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G = C4.10D16  order 128 = 27

2nd non-split extension by C4 of D16 acting via D16/D8=C2

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

Aliases: C4.10D16, C8.13Q16, C8.24SD16, C4.12SD32, C42.39D4, C4⋊C16.5C2, C8⋊1C8.3C2, C2.D8.4C4, (C2×C4).121D8, (C2×C8).335D4, C8⋊2Q8.1C2, (C4×C8).35C22, (C2×C4).19SD16, C2.5(C2.D16), C4.2(Q8⋊C4), C2.5(C8.17D4), C4.2(C4.10D4), C2.4(C4.10D8), C22.63(D4⋊C4), (C2×C8).25(C2×C4), (C2×C4).225(C22⋊C4), SmallGroup(128,96)

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

Derived series
C1 — C2×C8 — C4.10D16
Chief series
C1 — C2 — C4 — C2×C4 — C2×C8 — C4×C8 — C8⋊2Q8 — C4.10D16
Lower central
C1 — C2 — C2×C4 — C2×C8 — C4.10D16
Upper central
C1 — C22 — C42 — C4×C8 — C4.10D16
Jennings
C1 — C2 — C2 — C2 — C2 — C2×C4 — C2×C4 — C4×C8 — C4.10D16

Generators and relations for C4.10D16
 G = < a,b,c | a4=b16=1, c2=bab-1=a-1, ac=ca, cbc-1=ab-1 >

C1
C2
C2
C2
C4
C22
C4
C4
C4
2C4
8C4
8C4
C8
C8
C2×C4
C2×C4
C2×C4
2C8
4C2×C4
4C2×C4
8Q8
8C8
8Q8
C2×C8
C42
C2×C8
2C4⋊C4
2C4⋊C4
4C2×Q8
4C2×C8
4C4⋊C4
4C16
C4×C8
C2.D8
C2.D8
2C4⋊C8
2C4⋊Q8
2C2.D8
2C2×C16
C4⋊C16
C8⋊1C8
C8⋊2Q8
C4.10D16

Character table of C4.10D16

 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
ρ1122-2-2-2200000√2-√2√2-√2√2-√20000ζ1615-ζ169ζ165-ζ163-ζ1615+ζ169ζ165-ζ163-ζ1615+ζ169ζ1615-ζ169-ζ165+ζ163-ζ165+ζ163    orthogonal lifted from D16
ρ1222-2-2-2200000-√2√2-√2√2-√2√20000-ζ165+ζ163ζ1615-ζ169ζ165-ζ163ζ1615-ζ169ζ165-ζ163-ζ165+ζ163-ζ1615+ζ169-ζ1615+ζ169    orthogonal lifted from D16
ρ13222222-2-2-2000000000000√2-√2√2-√2√2√2-√2-√2    orthogonal lifted from D8
ρ14222222-2-2-2000000000000-√2√2-√2√2-√2-√2√2√2    orthogonal lifted from D8
ρ1522-2-2-2200000-√2√2-√2√2-√2√20000ζ165-ζ163-ζ1615+ζ169-ζ165+ζ163-ζ1615+ζ169-ζ165+ζ163ζ165-ζ163ζ1615-ζ169ζ1615-ζ169    orthogonal lifted from D16
ρ1622-2-2-2200000√2-√2√2-√2√2-√20000-ζ1615+ζ169-ζ165+ζ163ζ1615-ζ169-ζ165+ζ163ζ1615-ζ169-ζ1615+ζ169ζ165-ζ163ζ165-ζ163    orthogonal lifted from D16
ρ172-2-22002-200022-2-200√2-√2√2-√200000000    symplectic lifted from Q16, Schur index 2
ρ182-2-22002-200022-2-200-√2√2-√2√200000000    symplectic lifted from Q16, Schur index 2
ρ192222-2-2-2-22000000000000-√-2-√-2√-2√-2-√-2√-2√-2-√-2    complex lifted from SD16
ρ202-2-22002-2000-2-22200-√-2√-2√-2-√-200000000    complex lifted from SD16
ρ212-2-22002-2000-2-22200√-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
ρ2322-2-22-200000-√2√2-√2√2√2-√20000ζ1613+ζ1611ζ1615+ζ169ζ1613+ζ1611ζ167+ζ16ζ165+ζ163ζ165+ζ163ζ1615+ζ169ζ167+ζ16    complex lifted from SD32
ρ2422-2-22-200000√2-√2√2-√2-√2√20000ζ1615+ζ169ζ165+ζ163ζ1615+ζ169ζ1613+ζ1611ζ167+ζ16ζ167+ζ16ζ165+ζ163ζ1613+ζ1611    complex lifted from SD32
ρ2522-2-22-200000-√2√2-√2√2√2-√20000ζ165+ζ163ζ167+ζ16ζ165+ζ163ζ1615+ζ169ζ1613+ζ1611ζ1613+ζ1611ζ167+ζ16ζ1615+ζ169    complex lifted from SD32
ρ2622-2-22-200000√2-√2√2-√2-√2√20000ζ167+ζ16ζ1613+ζ1611ζ167+ζ16ζ165+ζ163ζ1615+ζ169ζ1615+ζ169ζ1613+ζ1611ζ165+ζ163    complex lifted from SD32
ρ274-4-4400-44000000000000000000000    symplectic lifted from C4.10D4, Schur index 2
ρ284-44-40000000-2√22√22√2-2√200000000000000    symplectic lifted from C8.17D4, Schur index 2
ρ294-44-400000002√2-2√2-2√22√200000000000000    symplectic lifted from C8.17D4, Schur index 2

Smallest permutation representation of C4.10D16
►Regular action on 128 points
Generators in S128
(1 125 109 75)(2 76 110 126)(3 127 111 77)(4 78 112 128)(5 113 97 79)(6 80 98 114)(7 115 99 65)(8 66 100 116)(9 117 101 67)(10 68 102 118)(11 119 103 69)(12 70 104 120)(13 121 105 71)(14 72 106 122)(15 123 107 73)(16 74 108 124)(17 64 39 87)(18 88 40 49)(19 50 41 89)(20 90 42 51)(21 52 43 91)(22 92 44 53)(23 54 45 93)(24 94 46 55)(25 56 47 95)(26 96 48 57)(27 58 33 81)(28 82 34 59)(29 60 35 83)(30 84 36 61)(31 62 37 85)(32 86 38 63)

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

Matrix representation of C4.10D16 ►in GL4(𝔽17) generated by

1000
0100
0001
00160
,
6400
13600
00130
0004
,
4600
61300
00143
001414
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,0,16,0,0,1,0],[6,13,0,0,4,6,0,0,0,0,13,0,0,0,0,4],[4,6,0,0,6,13,0,0,0,0,14,14,0,0,3,14] >;
 

C4.10D16 in GAP, Magma, Sage, TeX 

C_4._{10}D_{16}
 
% in TeX
 
G:=Group("C4.10D16");
 
// GroupNames label
 
G:=SmallGroup(128,96);
 
// by ID
 
G=gap.SmallGroup(128,96);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,120,422,387,520,794,416,2028,124]);
 
// Polycyclic
 
G:=Group<a,b,c|a^4=b^16=1,c^2=b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a*b^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C4.10D16 in TeX
Character table of C4.10D16 in TeX

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