C8.9SD16 - GroupNames

G = C₈.₉SD₁₆ order 128 = 2⁷

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

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

Aliases: C₈.₉SD₁₆, C₄².₆₆₃C₂³, C₈⋊C₈.₄C₂, (C₂×C₈).₁₅₂D₄, C₈⋊₃Q₈.₇C₂, C₄.₇(C₂×SD₁₆), C₄⋊Q₈.₈₇C₂², C₂.₁₁(C₈⋊₅D₄), (C₄×C₈).₁₅₃C₂², C₄⋊Q₁₆.₁₄C₂, C₂.₇(C₈.₂D₄), C₄.₂(C₈.C₂²), C₄.SD₁₆.₁₂C₂, C₂².₆₄(C₄⋊₁D₄), (C₂×C₄).₇₂₀(C₂×D₄), SmallGroup(128,448)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series

C₁ — C₄² — C₈.₉SD₁₆

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄×C₈ — C₈⋊C₈ — C₈.₉SD₁₆

Lower central

C₁ — C₂² — C₄² — C₈.₉SD₁₆

Upper central

C₁ — C₂² — C₄² — C₈.₉SD₁₆

Jennings

C₁ — C₂² — C₂² — C₄² — C₈.₉SD₁₆

Generators and relations for C₈.₉SD₁₆
G = < a,b,c | a⁸=b⁸=1, c²=a⁴, bab^-1=a⁵, cac^-1=a^-1, cbc^-1=b³ >

Subgroups: 192 in 88 conjugacy classes, 40 normal (12 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₈, C₈, C₂×C₄, C₂×C₄, C₂×C₄, Q₈, C₄², C₄⋊C₄, C₂×C₈, Q₁₆, C₂×Q₈, C₄×C₈, C₄×C₈, Q₈⋊C₄, C₄.Q₈, C₄⋊Q₈, C₄⋊Q₈, C₂×Q₁₆, C₈⋊C₈, C₄.SD₁₆, C₄⋊Q₁₆, C₈⋊₃Q₈, C₈⋊₃Q₈, C₈.₉SD₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, SD₁₆, C₂×D₄, C₄⋊₁D₄, C₂×SD₁₆, C₈.C₂², C₈⋊₅D₄, C₈.₂D₄, C₈.₉SD₁₆

Character table of C₈.₉SD₁₆

class	1	2A	2B	2C	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	8A	8B	8C	8D	8E	8F	8G	8H	8I	8J	8K	8L
size	1	1	1	1	2	2	2	2	2	2	16	16	16	16	4	4	4	4	4	4	4	4	4	4	4	4
ρ₁	1	1	1	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	-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	-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	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	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	-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	-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	1	-1	-1	linear of order 2
ρ₉	2	2	2	2	-2	2	-2	-2	2	-2	0	0	0	0	0	0	0	0	2	0	0	-2	0	0	-2	2	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	2	-2	2	-2	-2	-2	0	0	0	0	2	-2	0	0	0	-2	0	0	2	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	-2	-2	-2	2	-2	2	0	0	0	0	0	0	-2	-2	0	0	2	0	0	2	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	-2	-2	-2	2	-2	2	0	0	0	0	0	0	2	2	0	0	-2	0	0	-2	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	2	2	-2	2	-2	-2	2	-2	0	0	0	0	0	0	0	0	-2	0	0	2	0	0	2	-2	orthogonal lifted from D₄
ρ₁₄	2	2	2	2	2	-2	2	-2	-2	-2	0	0	0	0	-2	2	0	0	0	2	0	0	-2	0	0	0	orthogonal lifted from D₄
ρ₁₅	2	-2	2	-2	0	-2	0	0	2	0	0	0	0	0	√-2	√-2	√-2	-√-2	0	-√-2	-√-2	-2	-√-2	√-2	2	0	complex lifted from SD₁₆
ρ₁₆	2	-2	2	-2	0	2	0	0	-2	0	0	0	0	0	-√-2	√-2	-√-2	√-2	2	-√-2	-√-2	0	√-2	√-2	0	-2	complex lifted from SD₁₆
ρ₁₇	2	-2	2	-2	0	2	0	0	-2	0	0	0	0	0	-√-2	√-2	√-2	-√-2	-2	-√-2	√-2	0	√-2	-√-2	0	2	complex lifted from SD₁₆
ρ₁₈	2	-2	2	-2	0	-2	0	0	2	0	0	0	0	0	-√-2	-√-2	√-2	-√-2	0	√-2	-√-2	2	√-2	√-2	-2	0	complex lifted from SD₁₆
ρ₁₉	2	-2	2	-2	0	2	0	0	-2	0	0	0	0	0	√-2	-√-2	√-2	-√-2	2	√-2	√-2	0	-√-2	-√-2	0	-2	complex lifted from SD₁₆
ρ₂₀	2	-2	2	-2	0	-2	0	0	2	0	0	0	0	0	-√-2	-√-2	-√-2	√-2	0	√-2	√-2	-2	√-2	-√-2	2	0	complex lifted from SD₁₆
ρ₂₁	2	-2	2	-2	0	-2	0	0	2	0	0	0	0	0	√-2	√-2	-√-2	√-2	0	-√-2	√-2	2	-√-2	-√-2	-2	0	complex lifted from SD₁₆
ρ₂₂	2	-2	2	-2	0	2	0	0	-2	0	0	0	0	0	√-2	-√-2	-√-2	√-2	-2	√-2	-√-2	0	-√-2	√-2	0	2	complex lifted from SD₁₆
ρ₂₃	4	-4	-4	4	4	0	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from C₈.C₂², Schur index 2
ρ₂₄	4	4	-4	-4	0	0	0	-4	0	4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from C₈.C₂², Schur index 2
ρ₂₅	4	-4	-4	4	-4	0	4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from C₈.C₂², Schur index 2
ρ₂₆	4	4	-4	-4	0	0	0	4	0	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from C₈.C₂², Schur index 2

