C8:8Q16 - GroupNames

G = C₈⋊₈Q₁₆ order 128 = 2⁷

2^nd semidirect product of C₈ and Q₁₆ acting via Q₁₆/Q₈=C₂

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

Aliases: C₈⋊₈Q₁₆, Q₈.₁SD₁₆, C₄².₂₂₃C₂³, C₈⋊₂C₈.₂C₂, C₄⋊C₄.₁₉₈D₄, (C₂×C₈).₃₆₆D₄, (C₈×Q₈).₁₂C₂, C₄.₃₉(C₂×Q₁₆), C₈⋊₃Q₈.₅C₂, C₄.₆₇(C₄○D₈), C₄⋊C₈.₂₁C₂², (C₂×Q₈).₁₄₉D₄, C₄.Q₁₆.₃C₂, C₄⋊₂Q₁₆.₄C₂, C₄.₃₉(C₂×SD₁₆), C₄⋊Q₈.₄₆C₂², C₂.₁₁(C₈⋊₈D₄), (C₄×C₈).₂₅₂C₂², C₂.₆(C₄⋊₂Q₁₆), C₄.₁₀D₈.₂C₂, C₄.₆Q₁₆.₂C₂, (C₄×Q₈).₂₆₄C₂², C₂.₁₁(D₄.₃D₄), C₄.₁₁₁(C₈.C₂²), C₂².₁₈₄(C₄⋊D₄), (C₂×C₄).₈(C₄○D₄), (C₂×C₄).₁₂₅₈(C₂×D₄), SmallGroup(128,404)

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

Derived series

C₁ — C₄² — C₈⋊₈Q₁₆

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄×Q₈ — C₈×Q₈ — C₈⋊₈Q₁₆

Lower central

C₁ — C₂² — C₄² — C₈⋊₈Q₁₆

Upper central

C₁ — C₂² — C₄² — C₈⋊₈Q₁₆

Jennings

C₁ — C₂² — C₂² — C₄² — C₈⋊₈Q₁₆

Generators and relations for C₈⋊₈Q₁₆
G = < a,b,c | a⁸=b⁸=1, c²=b⁴, bab^-1=a³, ac=ca, cbc^-1=b^-1 >

Subgroups: 152 in 75 conjugacy classes, 36 normal (32 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.Q₈, C₂.D₈, C₄×Q₈, C₄⋊Q₈, C₂×Q₁₆, C₄.₁₀D₈, C₄.₆Q₁₆, C₈⋊₂C₈, C₈×Q₈, C₄⋊₂Q₁₆, C₄.Q₁₆, C₈⋊₃Q₈, C₈⋊₈Q₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, SD₁₆, Q₁₆, C₂×D₄, C₄○D₄, C₄⋊D₄, C₂×SD₁₆, C₂×Q₁₆, C₄○D₈, C₈.C₂², C₄⋊₂Q₁₆, C₈⋊₈D₄, D₄.₃D₄, C₈⋊₈Q₁₆

Character table of C₈⋊₈Q₁₆

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	-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
ρ₁₄	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
ρ₁₅	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
ρ₁₆	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
ρ₁₇	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	complex lifted from SD₁₆
ρ₁₈	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
ρ₁₉	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
ρ₂₀	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	complex lifted from SD₁₆
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	0	-2	2	0	0	0	0	√-2	-√-2	-√-2	√-2	-√-2	√-2	√-2	-√-2	√-2	-√-2	0	0	0	0	complex lifted from SD₁₆
ρ₂₅	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	complex lifted from SD₁₆
ρ₂₆	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
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	complex lifted from D₄.₃D₄
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	complex lifted from D₄.₃D₄

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

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

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

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

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

Matrix representation of C₈⋊₈Q₁₆ ►in GL₄(𝔽₁₇) generated by

1	0	0	0
0	1	0	0
0	0	0	10
0	0	12	10

9	0	0	0
0	2	0	0
0	0	4	9
0	0	0	13

0	1	0	0
16	0	0	0
0	0	13	8
0	0	13	4

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

C₈⋊₈Q₁₆ in GAP, Magma, Sage, TeX

C_8\rtimes_8Q_{16}

% in TeX

G:=Group("C8:8Q16");

// GroupNames label

G:=SmallGroup(128,404);

// by ID

G=gap.SmallGroup(128,404);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,64,422,352,1123,136,2804,718,172]);

// Polycyclic

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

// generators/relations

Export

Character table of C₈⋊₈Q₁₆ in TeX

G = C8⋊8Q16 order 128 = 27

2nd semidirect product of C8 and Q16 acting via Q16/Q8=C2

G = C₈⋊₈Q₁₆ order 128 = 2⁷

2^nd semidirect product of C₈ and Q₁₆ acting via Q₁₆/Q₈=C₂