C8.3Q16 - GroupNames

G = C₈.₃Q₁₆ order 128 = 2⁷

3^rd non-split extension by C₈ of Q₁₆ acting via Q₁₆/C₄=C₂²

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

Aliases: C₈.₃Q₁₆, C₄².₂₄₇C₂³, C₄⋊C₄.₆₉D₄, (C₂×C₈).₉₉D₄, (C₂×Q₈).₆₀D₄, C₄.₄₄(C₂×Q₁₆), C₈⋊₄Q₈.₄C₂, C₈⋊₂C₈.₁₀C₂, C₄⋊C₈.₃₃C₂², C₄.Q₁₆.₇C₂, C₈⋊₂Q₈.₁₉C₂, C₄⋊Q₈.₆₈C₂², (C₄×C₈).₁₅₀C₂², C₂.₉(C₈.D₄), C₄.₆Q₁₆.₆C₂, (C₄×Q₈).₄₈C₂², C₂.₁₁(C₄⋊₂Q₁₆), C₄.₄₅(C₈.C₂²), C₂.₁₅(D₄.₄D₄), C₂².₂₀₈(C₄⋊D₄), (C₂×C₄).₃₂(C₄○D₄), (C₂×C₄).₁₂₈₂(C₂×D₄), SmallGroup(128,428)

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

Derived series

C₁ — C₄² — C₈.₃Q₁₆

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄×Q₈ — C₈⋊₄Q₈ — C₈.₃Q₁₆

Lower central

C₁ — C₂² — C₄² — C₈.₃Q₁₆

Upper central

C₁ — C₂² — C₄² — C₈.₃Q₁₆

Jennings

C₁ — C₂² — C₂² — C₄² — C₈.₃Q₁₆

Generators and relations for C₈.₃Q₁₆
G = < a,b,c | a⁸=b⁸=1, c²=b⁴, bab^-1=a³, cac^-1=a⁵, cbc^-1=a⁴b^-1 >

Subgroups: 144 in 70 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, C₂×Q₈, C₂×Q₈, C₄×C₈, C₈⋊C₄, Q₈⋊C₄, C₄⋊C₈, C₄⋊C₈, C₄⋊C₈, C₂.D₈, C₄×Q₈, C₄⋊Q₈, C₄.₆Q₁₆, C₈⋊₂C₈, C₈⋊₄Q₈, C₄.Q₁₆, C₈⋊₂Q₈, C₈.₃Q₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, Q₁₆, C₂×D₄, C₄○D₄, C₄⋊D₄, C₂×Q₁₆, C₈.C₂², C₄⋊₂Q₁₆, C₈.D₄, D₄.₄D₄, C₈.₃Q₁₆

Character table of C₈.₃Q₁₆

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	0	0	0	0	0	0	0	0	-2√2	0	0	2√2	0	0	0	0	0	0	orthogonal lifted from D₄.₄D₄
ρ₂₀	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	orthogonal lifted from D₄.₄D₄
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	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
ρ₂₃	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

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

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

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

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

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

Matrix representation of C₈.₃Q₁₆ ►in GL₆(𝔽₁₇)

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

0	6	0	0	0	0
14	6	0	0	0	0
0	0	13	10	5	3
0	0	7	13	14	5
0	0	5	3	4	7
0	0	14	5	10	4

0	7	0	0	0	0
12	0	0	0	0	0
0	0	11	8	7	1
0	0	8	6	1	10
0	0	10	16	11	8
0	0	16	7	8	6

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,14,0,3,0,0,3,0,14,0,0,0,0,14,0,14,0,0,3,0,3,0],[0,14,0,0,0,0,6,6,0,0,0,0,0,0,13,7,5,14,0,0,10,13,3,5,0,0,5,14,4,10,0,0,3,5,7,4],[0,12,0,0,0,0,7,0,0,0,0,0,0,0,11,8,10,16,0,0,8,6,16,7,0,0,7,1,11,8,0,0,1,10,8,6] >;

C₈.₃Q₁₆ in GAP, Magma, Sage, TeX

C_8._3Q_{16}

% in TeX

G:=Group("C8.3Q16");

// GroupNames label

G:=SmallGroup(128,428);

// by ID

G=gap.SmallGroup(128,428);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,672,141,288,422,387,352,1123,136,2804,718,172]);

// 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=a^4*b^-1>;

// generators/relations

Export

Character table of C₈.₃Q₁₆ in TeX

G = C8.3Q16 order 128 = 27

3rd non-split extension by C8 of Q16 acting via Q16/C4=C22

G = C₈.₃Q₁₆ order 128 = 2⁷

3^rd non-split extension by C₈ of Q₁₆ acting via Q₁₆/C₄=C₂²