C8:13SD16 - GroupNames

G = C₈⋊₁₃SD₁₆ order 128 = 2⁷

1^st semidirect product of C₈ and SD₁₆ acting via SD₁₆/Q₈=C₂

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

Aliases: Q₈⋊₁D₈, C₈⋊₁₃SD₁₆, C₄².₂₁₉C₂³, (C₈×Q₈)⋊₅C₂, C₈⋊₁C₈⋊₅C₂, C₄.₃₂(C₂×D₈), C₄⋊C₄.₁₉₄D₄, (C₂×C₈).₃₄₀D₄, C₈⋊₄D₄.₄C₂, C₄⋊SD₁₆⋊₃₄C₂, C₄.₈₅(C₄○D₈), C₄.D₈⋊₁₄C₂, C₂.₇(C₈⋊₇D₄), (C₄×C₈).₅₀C₂², (C₂×Q₈).₁₄₇D₄, C₄.₇₉(C₂×SD₁₆), C₄⋊C₈.₁₇₄C₂², C₂.₇(C₄⋊SD₁₆), C₄.₁₁₇(C₈⋊C₂²), C₂.₇(D₄.₄D₄), C₄⋊₁D₄.₂₇C₂², (C₄×Q₈).₂₆₂C₂², C₂².₁₈₀(C₄⋊D₄), (C₂×C₄).₄(C₄○D₄), (C₂×C₄).₁₂₅₄(C₂×D₄), SmallGroup(128,400)

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

Derived series

C₁ — C₄² — C₈⋊₁₃SD₁₆

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄×Q₈ — C₈×Q₈ — C₈⋊₁₃SD₁₆

Lower central

C₁ — C₂² — C₄² — C₈⋊₁₃SD₁₆

Upper central

C₁ — C₂² — C₄² — C₈⋊₁₃SD₁₆

Jennings

C₁ — C₂² — C₂² — C₄² — C₈⋊₁₃SD₁₆

Generators and relations for C₈⋊₁₃SD₁₆
G = < a,b,c | a⁸=b⁸=c²=1, bab^-1=cac=a^-1, cbc=b³ >

Subgroups: 272 in 94 conjugacy classes, 36 normal (24 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, D₄, Q₈, Q₈, C₂³, C₄², C₄², C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, D₈, SD₁₆, C₂×D₄, C₂×Q₈, C₄×C₈, C₄×C₈, D₄⋊C₄, C₄⋊C₈, C₄⋊C₈, C₄⋊C₈, C₄×Q₈, C₄⋊₁D₄, C₂×D₈, C₂×SD₁₆, C₄.D₈, C₈⋊₁C₈, C₈×Q₈, C₄⋊SD₁₆, C₈⋊₄D₄, C₈⋊₁₃SD₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, D₈, SD₁₆, C₂×D₄, C₄○D₄, C₄⋊D₄, C₂×D₈, C₂×SD₁₆, C₄○D₈, C₈⋊C₂², C₄⋊SD₁₆, C₈⋊₇D₄, D₄.₄D₄, C₈⋊₁₃SD₁₆

Character table of C₈⋊₁₃SD₁₆

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	16	16	2	2	2	2	4	4	4	4	4	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	0	0	-2	-2	2	2	2	-2	-2	2	-2	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	0	-2	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	-2	2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	0	0	2	2	-2	-2	0	0	-2	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	0	0	0	0	-2	2	0	2	0	0	-2	-√2	√2	√2	-√2	√2	-√2	-√2	√2	-√2	√2	0	0	0	0	orthogonal lifted from D₈
ρ₁₄	2	2	-2	-2	0	0	0	0	-2	2	0	-2	0	0	2	-√2	√2	√2	-√2	-√2	√2	√2	√2	-√2	-√2	0	0	0	0	orthogonal lifted from D₈
ρ₁₅	2	2	-2	-2	0	0	0	0	-2	2	0	2	0	0	-2	√2	-√2	-√2	√2	-√2	√2	√2	-√2	√2	-√2	0	0	0	0	orthogonal lifted from D₈
ρ₁₆	2	2	-2	-2	0	0	0	0	-2	2	0	-2	0	0	2	√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	0	2	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	0	2	-2	2i	0	0	-2i	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	0	0	2	-2	-2i	0	0	2i	0	√2	-√2	-√2	√2	√-2	-√-2	√-2	√2	-√2	-√-2	0	0	0	0	complex lifted from C₄○D₈
ρ₂₀	2	-2	2	-2	0	0	-2	2	0	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	2	0	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	0	0	0	0	0	0	0	2	-2	2	-2	0	0	0	0	0	0	-√-2	-√-2	√-2	√-2	complex lifted from SD₁₆
ρ₂₃	2	2	-2	-2	0	0	0	0	2	-2	2i	0	0	-2i	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	-2	2	0	0	0	0	0	0	0	-2	2	-2	2	0	0	0	0	0	0	√-2	-√-2	-√-2	√-2	complex lifted from SD₁₆
ρ₂₅	2	2	-2	-2	0	0	0	0	2	-2	-2i	0	0	2i	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	0	0	-2	-2	-2	-2	0	0	2	0	0	0	0	0	0	-2i	-2i	2i	0	0	2i	0	0	0	0	complex lifted from C₄○D₄
ρ₂₇	4	-4	4	-4	0	0	4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₈⋊C₂²
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	orthogonal lifted from D₄.₄D₄
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	orthogonal lifted from D₄.₄D₄

Smallest permutation representation of C₈⋊₁₃SD₁₆
►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 44 57 53 22 12 36 31)(2 43 58 52 23 11 37 30)(3 42 59 51 24 10 38 29)(4 41 60 50 17 9 39 28)(5 48 61 49 18 16 40 27)(6 47 62 56 19 15 33 26)(7 46 63 55 20 14 34 25)(8 45 64 54 21 13 35 32)

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

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

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

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

Matrix representation of C₈⋊₁₃SD₁₆ ►in GL₄(𝔽₁₇) generated by

0	6	0	0
14	6	0	0
0	0	1	0
0	0	0	1

1	0	0	0
1	16	0	0
0	0	7	10
0	0	12	0

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

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

C₈⋊₁₃SD₁₆ in GAP, Magma, Sage, TeX

C_8\rtimes_{13}{\rm SD}_{16}

% in TeX

G:=Group("C8:13SD16");

// GroupNames label

G:=SmallGroup(128,400);

// by ID

G=gap.SmallGroup(128,400);

# by ID

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

// Polycyclic

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

// generators/relations

Export

Character table of C₈⋊₁₃SD₁₆ in TeX

G = C8⋊13SD16 order 128 = 27

1st semidirect product of C8 and SD16 acting via SD16/Q8=C2

G = C₈⋊₁₃SD₁₆ order 128 = 2⁷

1^st semidirect product of C₈ and SD₁₆ acting via SD₁₆/Q₈=C₂