C8:7Q16 - GroupNames

G = C₈⋊₇Q₁₆ order 128 = 2⁷

1^st semidirect product of C₈ and Q₁₆ acting via Q₁₆/Q₈=C₂

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

Aliases: C₈⋊₇Q₁₆, Q₈.₁D₈, C₄².₂₂₅C₂³, C₄.₃₄(C₂×D₈), (C₈×Q₈).₅C₂, C₈⋊₁C₈.₇C₂, C₄⋊C₄.₂₀₀D₄, (C₂×C₈).₃₄₄D₄, C₈⋊₂Q₈.₄C₂, C₄.₄₀(C₂×Q₁₆), C₄.₈₇(C₄○D₈), C₂.₉(C₈⋊₇D₄), C₄⋊C₈.₂₂C₂², (C₄×C₈).₅₄C₂², (C₂×Q₈).₁₅₀D₄, C₄⋊₂Q₁₆.₅C₂, C₄⋊Q₈.₄₈C₂², C₂.₇(C₄⋊₂Q₁₆), C₄.₁₀D₈.₃C₂, C₂.₉(D₄.₅D₄), (C₄×Q₈).₂₆₅C₂², C₄.₁₁₃(C₈.C₂²), C₂².₁₈₆(C₄⋊D₄), (C₂×C₄).₁₀(C₄○D₄), (C₂×C₄).₁₂₆₀(C₂×D₄), SmallGroup(128,406)

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

Derived series

C₁ — C₄² — C₈⋊₇Q₁₆

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄×Q₈ — C₈×Q₈ — C₈⋊₇Q₁₆

Lower central

C₁ — C₂² — C₄² — C₈⋊₇Q₁₆

Upper central

C₁ — C₂² — C₄² — C₈⋊₇Q₁₆

Jennings

C₁ — C₂² — C₂² — C₄² — C₈⋊₇Q₁₆

Generators and relations for C₈⋊₇Q₁₆
G = < a,b,c | a⁸=b⁸=1, c²=b⁴, bab^-1=a^-1, ac=ca, cbc^-1=b^-1 >

Subgroups: 160 in 78 conjugacy classes, 36 normal (24 characteristic)
C₁, C₂, C₄, C₄, C₂², C₈, C₈, C₂×C₄, C₂×C₄, Q₈, Q₈, C₄², C₄², C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, Q₁₆, C₂×Q₈, C₂×Q₈, C₄×C₈, C₄×C₈, Q₈⋊C₄, C₄⋊C₈, C₄⋊C₈, C₄⋊C₈, C₂.D₈, C₄×Q₈, C₄⋊Q₈, C₂×Q₁₆, C₄.₁₀D₈, C₈⋊₁C₈, C₈×Q₈, C₄⋊₂Q₁₆, C₈⋊₂Q₈, C₈⋊₇Q₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, D₈, Q₁₆, C₂×D₄, C₄○D₄, C₄⋊D₄, C₂×D₈, C₂×Q₁₆, C₄○D₈, C₈.C₂², C₄⋊₂Q₁₆, C₈⋊₇D₄, D₄.₅D₄, C₈⋊₇Q₁₆

