C8.SD16 - GroupNames

G = C₈.SD₁₆ order 128 = 2⁷

7^th 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, C₈⋊₂C₈.₈C₂, (C₂×Q₈).₅₇D₄, C₈⋊₄Q₈.₁C₂, 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

Derived series

C₁ — C₄² — C₈.SD₁₆

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄×Q₈ — C₈⋊₄Q₈ — C₈.SD₁₆

Lower central

C₁ — C₂² — C₄² — C₈.SD₁₆

Upper central

C₁ — C₂² — C₄² — C₈.SD₁₆

Jennings

C₁ — C₂² — C₂² — C₄² — 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, C₄, C₄, C₄, C₂², C₈, C₈, C₂×C₄, C₂×C₄, Q₈, C₄², C₄², C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, Q₁₆, C₂×Q₈, C₂×Q₈, C₄×C₈, C₈⋊C₄, Q₈⋊C₄, C₄⋊C₈, C₄⋊C₈, C₄⋊C₈, C₄.Q₈, C₄×Q₈, C₄⋊Q₈, C₂×Q₁₆, C₄.₁₀D₈, C₈⋊₂C₈, C₈⋊₄Q₈, Q₈⋊Q₈, C₄⋊Q₁₆, C₈.SD₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, SD₁₆, C₂×D₄, C₄○D₄, C₄⋊D₄, C₂×SD₁₆, C₈⋊C₂², C₈.C₂², C₄⋊SD₁₆, C₈.D₄, D₄.₅D₄, C₈.SD₁₆

Character table of C₈.SD₁₆

class	1	2A	2B	2C	4A	4B	4C	4D	4E	4F	4G	4H	4I	8A	8B	8C	8D	8E	8F	8G	8H	8I	8J
size	1	1	1	1	2	2	2	2	4	8	8	16	16	4	4	4	4	8	8	8	8	8	8
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	1	1	1	1	1	1	1	-1	1	-1	-1	-1	-1	1	-1	1	-1	-1	-1	linear of order 2
ρ₃	1	1	1	1	1	1	1	1	1	-1	-1	-1	1	-1	-1	-1	-1	-1	1	-1	1	1	1	linear of order 2
ρ₄	1	1	1	1	1	1	1	1	1	-1	-1	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	linear of order 2
ρ₅	1	1	1	1	1	1	1	1	1	1	1	-1	-1	1	1	1	1	-1	-1	-1	1	1	-1	linear of order 2
ρ₆	1	1	1	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	1	-1	-1	-1	1	linear of order 2
ρ₇	1	1	1	1	1	1	1	1	1	-1	-1	1	-1	-1	-1	-1	-1	1	-1	1	1	1	-1	linear of order 2
ρ₈	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	1	1	1	1	1	1	1	-1	-1	1	linear of order 2
ρ₉	2	2	2	2	-2	2	-2	2	-2	2	-2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	-2	2	-2	2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	2	-2	2	-2	-2	0	0	0	0	2	-2	-2	2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	2	-2	2	-2	-2	0	0	0	0	-2	2	2	-2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	2	2	-2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	2i	-2i	0	complex lifted from C₄○D₄
ρ₁₄	2	2	2	2	-2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	-2i	2i	0	complex lifted from C₄○D₄
ρ₁₅	2	-2	-2	2	-2	0	2	0	0	0	0	0	0	0	2	-2	0	-√-2	√-2	√-2	0	0	-√-2	complex lifted from SD₁₆
ρ₁₆	2	-2	-2	2	-2	0	2	0	0	0	0	0	0	0	2	-2	0	√-2	-√-2	-√-2	0	0	√-2	complex lifted from SD₁₆
ρ₁₇	2	-2	-2	2	-2	0	2	0	0	0	0	0	0	0	-2	2	0	-√-2	-√-2	√-2	0	0	√-2	complex lifted from SD₁₆
ρ₁₈	2	-2	-2	2	-2	0	2	0	0	0	0	0	0	0	-2	2	0	√-2	√-2	-√-2	0	0	-√-2	complex lifted from SD₁₆
ρ₁₉	4	-4	-4	4	4	0	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₈⋊C₂²
ρ₂₀	4	-4	4	-4	0	-4	0	4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from C₈.C₂², Schur index 2
ρ₂₁	4	-4	4	-4	0	4	0	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from C₈.C₂², Schur index 2
ρ₂₂	4	4	-4	-4	0	0	0	0	0	0	0	0	0	2√2	0	0	-2√2	0	0	0	0	0	0	symplectic lifted from D₄.₅D₄, Schur index 2
ρ₂₃	4	4	-4	-4	0	0	0	0	0	0	0	0	0	-2√2	0	0	2√2	0	0	0	0	0	0	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)

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

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

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

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

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

Matrix representation of C₈.SD₁₆ ►in GL₆(𝔽₁₇)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	7	10	0	0
0	0	12	0	0	0
0	0	0	0	10	7
0	0	0	0	5	0

8	0	0	0	0	0
0	2	0	0	0	0
0	0	5	11	0	0
0	0	2	12	0	0
0	0	0	0	3	10
0	0	0	0	8	14

0	9	0	0	0	0
2	0	0	0	0	0
0	0	0	0	12	6
0	0	0	0	15	5
0	0	3	10	0	0
0	0	8	14	0	0

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

G = C8.SD16 order 128 = 27

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

G = C₈.SD₁₆ order 128 = 2⁷

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