C8:SD16 - GroupNames

G = C₈⋊SD₁₆ order 128 = 2⁷

1^st semidirect product of C₈ and SD₁₆ acting via SD₁₆/C₄=C₂²

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

Aliases: C₈⋊₁SD₁₆, C₄².₂₃₇C₂³, C₄⋊C₄.₅₉D₄, C₈⋊₄Q₈⋊₁C₂, C₈⋊₁C₈⋊₂₀C₂, (C₂×C₈).₈₉D₄, C₈⋊₅D₄.₂C₂, (C₂×Q₈).₅₅D₄, Q₈⋊Q₈⋊₃₆C₂, C₂.₉(C₈⋊D₄), C₄⋊SD₁₆.₉C₂, C₄.₈₁(C₂×SD₁₆), C₄.D₈.₈C₂, C₄⋊Q₈.₆₀C₂², C₄.₁₀D₈⋊₂₇C₂, C₄⋊C₈.₁₈₃C₂², (C₄×C₈).₁₄₀C₂², C₂.₉(C₄⋊SD₁₆), (C₄×Q₈).₄₃C₂², C₄.₁₂₁(C₈⋊C₂²), C₄⋊₁D₄.₃₃C₂², C₄.₇₁(C₈.C₂²), C₂.₁₇(D₄.₃D₄), C₂².₁₉₈(C₄⋊D₄), (C₂×C₄).₂₂(C₄○D₄), (C₂×C₄).₁₂₇₂(C₂×D₄), SmallGroup(128,418)

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

Derived series

C₁ — C₄² — C₈⋊SD₁₆

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄×Q₈ — C₈⋊₄Q₈ — C₈⋊SD₁₆

Lower central

C₁ — C₂² — C₄² — C₈⋊SD₁₆

Upper central

C₁ — C₂² — C₄² — C₈⋊SD₁₆

Jennings

C₁ — C₂² — C₂² — C₄² — C₈⋊SD₁₆

Generators and relations for C₈⋊SD₁₆
G = < a,b,c | a⁸=b⁸=c²=1, bab^-1=a^-1, cac=a³, cbc=b³ >

Subgroups: 216 in 84 conjugacy classes, 34 normal (32 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₄², C₄², C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, SD₁₆, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄×C₈, C₈⋊C₄, D₄⋊C₄, Q₈⋊C₄, C₄⋊C₈, C₄⋊C₈, C₄.Q₈, C₄×Q₈, C₄⋊₁D₄, C₄⋊Q₈, C₂×SD₁₆, C₄.D₈, C₄.₁₀D₈, C₈⋊₁C₈, C₈⋊₄Q₈, C₄⋊SD₁₆, Q₈⋊Q₈, C₈⋊₅D₄, C₈⋊SD₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, SD₁₆, C₂×D₄, C₄○D₄, C₄⋊D₄, C₂×SD₁₆, C₈⋊C₂², C₈.C₂², C₄⋊SD₁₆, C₈⋊D₄, D₄.₃D₄, C₈⋊SD₁₆

Character table of C₈⋊SD₁₆

class	1	2A	2B	2C	2D	4A	4B	4C	4D	4E	4F	4G	4H	8A	8B	8C	8D	8E	8F	8G	8H	8I	8J
size	1	1	1	1	16	2	2	2	2	4	8	8	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	0	2	-2	2	-2	-2	0	0	0	2	-2	-2	2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	0	2	-2	2	-2	-2	0	0	0	-2	2	2	-2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	0	-2	2	-2	2	-2	2	-2	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	0	-2	2	-2	2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	2	2	0	-2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	-2i	2i	complex lifted from C₄○D₄
ρ₁₄	2	2	2	2	0	-2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	2i	-2i	complex lifted from C₄○D₄
ρ₁₅	2	-2	-2	2	0	-2	0	2	0	0	0	0	0	0	-2	2	0	-√-2	√-2	√-2	-√-2	0	0	complex lifted from SD₁₆
ρ₁₆	2	-2	-2	2	0	-2	0	2	0	0	0	0	0	0	2	-2	0	√-2	√-2	-√-2	-√-2	0	0	complex lifted from SD₁₆
ρ₁₇	2	-2	-2	2	0	-2	0	2	0	0	0	0	0	0	2	-2	0	-√-2	-√-2	√-2	√-2	0	0	complex lifted from SD₁₆
ρ₁₈	2	-2	-2	2	0	-2	0	2	0	0	0	0	0	0	-2	2	0	√-2	-√-2	-√-2	√-2	0	0	complex lifted from SD₁₆
ρ₁₉	4	-4	4	-4	0	0	4	0	-4	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	0	0	-4	0	4	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	complex lifted from D₄.₃D₄
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	complex lifted from D₄.₃D₄

