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G = C8.27D8  order 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, C8⋊1C8.2C2, (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

Derived series
C1 — C2×C8 — C8.27D8
Chief series
C1 — C2 — C4 — C2×C4 — C42 — C4×C8 — C4⋊Q16 — C8.27D8
Lower central
C1 — C2 — C2×C4 — C2×C8 — C8.27D8
Upper central
C1 — C22 — C42 — C4×C8 — C8.27D8
Jennings
C1 — C2 — C2 — C2 — C2 — C2×C4 — C2×C4 — C4×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 >

C1
C2
C2
C2
C22
C4
C4
C4
C4
2C4
8C4
8C4
C2×C4
C2×C4
C8
C8
C2×C4
2C8
4C2×C4
4Q8
4Q8
4C2×C4
4Q8
4Q8
8C8
C2×C8
C2×C8
C42
2C2×Q8
2C2×Q8
4C4⋊C4
4Q16
4Q16
4C16
4Q16
4C4⋊C4
4C2×C8
4Q16
C4×C8
C2×Q16
C2×Q16
2C4⋊Q8
2C2×C16
2C4⋊C8
2C2×Q16
C4⋊C16
C4⋊Q16
C8⋊1C8
C8.27D8

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
ρ112-22-20-2200002-2-2200√2-√2-√2√200000000    orthogonal lifted from D8
ρ1222222-2-22-2000000000000√2√2√2-√2-√2√2-√2-√2    orthogonal lifted from D8
ρ132-22-20-2200002-2-2200-√2√2√2-√200000000    orthogonal lifted from D8
ρ1422222-2-22-2000000000000-√2-√2-√2√2√2-√2√2√2    orthogonal lifted from D8
ρ152-2-22-2002000-√2√2-√2√2-√2√20000ζ167-ζ16ζ167-ζ16-ζ167+ζ16ζ165-ζ163ζ165-ζ163-ζ167+ζ16-ζ165+ζ163-ζ165+ζ163    symplectic lifted from Q32, Schur index 2
ρ162-2-22-2002000-√2√2-√2√2-√2√20000-ζ167+ζ16-ζ167+ζ16ζ167-ζ16-ζ165+ζ163-ζ165+ζ163ζ167-ζ16ζ165-ζ163ζ165-ζ163    symplectic lifted from Q32, Schur index 2
ρ172-2-22-2002000√2-√2√2-√2√2-√20000-ζ165+ζ163-ζ165+ζ163ζ165-ζ163ζ167-ζ16ζ167-ζ16ζ165-ζ163-ζ167+ζ16-ζ167+ζ16    symplectic lifted from Q32, Schur index 2
ρ182-2-22-2002000√2-√2√2-√2√2-√20000ζ165-ζ163ζ165-ζ163-ζ165+ζ163-ζ167+ζ16-ζ167+ζ16-ζ165+ζ163ζ167-ζ16ζ167-ζ16    symplectic lifted from Q32, Schur index 2
ρ192-2-22200-2000-√2√2-√2√2√2-√20000ζ167+ζ16ζ1615+ζ169ζ1615+ζ169ζ1613+ζ1611ζ165+ζ163ζ167+ζ16ζ1613+ζ1611ζ165+ζ163    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
ρ232-2-22200-2000√2-√2√2-√2-√2√20000ζ165+ζ163ζ1613+ζ1611ζ1613+ζ1611ζ167+ζ16ζ1615+ζ169ζ165+ζ163ζ167+ζ16ζ1615+ζ169    complex lifted from SD32
ρ242-2-22200-2000√2-√2√2-√2-√2√20000ζ1613+ζ1611ζ165+ζ163ζ165+ζ163ζ1615+ζ169ζ167+ζ16ζ1613+ζ1611ζ1615+ζ169ζ167+ζ16    complex lifted from SD32
ρ252-2-22200-2000-√2√2-√2√2√2-√20000ζ1615+ζ169ζ167+ζ16ζ167+ζ16ζ165+ζ163ζ1613+ζ1611ζ1615+ζ169ζ165+ζ163ζ1613+ζ1611    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
ρ2844-4-40000000-2√2-2√22√22√200000000000000    symplectic lifted from C8.17D4, Schur index 2
ρ2944-4-400000002√22√2-2√2-2√200000000000000    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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