Smallest permutation representation of C₈.₉SD₁₆
►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 59 22 71 78 29 54 98)(2 64 23 68 79 26 55 103)(3 61 24 65 80 31 56 100)(4 58 17 70 73 28 49 97)(5 63 18 67 74 25 50 102)(6 60 19 72 75 30 51 99)(7 57 20 69 76 27 52 104)(8 62 21 66 77 32 53 101)(9 110 84 125 44 91 115 37)(10 107 85 122 45 96 116 34)(11 112 86 127 46 93 117 39)(12 109 87 124 47 90 118 36)(13 106 88 121 48 95 119 33)(14 111 81 126 41 92 120 38)(15 108 82 123 42 89 113 35)(16 105 83 128 43 94 114 40)

(1 127 5 123)(2 126 6 122)(3 125 7 121)(4 124 8 128)(9 104 13 100)(10 103 14 99)(11 102 15 98)(12 101 16 97)(17 109 21 105)(18 108 22 112)(19 107 23 111)(20 106 24 110)(25 82 29 86)(26 81 30 85)(27 88 31 84)(28 87 32 83)(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 90 53 94)(50 89 54 93)(51 96 55 92)(52 95 56 91)(57 119 61 115)(58 118 62 114)(59 117 63 113)(60 116 64 120)

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

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

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

Matrix representation of C₈.₉SD₁₆ ►in GL₆(𝔽₁₇)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	2	10	11	7
0	0	10	8	3	13
0	0	7	13	8	3
0	0	13	4	14	16

2	0	0	0	0	0
1	8	0	0	0	0
0	0	0	0	1	0
0	0	16	16	16	16
0	0	16	0	0	0
0	0	2	2	0	1

3	1	0	0	0	0
9	14	0	0	0	0
0	0	7	0	1	0
0	0	1	11	1	1
0	0	1	0	10	0
0	0	15	14	12	6

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

C₈.₉SD₁₆ in GAP, Magma, Sage, TeX

C_8._9{\rm SD}_{16}

% in TeX

G:=Group("C8.9SD16");

// GroupNames label

G:=SmallGroup(128,448);

// by ID

G=gap.SmallGroup(128,448);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,288,422,387,100,1123,136,2804,172]);

// Polycyclic

G:=Group<a,b,c|a^8=b^8=1,c^2=a^4,b*a*b^-1=a^5,c*a*c^-1=a^-1,c*b*c^-1=b^3>;

// generators/relations

Export

Character table of C₈.₉SD₁₆ in TeX

G = C8.9SD16 order 128 = 27

9th non-split extension by C8 of SD16 acting via SD16/C4=C22

G = C₈.₉SD₁₆ order 128 = 2⁷

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