ16716ζ1615169    complex lifted from SD32
ρ242-2-22200-20002-22-2-220000ζ16131611ζ165163ζ165163ζ1615169ζ16716ζ16131611ζ1615169ζ16716    complex lifted from SD32
ρ252-2-22200-2000-22-222-20000ζ1615169ζ16716ζ16716ζ165163ζ16131611ζ1615169ζ165163ζ16131611    complex lifted from SD32
ρ262222-2-2-2-22000000000000--2-2--2-2--2-2--2-2    complex lifted from SD16
ρ274-44-404-40000000000000000000000    orthogonal lifted from C4.D4
ρ2844-4-40000000-22-22222200000000000000    symplectic lifted from C8.17D4, Schur index 2
ρ2944-4-400000002222-22-2200000000000000    symplectic lifted from C8.17D4, Schur index 2

Smallest permutation representation of C8.27D8
Regular action on 128 points
Generators in S128
(1 11 5 15 9 3 13 7)(2 12 6 16 10 4 14 8)(17 27 21 31 25 19 29 23)(18 28 22 32 26 20 30 24)(33 43 37 47 41 35 45 39)(34 44 38 48 42 36 46 40)(49 59 53 63 57 51 61 55)(50 60 54 64 58 52 62 56)(65 75 69 79 73 67 77 71)(66 76 70 80 74 68 78 72)(81 91 85 95 89 83 93 87)(82 92 86 96 90 84 94 88)(97 107 101 111 105 99 109 103)(98 108 102 112 106 100 110 104)(113 123 117 127 121 115 125 119)(114 124 118 128 122 116 126 120)
(1 73 34 53 30 111 92 114)(2 121 93 102 31 60 35 80)(3 71 36 51 32 109 94 128)(4 119 95 100 17 58 37 78)(5 69 38 49 18 107 96 126)(6 117 81 98 19 56 39 76)(7 67 40 63 20 105 82 124)(8 115 83 112 21 54 41 74)(9 65 42 61 22 103 84 122)(10 113 85 110 23 52 43 72)(11 79 44 59 24 101 86 120)(12 127 87 108 25 50 45 70)(13 77 46 57 26 99 88 118)(14 125 89 106 27 64 47 68)(15 75 48 55 28 97 90 116)(16 123 91 104 29 62 33 66)
(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)

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

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

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

Matrix representation of C8.27D8 in GL4(𝔽17) generated by

16000
01600
001414
00314
,
15000
0900
00143
0033
,
0900
15000
00110
0071
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,14,3,0,0,14,14],[15,0,0,0,0,9,0,0,0,0,14,3,0,0,3,3],[0,15,0,0,9,0,0,0,0,0,1,7,0,0,10,1] >;

C8.27D8 in GAP, Magma, Sage, TeX

C_8._{27}D_8
% in TeX

G:=Group("C8.27D8");
// GroupNames label

G:=SmallGroup(128,94);
// by ID

G=gap.SmallGroup(128,94);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,456,422,387,520,1690,416,2804,1411,172,4037,2028,124]);
// Polycyclic

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

Export

Subgroup lattice of C8.27D8 in TeX
Character table of C8.27D8 in TeX

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