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G = C8.3Q16order 128 = 27

3rd non-split extension by C8 of Q16 acting via Q16/C4=C22

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C8.3Q16, C42.247C23, C4⋊C4.69D4, (C2×C8).99D4, (C2×Q8).60D4, C4.44(C2×Q16), C84Q8.4C2, C82C8.10C2, C4⋊C8.33C22, C4.Q16.7C2, C82Q8.19C2, C4⋊Q8.68C22, (C4×C8).150C22, C2.9(C8.D4), C4.6Q16.6C2, (C4×Q8).48C22, C2.11(C42Q16), C4.45(C8.C22), C2.15(D4.4D4), C22.208(C4⋊D4), (C2×C4).32(C4○D4), (C2×C4).1282(C2×D4), SmallGroup(128,428)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C8.3Q16
C1C2C22C2×C4C42C4×Q8C84Q8 — C8.3Q16
C1C22C42 — C8.3Q16
C1C22C42 — C8.3Q16
C1C22C22C42 — C8.3Q16

Generators and relations for C8.3Q16
 G = < a,b,c | a8=b8=1, c2=b4, bab-1=a3, cac-1=a5, cbc-1=a4b-1 >

Subgroups: 144 in 70 conjugacy classes, 34 normal (22 characteristic)
C1, C2 [×3], C4 [×2], C4 [×2], C4 [×5], C22, C8 [×2], C8 [×4], C2×C4 [×3], C2×C4 [×4], Q8 [×4], C42, C42, C4⋊C4, C4⋊C4 [×5], C2×C8 [×2], C2×C8 [×3], C2×Q8, C2×Q8 [×2], C4×C8, C8⋊C4, Q8⋊C4 [×2], C4⋊C8, C4⋊C8 [×2], C4⋊C8, C2.D8 [×4], C4×Q8, C4⋊Q8 [×2], C4.6Q16 [×2], C82C8, C84Q8, C4.Q16 [×2], C82Q8, C8.3Q16
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, Q16 [×2], C2×D4 [×2], C4○D4, C4⋊D4, C2×Q16, C8.C22 [×3], C42Q16, C8.D4, D4.4D4, C8.3Q16

Character table of C8.3Q16

 class 12A2B2C4A4B4C4D4E4F4G4H4I8A8B8C8D8E8F8G8H8I8J
 size 1111222248816164444888888
ρ111111111111111111111111    trivial
ρ211111111111-11-1-1-1-11-11-1-1-1    linear of order 2
ρ3111111111-1-1-11-1-1-1-1-11-1111    linear of order 2
ρ4111111111-1-1111111-1-1-1-1-1-1    linear of order 2
ρ511111111111-1-11111-1-1-111-1    linear of order 2
ρ6111111111111-1-1-1-1-1-11-1-1-11    linear of order 2
ρ7111111111-1-11-1-1-1-1-11-1111-1    linear of order 2
ρ8111111111-1-1-1-11111111-1-11    linear of order 2
ρ92222-22-22-22-2000000000000    orthogonal lifted from D4
ρ102222-22-22-2-22000000000000    orthogonal lifted from D4
ρ1122222-22-2-200002-2-22000000    orthogonal lifted from D4
ρ1222222-22-2-20000-222-2000000    orthogonal lifted from D4
ρ132-2-22-20200000002-202-2-2002    symplectic lifted from Q16, Schur index 2
ρ142-2-22-2020000000-220-2-22002    symplectic lifted from Q16, Schur index 2
ρ152-2-22-20200000002-20-22200-2    symplectic lifted from Q16, Schur index 2
ρ162-2-22-2020000000-22022-200-2    symplectic lifted from Q16, Schur index 2
ρ172222-2-2-2-22000000000002i-2i0    complex lifted from C4○D4
ρ182222-2-2-2-2200000000000-2i2i0    complex lifted from C4○D4
ρ1944-4-4000000000-220022000000    orthogonal lifted from D4.4D4
ρ2044-4-40000000002200-22000000    orthogonal lifted from D4.4D4
ρ214-44-40-404000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ224-44-4040-4000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ234-4-4440-40000000000000000    symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of C8.3Q16
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 43 76 82 26 12 19 93)(2 46 77 85 27 15 20 96)(3 41 78 88 28 10 21 91)(4 44 79 83 29 13 22 94)(5 47 80 86 30 16 23 89)(6 42 73 81 31 11 24 92)(7 45 74 84 32 14 17 95)(8 48 75 87 25 9 18 90)(33 57 111 52 127 66 113 101)(34 60 112 55 128 69 114 104)(35 63 105 50 121 72 115 99)(36 58 106 53 122 67 116 102)(37 61 107 56 123 70 117 97)(38 64 108 51 124 65 118 100)(39 59 109 54 125 68 119 103)(40 62 110 49 126 71 120 98)
(1 56 26 97)(2 53 27 102)(3 50 28 99)(4 55 29 104)(5 52 30 101)(6 49 31 98)(7 54 32 103)(8 51 25 100)(9 114 48 112)(10 119 41 109)(11 116 42 106)(12 113 43 111)(13 118 44 108)(14 115 45 105)(15 120 46 110)(16 117 47 107)(17 68 74 59)(18 65 75 64)(19 70 76 61)(20 67 77 58)(21 72 78 63)(22 69 79 60)(23 66 80 57)(24 71 73 62)(33 93 127 82)(34 90 128 87)(35 95 121 84)(36 92 122 81)(37 89 123 86)(38 94 124 83)(39 91 125 88)(40 96 126 85)

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

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

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

Matrix representation of C8.3Q16 in GL6(𝔽17)

1600000
0160000
000303
00140140
0001403
0030140
,
060000
1460000
00131053
00713145
005347
00145104
,
070000
1200000
0011871
0086110
001016118
0016786

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,14,0,3,0,0,3,0,14,0,0,0,0,14,0,14,0,0,3,0,3,0],[0,14,0,0,0,0,6,6,0,0,0,0,0,0,13,7,5,14,0,0,10,13,3,5,0,0,5,14,4,10,0,0,3,5,7,4],[0,12,0,0,0,0,7,0,0,0,0,0,0,0,11,8,10,16,0,0,8,6,16,7,0,0,7,1,11,8,0,0,1,10,8,6] >;

C8.3Q16 in GAP, Magma, Sage, TeX

C_8._3Q_{16}
% in TeX

G:=Group("C8.3Q16");
// GroupNames label

G:=SmallGroup(128,428);
// by ID

G=gap.SmallGroup(128,428);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,672,141,288,422,387,352,1123,136,2804,718,172]);
// Polycyclic

G:=Group<a,b,c|a^8=b^8=1,c^2=b^4,b*a*b^-1=a^3,c*a*c^-1=a^5,c*b*c^-1=a^4*b^-1>;
// generators/relations

Export

Character table of C8.3Q16 in TeX

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