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^[×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], C₈⋊₂C₈, C₈⋊₄Q₈, 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	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 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)

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