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G = C4.10D16order 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, C81C8.3C2, C2.D8.4C4, (C2×C4).121D8, (C2×C8).335D4, C82Q8.1C2, (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

C1C2×C8 — C4.10D16
C1C2C4C2×C4C2×C8C4×C8C82Q8 — C4.10D16
C1C2C2×C4C2×C8 — C4.10D16
C1C22C42C4×C8 — C4.10D16
C1C2C2C2C2C2×C4C2×C4C4×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 >

2C4
8C4
8C4
2C8
4C2×C4
4C2×C4
8Q8
8C8
8Q8
2C4⋊C4
2C4⋊C4
4C2×Q8
4C2×C8
4C4⋊C4
4C16
2C4⋊C8
2C4⋊Q8
2C2.D8
2C2×C16

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
ρ1122-2-2-22000002-22-22-20000ζ1615169ζ1651631615169ζ1651631615169ζ1615169165163165163    orthogonal lifted from D16
ρ1222-2-2-2200000-22-22-220000165163ζ1615169ζ165163ζ1615169ζ16516316516316151691615169    orthogonal lifted from D16
ρ13222222-2-2-20000000000002-22-222-2-2    orthogonal lifted from D8
ρ14222222-2-2-2000000000000-22-22-2-222    orthogonal lifted from D8
ρ1522-2-2-2200000-22-22-220000ζ16516316151691651631615169165163ζ165163ζ1615169ζ1615169    orthogonal lifted from D16
ρ1622-2-2-22000002-22-22-200001615169165163ζ1615169165163ζ16151691615169ζ165163ζ165163    orthogonal lifted from D16
ρ172-2-22002-200022-2-2002-22-200000000    symplectic lifted from Q16, Schur index 2
ρ182-2-22002-200022-2-200-22-2200000000    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
ρ2322-2-22-200000-22-222-20000ζ16131611ζ1615169ζ16131611ζ16716ζ165163ζ165163ζ1615169ζ16716    complex lifted from SD32
ρ2422-2-22-2000002-22-2-220000ζ1615169ζ165163ζ1615169ζ16131611ζ16716ζ16716ζ165163ζ16131611    complex lifted from SD32
ρ2522-2-22-200000-22-222-20000ζ165163ζ16716ζ165163ζ1615169ζ16131611ζ16131611ζ16716ζ1615169    complex lifted from SD32
ρ2622-2-22-2000002-22-2-220000ζ16716ζ16131611ζ16716ζ165163ζ1615169ζ1615169ζ16131611ζ165163    complex lifted from SD32
ρ274-4-4400-44000000000000000000000    symplectic lifted from C4.10D4, Schur index 2
ρ284-44-40000000-222222-2200000000000000    symplectic lifted from C8.17D4, Schur index 2
ρ294-44-4000000022-22-222200000000000000    symplectic lifted from C8.17D4, Schur index 2

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

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

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

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

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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