Copied to
clipboard

G = C8.SD16order 128 = 27

7th non-split extension by C8 of SD16 acting via SD16/C4=C22

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

Aliases: C8.7SD16, C42.241C23, C4⋊C4.63D4, (C2×C8).93D4, C82C8.8C2, (C2×Q8).57D4, C84Q8.1C2, Q8⋊Q8.9C2, C4.83(C2×SD16), C4⋊Q8.62C22, C4⋊C8.185C22, (C4×C8).144C22, C4⋊Q16.13C2, C2.7(C8.D4), (C4×Q8).45C22, C4.123(C8⋊C22), C2.11(C4⋊SD16), C4.10D8.12C2, C4.43(C8.C22), C2.13(D4.5D4), C22.202(C4⋊D4), (C2×C4).26(C4○D4), (C2×C4).1276(C2×D4), SmallGroup(128,422)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C8.SD16
C1C2C22C2×C4C42C4×Q8C84Q8 — C8.SD16
C1C22C42 — C8.SD16
C1C22C42 — C8.SD16
C1C22C22C42 — C8.SD16

Generators and relations for C8.SD16
 G = < a,b,c | a8=b8=1, c2=a4, bab-1=a3, cac-1=a-1, cbc-1=a4b3 >

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, C4.Q8 [×2], C4×Q8, C4⋊Q8 [×2], C2×Q16 [×2], C4.10D8 [×2], C82C8, C84Q8, Q8⋊Q8 [×2], C4⋊Q16, C8.SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, SD16 [×2], C2×D4 [×2], C4○D4, C4⋊D4, C2×SD16, C8⋊C22, C8.C22 [×2], C4⋊SD16, C8.D4, D4.5D4, C8.SD16

Character table of C8.SD16

 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
ρ132222-2-2-2-22000000000002i-2i0    complex lifted from C4○D4
ρ142222-2-2-2-2200000000000-2i2i0    complex lifted from C4○D4
ρ152-2-22-20200000002-20--2-2-200--2    complex lifted from SD16
ρ162-2-22-20200000002-20-2--2--200-2    complex lifted from SD16
ρ172-2-22-2020000000-220--2--2-200-2    complex lifted from SD16
ρ182-2-22-2020000000-220-2-2--200--2    complex lifted from SD16
ρ194-4-4440-40000000000000000    orthogonal lifted from C8⋊C22
ρ204-44-40-404000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ214-44-4040-4000000000000000    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 C8.SD16
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 86 103 33 18 119 52 47)(2 81 104 36 19 114 53 42)(3 84 97 39 20 117 54 45)(4 87 98 34 21 120 55 48)(5 82 99 37 22 115 56 43)(6 85 100 40 23 118 49 46)(7 88 101 35 24 113 50 41)(8 83 102 38 17 116 51 44)(9 77 109 30 122 64 91 71)(10 80 110 25 123 59 92 66)(11 75 111 28 124 62 93 69)(12 78 112 31 125 57 94 72)(13 73 105 26 126 60 95 67)(14 76 106 29 127 63 96 70)(15 79 107 32 128 58 89 65)(16 74 108 27 121 61 90 68)
(1 106 5 110)(2 105 6 109)(3 112 7 108)(4 111 8 107)(9 104 13 100)(10 103 14 99)(11 102 15 98)(12 101 16 97)(17 89 21 93)(18 96 22 92)(19 95 23 91)(20 94 24 90)(25 37 29 33)(26 36 30 40)(27 35 31 39)(28 34 32 38)(41 72 45 68)(42 71 46 67)(43 70 47 66)(44 69 48 65)(49 122 53 126)(50 121 54 125)(51 128 55 124)(52 127 56 123)(57 84 61 88)(58 83 62 87)(59 82 63 86)(60 81 64 85)(73 114 77 118)(74 113 78 117)(75 120 79 116)(76 119 80 115)

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

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

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

Matrix representation of C8.SD16 in GL6(𝔽17)

100000
010000
0071000
0012000
0000107
000050
,
800000
020000
0051100
0021200
0000310
0000814
,
090000
200000
0000126
0000155
0031000
0081400

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,12,0,0,0,0,10,0,0,0,0,0,0,0,10,5,0,0,0,0,7,0],[8,0,0,0,0,0,0,2,0,0,0,0,0,0,5,2,0,0,0,0,11,12,0,0,0,0,0,0,3,8,0,0,0,0,10,14],[0,2,0,0,0,0,9,0,0,0,0,0,0,0,0,0,3,8,0,0,0,0,10,14,0,0,12,15,0,0,0,0,6,5,0,0] >;

C8.SD16 in GAP, Magma, Sage, TeX

C_8.{\rm SD}_{16}
% in TeX

G:=Group("C8.SD16");
// GroupNames label

G:=SmallGroup(128,422);
// by ID

G=gap.SmallGroup(128,422);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,288,422,387,352,1123,136,2804,718,172]);
// Polycyclic

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

Export

Character table of C8.SD16 in TeX

׿
×
𝔽