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G = C8.16Q16  order 128 = 27

4th non-split extension by C8 of Q16 acting via Q16/Q8=C2

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

Aliases: C8.16Q16, C8.27SD16, C42.38D4, C4⋊C16.4C2, C8⋊2C8.1C2, C2.D8.3C4, (C2×C8).360D4, (C2×C4).105D8, (C2×C4).18SD16, C8.5Q8.1C2, C2.7(D8⋊2C4), (C4×C8).100C22, C4.1(Q8⋊C4), C2.7(D8.C4), C2.3(C4.10D8), C4.1(C4.10D4), C22.62(D4⋊C4), (C2×C8).24(C2×C4), (C2×C4).224(C22⋊C4), SmallGroup(128,95)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C8 — C8.16Q16
Chief series
C1 — C2 — C4 — C2×C4 — C2×C8 — C4×C8 — C8.5Q8 — C8.16Q16
Lower central
C1 — C2 — C2×C4 — C2×C8 — C8.16Q16
Upper central
C1 — C22 — C42 — C4×C8 — C8.16Q16
Jennings
C1 — C2 — C2 — C2 — C2 — C2×C4 — C2×C4 — C4×C8 — C8.16Q16

Generators and relations for C8.16Q16
 G = < a,b,c | a8=b8=1, c2=a4b4, bab-1=cac-1=a3, cbc-1=ab-1 >

C1
C2
C2
C2
C4
C22
C4
2C4
2C4
8C4
8C4
C8
C2×C4
C8
C2×C4
C2×C4
2C8
4C2×C4
4C2×C4
8C8
C2×C8
C2×C8
C42
2C4⋊C4
2C4⋊C4
4C16
4C4⋊C4
4C2×C8
4C4⋊C4
C2.D8
C4×C8
C2.D8
2C4⋊C8
2C2×C16
2C42.C2
2C4.Q8
C4⋊C16
C8⋊2C8
C8.5Q8
C8.16Q16

Character table of C8.16Q16

 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
ρ51111-1-111-11-1-1-1-1-111ii-i-ii-i-iii-ii-i    linear of order 4
ρ61111-1-111-1-11-1-1-1-111ii-i-i-iii-i-ii-ii    linear of order 4
ρ71111-1-111-1-11-1-1-1-111-i-iiii-i-iii-ii-i    linear of order 4
ρ81111-1-111-11-1-1-1-1-111-i-iii-iii-i-ii-ii    linear of order 4
ρ92222-2-222-2002222-2-2000000000000    orthogonal lifted from D4
ρ1022222222200-2-2-2-2-2-2000000000000    orthogonal lifted from D4
ρ11222222-2-2-2000000000000√2-√2√2-√2√2√2-√2-√2    orthogonal lifted from D8
ρ12222222-2-2-2000000000000-√2√2-√2√2-√2-√2√2√2    orthogonal lifted from D8
ρ132-2-22002-200022-2-200√2-√2√2-√200000000    symplectic lifted from Q16, Schur index 2
ρ142-2-22002-200022-2-200-√2√2-√2√200000000    symplectic lifted from Q16, Schur index 2
ρ152222-2-2-2-22000000000000-√-2-√-2√-2√-2-√-2√-2√-2-√-2    complex lifted from SD16
ρ162-2-22002-2000-2-22200-√-2√-2√-2-√-200000000    complex lifted from SD16
ρ172-2-22002-2000-2-22200√-2-√-2-√-2√-200000000    complex lifted from SD16
ρ182222-2-2-2-22000000000000√-2√-2-√-2-√-2√-2-√-2-√-2√-2    complex lifted from SD16
ρ1922-2-22i-2i00000-√-2√-2-√-2√-2√2-√20000ζ167+ζ165ζ1613+ζ167ζ1611+ζ169ζ1611+ζ16ζ1615+ζ1613ζ163+ζ16ζ169+ζ163ζ1615+ζ165    complex lifted from D8.C4
ρ2022-2-2-2i2i00000√-2-√-2√-2-√-2√2-√20000ζ163+ζ16ζ1611+ζ16ζ1615+ζ1613ζ1613+ζ167ζ1611+ζ169ζ167+ζ165ζ1615+ζ165ζ169+ζ163    complex lifted from D8.C4
ρ2122-2-22i-2i00000√-2-√-2√-2-√-2-√2√20000ζ1611+ζ16ζ1611+ζ169ζ1615+ζ165ζ1615+ζ1613ζ169+ζ163ζ1613+ζ167ζ167+ζ165ζ163+ζ16    complex lifted from D8.C4
ρ2222-2-22i-2i00000-√-2√-2-√-2√-2√2-√20000ζ1615+ζ1613ζ1615+ζ165ζ163+ζ16ζ169+ζ163ζ167+ζ165ζ1611+ζ169ζ1611+ζ16ζ1613+ζ167    complex lifted from D8.C4
ρ2322-2-2-2i2i00000-√-2√-2-√-2√-2-√2√20000ζ1615+ζ165ζ167+ζ165ζ1611+ζ16ζ163+ζ16ζ1613+ζ167ζ169+ζ163ζ1611+ζ169ζ1615+ζ1613    complex lifted from D8.C4
ρ2422-2-22i-2i00000√-2-√-2√-2-√-2-√2√20000ζ169+ζ163ζ163+ζ16ζ1613+ζ167ζ167+ζ165ζ1611+ζ16ζ1615+ζ165ζ1615+ζ1613ζ1611+ζ169    complex lifted from D8.C4
ρ2522-2-2-2i2i00000√-2-√-2√-2-√-2√2-√20000ζ1611+ζ169ζ169+ζ163ζ167+ζ165ζ1615+ζ165ζ163+ζ16ζ1615+ζ1613ζ1613+ζ167ζ1611+ζ16    complex lifted from D8.C4
ρ2622-2-2-2i2i00000-√-2√-2-√-2√-2-√2√20000ζ1613+ζ167ζ1615+ζ1613ζ169+ζ163ζ1611+ζ169ζ1615+ζ165ζ1611+ζ16ζ163+ζ16ζ167+ζ165    complex lifted from D8.C4
ρ274-4-4400-44000000000000000000000    symplectic lifted from C4.10D4, Schur index 2
ρ284-44-40000000-2√-22√-22√-2-2√-200000000000000    complex lifted from D8⋊2C4
ρ294-44-400000002√-2-2√-2-2√-22√-200000000000000    complex lifted from D8⋊2C4

Smallest permutation representation of C8.16Q16
►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)

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

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

Matrix representation of C8.16Q16 ►in GL4(𝔽17) generated by

121200
51200
0010
0001
,
01500
15000
0080
00515
,
131100
11400
00115
00116
G:=sub<GL(4,GF(17))| [12,5,0,0,12,12,0,0,0,0,1,0,0,0,0,1],[0,15,0,0,15,0,0,0,0,0,8,5,0,0,0,15],[13,11,0,0,11,4,0,0,0,0,1,1,0,0,15,16] >;
 

C8.16Q16 in GAP, Magma, Sage, TeX 

C_8._{16}Q_{16}
 
% in TeX
 
G:=Group("C8.16Q16");
 
// GroupNames label
 
G:=SmallGroup(128,95);
 
// by ID
 
G=gap.SmallGroup(128,95);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,120,422,387,520,794,192,2028,124]);
 
// Polycyclic
 
G:=Group<a,b,c|a^8=b^8=1,c^2=a^4*b^4,b*a*b^-1=c*a*c^-1=a^3,c*b*c^-1=a*b^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C8.16Q16 in TeX
Character table of C8.16Q16 in TeX

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