Character table of C₈⋊₇Q₁₆

class	1	2A	2B	2C	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	8A	8B	8C	8D	8E	8F	8G	8H	8I	8J	8K	8L	8M	8N
size	1	1	1	1	2	2	2	2	4	4	4	4	4	16	16	2	2	2	2	4	4	4	4	4	4	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	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	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	-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	-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	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	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	-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	-1	1	-1	linear of order 2
ρ₉	2	2	2	2	2	-2	2	-2	0	0	0	-2	0	0	0	-2	-2	-2	-2	0	0	0	2	2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	-2	2	-2	2	2	-2	-2	-2	2	0	0	0	0	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	2	2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	2	-2	2	-2	0	0	0	-2	0	0	0	2	2	2	2	0	0	0	-2	-2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	-2	-2	0	2	0	-2	0	-2	2	0	0	0	0	-√2	√2	√2	-√2	√2	-√2	-√2	√2	-√2	√2	0	0	0	0	orthogonal lifted from D₈
ρ₁₄	2	2	-2	-2	0	2	0	-2	0	2	-2	0	0	0	0	√2	-√2	-√2	√2	√2	-√2	-√2	-√2	√2	√2	0	0	0	0	orthogonal lifted from D₈
ρ₁₅	2	2	-2	-2	0	2	0	-2	0	2	-2	0	0	0	0	-√2	√2	√2	-√2	-√2	√2	√2	√2	-√2	-√2	0	0	0	0	orthogonal lifted from D₈
ρ₁₆	2	2	-2	-2	0	2	0	-2	0	-2	2	0	0	0	0	√2	-√2	-√2	√2	-√2	√2	√2	-√2	√2	-√2	0	0	0	0	orthogonal lifted from D₈
ρ₁₇	2	-2	2	-2	-2	0	2	0	0	0	0	0	0	0	0	-2	2	-2	2	0	0	0	0	0	0	√2	-√2	-√2	√2	symplectic lifted from Q₁₆, Schur index 2
ρ₁₈	2	-2	2	-2	-2	0	2	0	0	0	0	0	0	0	0	2	-2	2	-2	0	0	0	0	0	0	-√2	-√2	√2	√2	symplectic lifted from Q₁₆, Schur index 2
ρ₁₉	2	-2	2	-2	-2	0	2	0	0	0	0	0	0	0	0	2	-2	2	-2	0	0	0	0	0	0	√2	√2	-√2	-√2	symplectic lifted from Q₁₆, Schur index 2
ρ₂₀	2	-2	2	-2	-2	0	2	0	0	0	0	0	0	0	0	-2	2	-2	2	0	0	0	0	0	0	-√2	√2	√2	-√2	symplectic lifted from Q₁₆, Schur index 2
ρ₂₁	2	2	-2	-2	0	-2	0	2	2i	0	0	0	-2i	0	0	√2	-√2	-√2	√2	√-2	-√-2	√-2	√2	-√2	-√-2	0	0	0	0	complex lifted from C₄○D₈
ρ₂₂	2	2	-2	-2	0	-2	0	2	-2i	0	0	0	2i	0	0	-√2	√2	√2	-√2	√-2	-√-2	√-2	-√2	√2	-√-2	0	0	0	0	complex lifted from C₄○D₈
ρ₂₃	2	2	-2	-2	0	-2	0	2	2i	0	0	0	-2i	0	0	-√2	√2	√2	-√2	-√-2	√-2	-√-2	-√2	√2	√-2	0	0	0	0	complex lifted from C₄○D₈
ρ₂₄	2	2	-2	-2	0	-2	0	2	-2i	0	0	0	2i	0	0	√2	-√2	-√2	√2	-√-2	√-2	-√-2	√2	-√2	√-2	0	0	0	0	complex lifted from C₄○D₈
ρ₂₅	2	2	2	2	-2	-2	-2	-2	0	0	0	2	0	0	0	0	0	0	0	2i	2i	-2i	0	0	-2i	0	0	0	0	complex lifted from C₄○D₄
ρ₂₆	2	2	2	2	-2	-2	-2	-2	0	0	0	2	0	0	0	0	0	0	0	-2i	-2i	2i	0	0	2i	0	0	0	0	complex lifted from C₄○D₄
ρ₂₇	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	0	0	0	symplectic lifted from C₈.C₂², Schur index 2
ρ₂₈	4	-4	-4	4	0	0	0	0	0	0	0	0	0	0	0	2√2	2√2	-2√2	-2√2	0	0	0	0	0	0	0	0	0	0	symplectic lifted from D₄.₅D₄, Schur index 2
ρ₂₉	4	-4	-4	4	0	0	0	0	0	0	0	0	0	0	0	-2√2	-2√2	2√2	2√2	0	0	0	0	0	0	0	0	0	0	symplectic lifted from D₄.₅D₄, Schur index 2

Smallest permutation representation of C₈⋊₇Q₁₆
►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 16 53 87 61 46 73 106)(2 15 54 86 62 45 74 105)(3 14 55 85 63 44 75 112)(4 13 56 84 64 43 76 111)(5 12 49 83 57 42 77 110)(6 11 50 82 58 41 78 109)(7 10 51 81 59 48 79 108)(8 9 52 88 60 47 80 107)(17 126 68 120 102 39 30 92)(18 125 69 119 103 38 31 91)(19 124 70 118 104 37 32 90)(20 123 71 117 97 36 25 89)(21 122 72 116 98 35 26 96)(22 121 65 115 99 34 27 95)(23 128 66 114 100 33 28 94)(24 127 67 113 101 40 29 93)

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

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

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

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

Matrix representation of C₈⋊₇Q₁₆ ►in GL₄(𝔽₁₇) generated by

2	0	0	0
0	9	0	0
0	0	16	0
0	0	0	16

0	1	0	0
16	0	0	0
0	0	11	11
0	0	3	0

1	0	0	0
0	16	0	0
0	0	11	3
0	0	16	6

G:=sub<GL(4,GF(17))| [2,0,0,0,0,9,0,0,0,0,16,0,0,0,0,16],[0,16,0,0,1,0,0,0,0,0,11,3,0,0,11,0],[1,0,0,0,0,16,0,0,0,0,11,16,0,0,3,6] >;

C₈⋊₇Q₁₆ in GAP, Magma, Sage, TeX

C_8\rtimes_7Q_{16}

% in TeX

G:=Group("C8:7Q16");

// GroupNames label

G:=SmallGroup(128,406);

// by ID

G=gap.SmallGroup(128,406);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,288,422,352,1123,136,2804,718,172]);

// Polycyclic

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

// generators/relations

Export

Character table of C₈⋊₇Q₁₆ in TeX

G = C8⋊7Q16 order 128 = 27

1st semidirect product of C8 and Q16 acting via Q16/Q8=C2

G = C₈⋊₇Q₁₆ order 128 = 2⁷

1^st semidirect product of C₈ and Q₁₆ acting via Q₁₆/Q₈=C₂