Smallest permutation representation of C₈⋊SD₁₆
►On 64 points

Generators in S₆₄

(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)

(1 29 35 59 55 43 21 12)(2 28 36 58 56 42 22 11)(3 27 37 57 49 41 23 10)(4 26 38 64 50 48 24 9)(5 25 39 63 51 47 17 16)(6 32 40 62 52 46 18 15)(7 31 33 61 53 45 19 14)(8 30 34 60 54 44 20 13)

(2 4)(3 7)(6 8)(9 42)(10 45)(11 48)(12 43)(13 46)(14 41)(15 44)(16 47)(17 39)(18 34)(19 37)(20 40)(21 35)(22 38)(23 33)(24 36)(25 63)(26 58)(27 61)(28 64)(29 59)(30 62)(31 57)(32 60)(49 53)(50 56)(52 54)

G:=sub<Sym(64)| (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), (1,29,35,59,55,43,21,12)(2,28,36,58,56,42,22,11)(3,27,37,57,49,41,23,10)(4,26,38,64,50,48,24,9)(5,25,39,63,51,47,17,16)(6,32,40,62,52,46,18,15)(7,31,33,61,53,45,19,14)(8,30,34,60,54,44,20,13), (2,4)(3,7)(6,8)(9,42)(10,45)(11,48)(12,43)(13,46)(14,41)(15,44)(16,47)(17,39)(18,34)(19,37)(20,40)(21,35)(22,38)(23,33)(24,36)(25,63)(26,58)(27,61)(28,64)(29,59)(30,62)(31,57)(32,60)(49,53)(50,56)(52,54)>;

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), (1,29,35,59,55,43,21,12)(2,28,36,58,56,42,22,11)(3,27,37,57,49,41,23,10)(4,26,38,64,50,48,24,9)(5,25,39,63,51,47,17,16)(6,32,40,62,52,46,18,15)(7,31,33,61,53,45,19,14)(8,30,34,60,54,44,20,13), (2,4)(3,7)(6,8)(9,42)(10,45)(11,48)(12,43)(13,46)(14,41)(15,44)(16,47)(17,39)(18,34)(19,37)(20,40)(21,35)(22,38)(23,33)(24,36)(25,63)(26,58)(27,61)(28,64)(29,59)(30,62)(31,57)(32,60)(49,53)(50,56)(52,54) );

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)], [(1,29,35,59,55,43,21,12),(2,28,36,58,56,42,22,11),(3,27,37,57,49,41,23,10),(4,26,38,64,50,48,24,9),(5,25,39,63,51,47,17,16),(6,32,40,62,52,46,18,15),(7,31,33,61,53,45,19,14),(8,30,34,60,54,44,20,13)], [(2,4),(3,7),(6,8),(9,42),(10,45),(11,48),(12,43),(13,46),(14,41),(15,44),(16,47),(17,39),(18,34),(19,37),(20,40),(21,35),(22,38),(23,33),(24,36),(25,63),(26,58),(27,61),(28,64),(29,59),(30,62),(31,57),(32,60),(49,53),(50,56),(52,54)]])

Matrix representation of C₈⋊SD₁₆ ►in GL₈(𝔽₁₇)

8	11	16	16	0	0	0	0
5	8	0	16	0	0	0	0
6	5	14	11	0	0	0	0
12	1	12	4	0	0	0	0
0	0	0	0	0	13	6	6
0	0	0	0	0	0	11	0
0	0	0	0	11	14	0	0
0	0	0	0	3	0	13	0

0	0	1	0	0	0	0	0
1	1	1	2	0	0	0	0
0	16	0	0	0	0	0	0
0	0	16	16	0	0	0	0
0	0	0	0	0	0	1	0
0	0	0	0	0	7	15	15
0	0	0	0	1	0	0	0
0	0	0	0	16	7	10	10

1	0	0	0	0	0	0	0
0	16	0	0	0	0	0	0
16	16	16	15	0	0	0	0
0	1	0	1	0	0	0	0
0	0	0	0	1	0	0	0
0	0	0	0	15	16	0	0
0	0	0	0	0	0	1	0
0	0	0	0	10	0	16	16

