C8:10SD16 - GroupNames

G = C₈⋊₁₀SD₁₆ order 128 = 2⁷

1^st semidirect product of C₈ and SD₁₆ acting via SD₁₆/D₄=C₂

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

Aliases: D₄.₁D₈, C₈⋊₁₀SD₁₆, C₄².₂₂₄C₂³, C₈⋊₁C₈⋊₆C₂, C₈⋊₂Q₈⋊₄C₂, (C₈×D₄).₅C₂, C₄.₃₃(C₂×D₈), C₄⋊C₄.₁₉₉D₄, (C₂×C₈).₃₄₃D₄, (C₂×D₄).₁₉₀D₄, C₄.₈₆(C₄○D₈), C₂.₈(C₈⋊₇D₄), (C₄×C₈).₅₃C₂², C₄.₅₈(C₂×SD₁₆), C₄⋊Q₈.₄₇C₂², C₄.₁₀D₈⋊₂₄C₂, C₄⋊C₈.₁₇₇C₂², D₄.D₄.₇C₂, C₂.₇(D₄.D₄), C₂.₈(D₄.₅D₄), (C₄×D₄).₂₇₈C₂², C₄.₁₁₂(C₈.C₂²), C₂².₁₈₅(C₄⋊D₄), (C₂×C₄).₉(C₄○D₄), (C₂×C₄).₁₂₅₉(C₂×D₄), SmallGroup(128,405)

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

Derived series

C₁ — C₄² — C₈⋊₁₀SD₁₆

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄×D₄ — C₈×D₄ — C₈⋊₁₀SD₁₆

Lower central

C₁ — C₂² — C₄² — C₈⋊₁₀SD₁₆

Upper central

C₁ — C₂² — C₄² — C₈⋊₁₀SD₁₆

Jennings

C₁ — C₂² — C₂² — C₄² — C₈⋊₁₀SD₁₆

Generators and relations for C₈⋊₁₀SD₁₆
G = < a,b,c | a⁸=b⁸=c²=1, bab^-1=a^-1, ac=ca, cbc=b³ >

Subgroups: 192 in 86 conjugacy classes, 36 normal (24 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, SD₁₆, C₂²×C₄, C₂×D₄, C₂×Q₈, C₄×C₈, C₂²⋊C₈, Q₈⋊C₄, C₄⋊C₈, C₄⋊C₈, C₂.D₈, C₄×D₄, C₄⋊Q₈, C₂²×C₈, C₂×SD₁₆, C₄.₁₀D₈, C₈⋊₁C₈, C₈×D₄, D₄.D₄, C₈⋊₂Q₈, C₈⋊₁₀SD₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, D₈, SD₁₆, C₂×D₄, C₄○D₄, C₄⋊D₄, C₂×D₈, C₂×SD₁₆, C₄○D₈, C₈.C₂², D₄.D₄, C₈⋊₇D₄, D₄.₅D₄, C₈⋊₁₀SD₁₆

Character table of C₈⋊₁₀SD₁₆

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	4I	8A	8B	8C	8D	8E	8F	8G	8H	8I	8J	8K	8L	8M	8N
size	1	1	1	1	4	4	2	2	2	2	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	-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
ρ₁₀	2	2	2	2	0	0	2	-2	2	-2	0	-2	0	0	0	2	2	2	2	0	0	0	-2	-2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	0	0	2	-2	2	-2	0	-2	0	0	0	-2	-2	-2	-2	0	0	0	2	2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	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
ρ₁₃	2	2	-2	-2	2	-2	0	2	0	-2	0	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	-2	2	0	2	0	-2	0	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	2	-2	0	2	0	-2	0	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	-2	2	0	2	0	-2	0	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	0	0	-2	-2	-2	-2	0	2	0	0	0	0	0	0	0	-2i	-2i	2i	0	0	2i	0	0	0	0	complex lifted from C₄○D₄
ρ₁₈	2	2	-2	-2	0	0	0	-2	0	2	-2i	0	2i	0	0	√2	-√2	-√2	√2	-√-2	√-2	-√-2	√2	-√2	√-2	0	0	0	0	complex lifted from C₄○D₈
ρ₁₉	2	-2	2	-2	0	0	-2	0	2	0	0	0	0	0	0	2	-2	2	-2	0	0	0	0	0	0	√-2	√-2	-√-2	-√-2	complex lifted from SD₁₆
ρ₂₀	2	-2	2	-2	0	0	-2	0	2	0	0	0	0	0	0	-2	2	-2	2	0	0	0	0	0	0	-√-2	√-2	√-2	-√-2	complex lifted from SD₁₆
ρ₂₁	2	-2	2	-2	0	0	-2	0	2	0	0	0	0	0	0	2	-2	2	-2	0	0	0	0	0	0	-√-2	-√-2	√-2	√-2	complex lifted from SD₁₆
ρ₂₂	2	2	2	2	0	0	-2	-2	-2	-2	0	2	0	0	0	0	0	0	0	2i	2i	-2i	0	0	-2i	0	0	0	0	complex lifted from C₄○D₄
ρ₂₃	2	2	-2	-2	0	0	0	-2	0	2	2i	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	0	0	-2	0	2	2i	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	0	0	-2	0	2	0	0	0	0	0	0	-2	2	-2	2	0	0	0	0	0	0	√-2	-√-2	-√-2	√-2	complex lifted from SD₁₆
ρ₂₆	2	2	-2	-2	0	0	0	-2	0	2	-2i	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	0	0	4	0	-4	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⋊₁₀SD₁₆
►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 59 35 16 49 30 43 18)(2 58 36 15 50 29 44 17)(3 57 37 14 51 28 45 24)(4 64 38 13 52 27 46 23)(5 63 39 12 53 26 47 22)(6 62 40 11 54 25 48 21)(7 61 33 10 55 32 41 20)(8 60 34 9 56 31 42 19)

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

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

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

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

Matrix representation of C₈⋊₁₀SD₁₆ ►in GL₄(𝔽₁₇) generated by

15	0	0	0
12	8	0	0
0	0	1	0
0	0	0	1

16	2	0	0
16	1	0	0
0	0	0	10
0	0	12	10

1	0	0	0
1	16	0	0
0	0	1	0
0	0	1	16

G:=sub<GL(4,GF(17))| [15,12,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[16,16,0,0,2,1,0,0,0,0,0,12,0,0,10,10],[1,1,0,0,0,16,0,0,0,0,1,1,0,0,0,16] >;

C₈⋊₁₀SD₁₆ in GAP, Magma, Sage, TeX

C_8\rtimes_{10}{\rm SD}_{16}

% in TeX

G:=Group("C8:10SD16");

// GroupNames label

G:=SmallGroup(128,405);

// by ID

G=gap.SmallGroup(128,405);

# by ID

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

// Polycyclic

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

// generators/relations

Export

Character table of C₈⋊₁₀SD₁₆ in TeX

G = C8⋊10SD16 order 128 = 27

1st semidirect product of C8 and SD16 acting via SD16/D4=C2

G = C₈⋊₁₀SD₁₆ order 128 = 2⁷

1^st semidirect product of C₈ and SD₁₆ acting via SD₁₆/D₄=C₂