C8:2Q16 - GroupNames

G = C₈⋊₂Q₁₆ order 128 = 2⁷

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

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

Aliases: C₈⋊₂Q₁₆, C₄².₂₄₅C₂³, C₄⋊C₄.₆₇D₄, (C₂×C₈).₉₇D₄, C₈⋊₂C₈.₉C₂, (C₂×Q₈).₅₉D₄, C₈⋊₄Q₈.₃C₂, C₄.₄₃(C₂×Q₁₆), C₂.₉(C₈⋊₂D₄), C₄⋊C₈.₃₂C₂², C₈⋊₂Q₈.₁₈C₂, C₄⋊₂Q₁₆.₈C₂, C₄⋊Q₈.₆₆C₂², C₄.₇₂(C₈⋊C₂²), (C₄×C₈).₁₄₈C₂², C₄.₁₀D₈.₈C₂, (C₄×Q₈).₄₇C₂², C₂.₁₀(C₄⋊₂Q₁₆), 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

Derived series

C₁ — C₄² — C₈⋊₂Q₁₆

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄×Q₈ — C₈⋊₄Q₈ — C₈⋊₂Q₁₆

Lower central

C₁ — C₂² — C₄² — C₈⋊₂Q₁₆

Upper central

C₁ — C₂² — C₄² — C₈⋊₂Q₁₆

Jennings

C₁ — C₂² — C₂² — C₄² — C₈⋊₂Q₁₆

Generators and relations for C₈⋊₂Q₁₆
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, 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, C₂.D₈, C₄×Q₈, C₄⋊Q₈, C₂×Q₁₆, C₄.₁₀D₈, C₈⋊₂C₈, C₈⋊₄Q₈, C₄⋊₂Q₁₆, C₈⋊₂Q₈, C₈⋊₂Q₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, Q₁₆, C₂×D₄, C₄○D₄, C₄⋊D₄, C₂×Q₁₆, C₈⋊C₂², C₈.C₂², C₄⋊₂Q₁₆, C₈⋊₂D₄, D₄.₅D₄, C₈⋊₂Q₁₆

Character table of C₈⋊₂Q₁₆

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	0	2	0	0	0	0	0	0	0	2	-2	0	√2	-√2	-√2	0	0	√2	symplectic lifted from Q₁₆, Schur index 2
ρ₁₄	2	-2	-2	2	-2	0	2	0	0	0	0	0	0	0	-2	2	0	-√2	-√2	√2	0	0	√2	symplectic lifted from Q₁₆, Schur index 2
ρ₁₅	2	-2	-2	2	-2	0	2	0	0	0	0	0	0	0	2	-2	0	-√2	√2	√2	0	0	-√2	symplectic lifted from Q₁₆, Schur index 2
ρ₁₆	2	-2	-2	2	-2	0	2	0	0	0	0	0	0	0	-2	2	0	√2	√2	-√2	0	0	-√2	symplectic lifted from Q₁₆, Schur index 2
ρ₁₇	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
ρ₁₈	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
ρ₁₉	4	-4	4	-4	0	-4	0	4	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	orthogonal lifted from C₈⋊C₂²
ρ₂₁	4	-4	-4	4	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
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⋊₂Q₁₆
►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 82 22 16 63 92)(2 42 73 85 23 11 64 95)(3 45 74 88 24 14 57 90)(4 48 75 83 17 9 58 93)(5 43 76 86 18 12 59 96)(6 46 77 81 19 15 60 91)(7 41 78 84 20 10 61 94)(8 44 79 87 21 13 62 89)(25 106 50 127 66 117 101 33)(26 109 51 122 67 120 102 36)(27 112 52 125 68 115 103 39)(28 107 53 128 69 118 104 34)(29 110 54 123 70 113 97 37)(30 105 55 126 71 116 98 40)(31 108 56 121 72 119 99 35)(32 111 49 124 65 114 100 38)

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

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

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

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

Matrix representation of C₈⋊₂Q₁₆ ►in GL₆(𝔽₁₇)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	7	13	12	14
0	0	4	7	3	12
0	0	5	3	10	4
0	0	14	5	13	10

0	6	0	0	0	0
14	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	3	0	0	0	0
16	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

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] >;

C₈⋊₂Q₁₆ 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 C₈⋊₂Q₁₆ in TeX

G = C8⋊2Q16 order 128 = 27

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

G = C₈⋊₂Q₁₆ order 128 = 2⁷

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