G:=sub<GL(8,GF(17))| [8,5,6,12,0,0,0,0,11,8,5,1,0,0,0,0,16,0,14,12,0,0,0,0,16,16,11,4,0,0,0,0,0,0,0,0,0,0,11,3,0,0,0,0,13,0,14,0,0,0,0,0,6,11,0,13,0,0,0,0,6,0,0,0],[0,1,0,0,0,0,0,0,0,1,16,0,0,0,0,0,1,1,0,16,0,0,0,0,0,2,0,16,0,0,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,7,0,7,0,0,0,0,1,15,0,10,0,0,0,0,0,15,0,10],[1,0,16,0,0,0,0,0,0,16,16,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,1,15,0,10,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,0,16] >;

C₈⋊SD₁₆ in GAP, Magma, Sage, TeX

C_8\rtimes {\rm SD}_{16}

% in TeX

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

// GroupNames label

G:=SmallGroup(128,418);

// by ID

G=gap.SmallGroup(128,418);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,64,422,387,352,1123,136,2804,718,172]);

// Polycyclic

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

// generators/relations

Export

Character table of C₈⋊SD₁₆ in TeX

8	11	16	16	0	0	0	0
5	8	0	16	0	0	0	0
6	5	14	11	0	0	0	0
12	1	12	4	0	0	0	0
0	0	0	0	0	13	6	6
0	0	0	0	0	0	11	0
0	0	0	0	11	14	0	0
0	0	0	0	3	0	13	0

0	0	1	0	0	0	0	0
1	1	1	2	0	0	0	0
0	16	0	0	0	0	0	0
0	0	16	16	0	0	0	0
0	0	0	0	0	0	1	0
0	0	0	0	0	7	15	15
0	0	0	0	1	0	0	0
0	0	0	0	16	7	10	10

1	0	0	0	0	0	0	0
0	16	0	0	0	0	0	0
16	16	16	15	0	0	0	0
0	1	0	1	0	0	0	0
0	0	0	0	1	0	0	0
0	0	0	0	15	16	0	0
0	0	0	0	0	0	1	0
0	0	0	0	10	0	16	16

8	11	16	16	0	0	0	0
5	8	0	16	0	0	0	0
6	5	14	11	0	0	0	0
12	1	12	4	0	0	0	0
0	0	0	0	0	13	6	6
0	0	0	0	0	0	11	0
0	0	0	0	11	14	0	0
0	0	0	0	3	0	13	0

0	0	1	0	0	0	0	0
1	1	1	2	0	0	0	0
0	16	0	0	0	0	0	0
0	0	16	16	0	0	0	0
0	0	0	0	0	0	1	0
0	0	0	0	0	7	15	15
0	0	0	0	1	0	0	0
0	0	0	0	16	7	10	10

1	0	0	0	0	0	0	0
0	16	0	0	0	0	0	0
16	16	16	15	0	0	0	0
0	1	0	1	0	0	0	0
0	0	0	0	1	0	0	0
0	0	0	0	15	16	0	0
0	0	0	0	0	0	1	0
0	0	0	0	10	0	16	16

G = C8⋊SD16 order 128 = 27

1st semidirect product of C8 and SD16 acting via SD16/C4=C22

G = C₈⋊SD₁₆ order 128 = 2⁷

1^st semidirect product of C₈ and SD₁₆ acting via SD₁₆/C₄=C₂²

8	11	16	16	0	0	0	0
5	8	0	16	0	0	0	0
6	5	14	11	0	0	0	0
12	1	12	4	0	0	0	0
0	0	0	0	0	13	6	6
0	0	0	0	0	0	11	0
0	0	0	0	11	14	0	0
0	0	0	0	3	0	13	0

0	0	1	0	0	0	0	0
1	1	1	2	0	0	0	0
0	16	0	0	0	0	0	0
0	0	16	16	0	0	0	0
0	0	0	0	0	0	1	0
0	0	0	0	0	7	15	15
0	0	0	0	1	0	0	0
0	0	0	0	16	7	10	10

1	0	0	0	0	0	0	0
0	16	0	0	0	0	0	0
16	16	16	15	0	0	0	0
0	1	0	1	0	0	0	0
0	0	0	0	1	0	0	0
0	0	0	0	15	16	0	0
0	0	0	0	0	0	1	0
0	0	0	0	10	0	16	16