C8:11SD16 - GroupNames

G = C₈⋊₁₁SD₁₆ order 128 = 2⁷

2^nd semidirect product of C₈ and SD₁₆ acting via SD₁₆/D₄=C₂

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

Aliases: C₈⋊₁₁SD₁₆, D₄.₁SD₁₆, C₄².₂₂₂C₂³, C₈⋊₂C₈⋊₂₃C₂, C₄⋊C₄.₁₉₇D₄, C₈⋊₃Q₈⋊₁₅C₂, (C₈×D₄).₁₃C₂, (C₂×C₈).₃₆₅D₄, (C₂×D₄).₁₈₉D₄, C₄.₆₆(C₄○D₈), D₄⋊₂Q₈.₇C₂, C₄.₅₇(C₂×SD₁₆), C₄⋊Q₈.₄₅C₂², C₄.₁₀D₈⋊₂₃C₂, C₂.₁₀(C₈⋊₈D₄), C₄⋊C₈.₁₇₆C₂², (C₄×C₈).₂₅₁C₂², C₄.₆Q₁₆⋊₁₄C₂, D₄.D₄.₆C₂, C₂.₆(D₄.D₄), (C₄×D₄).₂₇₇C₂², C₂.₁₀(D₄.₃D₄), C₄.₁₁₀(C₈.C₂²), C₂².₁₈₃(C₄⋊D₄), (C₂×C₄).₇(C₄○D₄), (C₂×C₄).₁₂₅₇(C₂×D₄), SmallGroup(128,403)

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

Derived series

C₁ — C₄² — C₈⋊₁₁SD₁₆

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄×D₄ — C₈×D₄ — C₈⋊₁₁SD₁₆

Lower central

C₁ — C₂² — C₄² — C₈⋊₁₁SD₁₆

Upper central

C₁ — C₂² — C₄² — C₈⋊₁₁SD₁₆

Jennings

C₁ — C₂² — C₂² — C₄² — C₈⋊₁₁SD₁₆

Generators and relations for C₈⋊₁₁SD₁₆
G = < a,b,c | a⁸=b⁸=c²=1, bab^-1=a³, ac=ca, cbc=b³ >

Subgroups: 184 in 83 conjugacy classes, 36 normal (32 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, D₄⋊C₄, Q₈⋊C₄, C₄⋊C₈, C₄.Q₈, C₄×D₄, C₄⋊Q₈, C₂²×C₈, C₂×SD₁₆, C₄.₁₀D₈, C₄.₆Q₁₆, C₈⋊₂C₈, C₈×D₄, D₄.D₄, D₄⋊₂Q₈, C₈⋊₃Q₈, C₈⋊₁₁SD₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, SD₁₆, C₂×D₄, C₄○D₄, C₄⋊D₄, C₂×SD₁₆, C₄○D₈, C₈.C₂², D₄.D₄, C₈⋊₈D₄, D₄.₃D₄, C₈⋊₁₁SD₁₆

Character table of C₈⋊₁₁SD₁₆

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	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
ρ₁₄	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
ρ₁₅	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
ρ₁₆	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
ρ₁₇	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	complex lifted from SD₁₆
ρ₁₈	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
ρ₁₉	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	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	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	complex lifted from SD₁₆
ρ₂₃	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
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	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
ρ₂₆	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	complex lifted from SD₁₆
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	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⋊₁₁SD₁₆
►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 12 39 20 45 32 49 58)(2 15 40 23 46 27 50 61)(3 10 33 18 47 30 51 64)(4 13 34 21 48 25 52 59)(5 16 35 24 41 28 53 62)(6 11 36 19 42 31 54 57)(7 14 37 22 43 26 55 60)(8 9 38 17 44 29 56 63)

(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)(25 59)(26 60)(27 61)(28 62)(29 63)(30 64)(31 57)(32 58)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 49)(40 50)

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

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

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

Matrix representation of C₈⋊₁₁SD₁₆ ►in GL₄(𝔽₁₇) generated by

1	0	0	0
0	1	0	0
0	0	15	0
0	0	13	9

12	5	0	0
12	12	0	0
0	0	7	2
0	0	9	10

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

G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,15,13,0,0,0,9],[12,12,0,0,5,12,0,0,0,0,7,9,0,0,2,10],[1,0,0,0,0,16,0,0,0,0,1,10,0,0,0,16] >;

C₈⋊₁₁SD₁₆ in GAP, Magma, Sage, TeX

C_8\rtimes_{11}{\rm SD}_{16}

% in TeX

G:=Group("C8:11SD16");

// GroupNames label

G:=SmallGroup(128,403);

// by ID

G=gap.SmallGroup(128,403);

# by ID

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

// Polycyclic

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

// generators/relations

Export

Character table of C₈⋊₁₁SD₁₆ in TeX

G = C8⋊11SD16 order 128 = 27

2nd semidirect product of C8 and SD16 acting via SD16/D4=C2

G = C₈⋊₁₁SD₁₆ order 128 = 2⁷

2^nd semidirect product of C₈ and SD₁₆ acting via SD₁₆/D₄=C₂