Copied to
clipboard

G = C82Q16order 128 = 27

2nd semidirect product of C8 and Q16 acting via Q16/C4=C22

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

Aliases: C82Q16, C42.245C23, C4⋊C4.67D4, (C2×C8).97D4, C82C8.9C2, (C2×Q8).59D4, C84Q8.3C2, C4.43(C2×Q16), C2.9(C82D4), C4⋊C8.32C22, C82Q8.18C2, C42Q16.8C2, C4⋊Q8.66C22, C4.72(C8⋊C22), (C4×C8).148C22, C4.10D8.8C2, (C4×Q8).47C22, C2.10(C42Q16), C2.15(D4.5D4), C4.117(C8.C22), C22.206(C4⋊D4), (C2×C4).30(C4○D4), (C2×C4).1280(C2×D4), SmallGroup(128,426)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C82Q16
C1C2C22C2×C4C42C4×Q8C84Q8 — C82Q16
C1C22C42 — C82Q16
C1C22C42 — C82Q16
C1C22C22C42 — C82Q16

Generators and relations for C82Q16
 G = < a,b,c | a8=b8=1, c2=b4, bab-1=a3, cac-1=a5, cbc-1=b-1 >

Subgroups: 160 in 76 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 [×6], C42, C42, C4⋊C4, C4⋊C4 [×5], C2×C8 [×2], C2×C8 [×3], Q16 [×4], C2×Q8, C2×Q8 [×2], C4×C8, C8⋊C4, Q8⋊C4 [×2], C4⋊C8, C4⋊C8 [×2], C4⋊C8, C2.D8 [×2], C4×Q8, C4⋊Q8 [×2], C2×Q16 [×2], C4.10D8 [×2], C82C8, C84Q8, C42Q16 [×2], C82Q8, C82Q16
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, Q16 [×2], C2×D4 [×2], C4○D4, C4⋊D4, C2×Q16, C8⋊C22 [×2], C8.C22, C42Q16, C82D4, D4.5D4, C82Q16

Character table of C82Q16

 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
ρ194-44-40-404000000000000000    orthogonal lifted from C8⋊C22
ρ204-44-4040-4000000000000000    orthogonal lifted from C8⋊C22
ρ214-4-4440-40000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2244-4-40000000002200-22000000    symplectic lifted from D4.5D4, Schur index 2
ρ2344-4-4000000000-220022000000    symplectic lifted from D4.5D4, Schur index 2

Smallest permutation representation of C82Q16
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 47 80 120 53 85 23 16)(2 42 73 115 54 88 24 11)(3 45 74 118 55 83 17 14)(4 48 75 113 56 86 18 9)(5 43 76 116 49 81 19 12)(6 46 77 119 50 84 20 15)(7 41 78 114 51 87 21 10)(8 44 79 117 52 82 22 13)(25 127 59 91 66 112 97 33)(26 122 60 94 67 107 98 36)(27 125 61 89 68 110 99 39)(28 128 62 92 69 105 100 34)(29 123 63 95 70 108 101 37)(30 126 64 90 71 111 102 40)(31 121 57 93 72 106 103 35)(32 124 58 96 65 109 104 38)
(1 59 53 97)(2 64 54 102)(3 61 55 99)(4 58 56 104)(5 63 49 101)(6 60 50 98)(7 57 51 103)(8 62 52 100)(9 96 113 38)(10 93 114 35)(11 90 115 40)(12 95 116 37)(13 92 117 34)(14 89 118 39)(15 94 119 36)(16 91 120 33)(17 68 74 27)(18 65 75 32)(19 70 76 29)(20 67 77 26)(21 72 78 31)(22 69 79 28)(23 66 80 25)(24 71 73 30)(41 121 87 106)(42 126 88 111)(43 123 81 108)(44 128 82 105)(45 125 83 110)(46 122 84 107)(47 127 85 112)(48 124 86 109)

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

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

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

Matrix representation of C82Q16 in GL6(𝔽17)

100000
010000
007131214
0047312
0053104
001451310
,
060000
14110000
006004
0001140
000460
0040011
,
1130000
1660000
000010
000001
001000
000100

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

C82Q16 in GAP, Magma, Sage, TeX

C_8\rtimes_2Q_{16}
% in TeX

G:=Group("C8:2Q16");
// GroupNames label

G:=SmallGroup(128,426);
// by ID

G=gap.SmallGroup(128,426);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,288,422,387,352,1123,136,2804,718,172]);
// 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=b^-1>;
// generators/relations

Export

Character table of C82Q16 in TeX

׿
×
𝔽