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G = C16⋊Q8order 128 = 27

The semidirect product of C16 and Q8 acting via Q8/C2=C22

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

Aliases: C16⋊Q8, C8.4Q16, C42.158D4, (C2×C4).50D8, C4.8(C4⋊Q8), C8.21(C2×Q8), (C2×C8).140D4, C4.12(C2×Q16), C163C4.8C2, C165C4.2C2, C164C4.3C2, C82Q8.20C2, C2.9(C82Q8), C8.5Q8.7C2, (C2×C8).553C23, (C2×C16).31C22, (C4×C8).172C22, C22.139(C2×D8), C2.23(C16⋊C22), C2.D8.32C22, C2.23(Q32⋊C2), (C2×C4).821(C2×D4), SmallGroup(128,987)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — C16⋊Q8
C1C2C4C8C2×C8C2×C16C165C4 — C16⋊Q8
C1C2C4C2×C8 — C16⋊Q8
C1C22C42C4×C8 — C16⋊Q8
C1C2C2C2C2C4C4C2×C8 — C16⋊Q8

Generators and relations for C16⋊Q8
 G = < a,b,c | a16=b4=1, c2=b2, bab-1=a9, cac-1=a7, cbc-1=b-1 >

2C4
2C4
8C4
8C4
8C4
8C4
4C2×C4
4C2×C4
4C2×C4
4C2×C4
8Q8
8Q8
2C4⋊C4
2C4⋊C4
2C4⋊C4
2C4⋊C4
4C4⋊C4
4C4⋊C4
4C2×Q8
4C4⋊C4
2C4.Q8
2C2.D8
2C42.C2
2C4⋊Q8

Character table of C16⋊Q8

 class 12A2B2C4A4B4C4D4E4F4G4H8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 111122441616161622224444444444
ρ111111111111111111111111111    trivial
ρ2111111111-11-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ3111111-1-1-111-11111-1-1-1-111-111-1    linear of order 2
ρ4111111-1-1-1-1111111-1-111-1-11-1-11    linear of order 2
ρ5111111-1-111-1-11111-1-111-1-11-1-11    linear of order 2
ρ6111111-1-11-1-111111-1-1-1-111-111-1    linear of order 2
ρ711111111-11-11111111-1-1-1-1-1-1-1-1    linear of order 2
ρ811111111-1-1-1-111111111111111    linear of order 2
ρ92222-2-2-220000000000-22-2-2-2222    orthogonal lifted from D8
ρ10222222220000-2-2-2-2-2-200000000    orthogonal lifted from D4
ρ112222-2-2-2200000000002-2222-2-2-2    orthogonal lifted from D8
ρ12222222-2-20000-2-2-2-22200000000    orthogonal lifted from D4
ρ132222-2-22-200000000002-2-2-2222-2    orthogonal lifted from D8
ρ142222-2-22-20000000000-2222-2-2-22    orthogonal lifted from D8
ρ152-22-2-22000000-222-2002-200-2002    symplectic lifted from Q8, Schur index 2
ρ162-22-2-22000000-222-200-2200200-2    symplectic lifted from Q8, Schur index 2
ρ172-22-2-220000002-2-220000-2202-20    symplectic lifted from Q8, Schur index 2
ρ182-22-2-220000002-2-2200002-20-220    symplectic lifted from Q8, Schur index 2
ρ192-22-22-20000000000-22-2-22-222-22    symplectic lifted from Q16, Schur index 2
ρ202-22-22-200000000002-2-2-2-222-222    symplectic lifted from Q16, Schur index 2
ρ212-22-22-20000000000-2222-22-2-22-2    symplectic lifted from Q16, Schur index 2
ρ222-22-22-200000000002-2222-2-22-2-2    symplectic lifted from Q16, Schur index 2
ρ234-4-44000000002222-22-220000000000    orthogonal lifted from C16⋊C22
ρ244-4-4400000000-22-2222220000000000    orthogonal lifted from C16⋊C22
ρ2544-4-40000000022-2222-220000000000    symplectic lifted from Q32⋊C2, Schur index 2
ρ2644-4-400000000-2222-22220000000000    symplectic lifted from Q32⋊C2, Schur index 2

Smallest permutation representation of C16⋊Q8
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 70 89 30)(2 79 90 23)(3 72 91 32)(4 65 92 25)(5 74 93 18)(6 67 94 27)(7 76 95 20)(8 69 96 29)(9 78 81 22)(10 71 82 31)(11 80 83 24)(12 73 84 17)(13 66 85 26)(14 75 86 19)(15 68 87 28)(16 77 88 21)(33 53 122 108)(34 62 123 101)(35 55 124 110)(36 64 125 103)(37 57 126 112)(38 50 127 105)(39 59 128 98)(40 52 113 107)(41 61 114 100)(42 54 115 109)(43 63 116 102)(44 56 117 111)(45 49 118 104)(46 58 119 97)(47 51 120 106)(48 60 121 99)
(1 127 89 38)(2 118 90 45)(3 125 91 36)(4 116 92 43)(5 123 93 34)(6 114 94 41)(7 121 95 48)(8 128 96 39)(9 119 81 46)(10 126 82 37)(11 117 83 44)(12 124 84 35)(13 115 85 42)(14 122 86 33)(15 113 87 40)(16 120 88 47)(17 110 73 55)(18 101 74 62)(19 108 75 53)(20 99 76 60)(21 106 77 51)(22 97 78 58)(23 104 79 49)(24 111 80 56)(25 102 65 63)(26 109 66 54)(27 100 67 61)(28 107 68 52)(29 98 69 59)(30 105 70 50)(31 112 71 57)(32 103 72 64)

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

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

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

Matrix representation of C16⋊Q8 in GL6(𝔽17)

1600000
0160000
00111514
00151368
00167142
0061596
,
0160000
100000
0012816
0078124
00014610
001411122
,
170000
7160000
00151436
00215713
0037111
0016343

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,15,16,6,0,0,1,13,7,15,0,0,15,6,14,9,0,0,14,8,2,6],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,1,7,0,14,0,0,2,8,14,11,0,0,8,12,6,12,0,0,16,4,10,2],[1,7,0,0,0,0,7,16,0,0,0,0,0,0,15,2,3,16,0,0,14,15,7,3,0,0,3,7,1,4,0,0,6,13,11,3] >;

C16⋊Q8 in GAP, Magma, Sage, TeX

C_{16}\rtimes Q_8
% in TeX

G:=Group("C16:Q8");
// GroupNames label

G:=SmallGroup(128,987);
// by ID

G=gap.SmallGroup(128,987);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,56,141,64,422,723,268,1684,242,4037,124]);
// Polycyclic

G:=Group<a,b,c|a^16=b^4=1,c^2=b^2,b*a*b^-1=a^9,c*a*c^-1=a^7,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C16⋊Q8 in TeX
Character table of C16⋊Q8 in TeX

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