C8:3D8 - GroupNames

G = C₈⋊₃D₈ order 128 = 2⁷

3^rd semidirect product of C₈ and D₈ acting via D₈/C₄=C₂²

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

Aliases: C₈⋊₃D₈, C₄².₆₆₈C₂³, C₄.₅(C₂×D₈), C₈⋊₅D₄⋊₂C₂, C₈⋊₄D₄⋊₅C₂, C₈⋊C₈⋊₁₄C₂, (C₂×C₈).₃₇D₄, C₄.₄D₈⋊₁₀C₂, C₂.₈(C₈⋊₄D₄), C₄.₆(C₈⋊C₂²), C₄⋊Q₈.₉₂C₂², C₂.₁₁(C₈⋊₃D₄), (C₄×C₈).₁₅₄C₂², C₄⋊₁D₄.₅₁C₂², C₂².₆₉(C₄⋊₁D₄), (C₂×C₄).₇₂₅(C₂×D₄), SmallGroup(128,453)

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

Derived series

C₁ — C₄² — C₈⋊₃D₈

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄×C₈ — C₈⋊C₈ — C₈⋊₃D₈

Lower central

C₁ — C₂² — C₄² — C₈⋊₃D₈

Upper central

C₁ — C₂² — C₄² — C₈⋊₃D₈

Jennings

C₁ — C₂² — C₂² — C₄² — C₈⋊₃D₈

Generators and relations for C₈⋊₃D₈
G = < a,b,c | a⁸=b⁸=c²=1, bab^-1=a⁵, cac=a^-1, cbc=b^-1 >

Subgroups: 384 in 121 conjugacy classes, 40 normal (12 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₄², C₄⋊C₄, C₂×C₈, D₈, SD₁₆, C₂×D₄, C₂×Q₈, C₄×C₈, C₄×C₈, D₄⋊C₄, C₄⋊₁D₄, C₄⋊₁D₄, C₄⋊Q₈, C₂×D₈, C₂×SD₁₆, C₈⋊C₈, C₄.₄D₈, C₈⋊₅D₄, C₈⋊₄D₄, C₈⋊₄D₄, C₈⋊₃D₈
Quotients: C₁, C₂, C₂², D₄, C₂³, D₈, C₂×D₄, C₄⋊₁D₄, C₂×D₈, C₈⋊C₂², C₈⋊₄D₄, C₈⋊₃D₄, C₈⋊₃D₈

Character table of C₈⋊₃D₈

class	1	2A	2B	2C	2D	2E	2F	4A	4B	4C	4D	4E	4F	4G	8A	8B	8C	8D	8E	8F	8G	8H	8I	8J	8K	8L
size	1	1	1	1	16	16	16	2	2	2	2	2	2	16	4	4	4	4	4	4	4	4	4	4	4	4
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	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	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	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	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	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	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	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	linear of order 2
ρ₉	2	2	2	2	0	0	0	-2	2	-2	-2	2	-2	0	0	0	0	0	-2	0	0	2	0	0	2	-2	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	0	0	0	-2	-2	-2	2	-2	2	0	0	0	2	2	0	0	-2	0	0	-2	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	0	0	0	2	-2	2	-2	-2	-2	0	2	-2	0	0	0	-2	0	0	2	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	0	0	0	-2	-2	-2	2	-2	2	0	0	0	-2	-2	0	0	2	0	0	2	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	2	2	0	0	0	2	-2	2	-2	-2	-2	0	-2	2	0	0	0	2	0	0	-2	0	0	0	orthogonal lifted from D₄
ρ₁₄	2	2	2	2	0	0	0	-2	2	-2	-2	2	-2	0	0	0	0	0	2	0	0	-2	0	0	-2	2	orthogonal lifted from D₄
ρ₁₅	2	-2	2	-2	0	0	0	0	-2	0	0	2	0	0	√2	√2	-√2	√2	0	-√2	√2	2	-√2	-√2	-2	0	orthogonal lifted from D₈
ρ₁₆	2	-2	2	-2	0	0	0	0	-2	0	0	2	0	0	√2	√2	√2	-√2	0	-√2	-√2	-2	-√2	√2	2	0	orthogonal lifted from D₈
ρ₁₇	2	-2	2	-2	0	0	0	0	2	0	0	-2	0	0	-√2	√2	√2	-√2	-2	-√2	√2	0	√2	-√2	0	2	orthogonal lifted from D₈
ρ₁₈	2	-2	2	-2	0	0	0	0	2	0	0	-2	0	0	√2	-√2	√2	-√2	2	√2	√2	0	-√2	-√2	0	-2	orthogonal lifted from D₈
ρ₁₉	2	-2	2	-2	0	0	0	0	-2	0	0	2	0	0	-√2	-√2	-√2	√2	0	√2	√2	-2	√2	-√2	2	0	orthogonal lifted from D₈
ρ₂₀	2	-2	2	-2	0	0	0	0	-2	0	0	2	0	0	-√2	-√2	√2	-√2	0	√2	-√2	2	√2	√2	-2	0	orthogonal lifted from D₈
ρ₂₁	2	-2	2	-2	0	0	0	0	2	0	0	-2	0	0	-√2	√2	-√2	√2	2	-√2	-√2	0	√2	√2	0	-2	orthogonal lifted from D₈
ρ₂₂	2	-2	2	-2	0	0	0	0	2	0	0	-2	0	0	√2	-√2	-√2	√2	-2	√2	-√2	0	-√2	√2	0	2	orthogonal lifted from D₈
ρ₂₃	4	-4	-4	4	0	0	0	0	0	0	-4	0	4	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₈⋊C₂²
ρ₂₄	4	-4	-4	4	0	0	0	0	0	0	4	0	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₈⋊C₂²
ρ₂₅	4	4	-4	-4	0	0	0	-4	0	4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₈⋊C₂²
ρ₂₆	4	4	-4	-4	0	0	0	4	0	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₈⋊C₂²

Smallest permutation representation of C₈⋊₃D₈
►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 25 18 46 39 51 61 9)(2 30 19 43 40 56 62 14)(3 27 20 48 33 53 63 11)(4 32 21 45 34 50 64 16)(5 29 22 42 35 55 57 13)(6 26 23 47 36 52 58 10)(7 31 24 44 37 49 59 15)(8 28 17 41 38 54 60 12)

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

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

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

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

Matrix representation of C₈⋊₃D₈ ►in GL₆(𝔽₁₇)

16	0	0	0	0	0
0	16	0	0	0	0
0	0	16	16	3	14
0	0	1	16	3	3
0	0	3	14	1	1
0	0	3	3	16	1

3	14	0	0	0	0
3	3	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	16	0	0	0
0	0	0	16	0	0

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

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,1,3,3,0,0,16,16,14,3,0,0,3,3,1,16,0,0,14,3,1,1],[3,3,0,0,0,0,14,3,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;

C₈⋊₃D₈ in GAP, Magma, Sage, TeX

C_8\rtimes_3D_8

% in TeX

G:=Group("C8:3D8");

// GroupNames label

G:=SmallGroup(128,453);

// by ID

G=gap.SmallGroup(128,453);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,64,422,387,436,1123,136,2804,172]);

// Polycyclic

G:=Group<a,b,c|a^8=b^8=c^2=1,b*a*b^-1=a^5,c*a*c=a^-1,c*b*c=b^-1>;

// generators/relations

Export

Character table of C₈⋊₃D₈ in TeX

G = C8⋊3D8 order 128 = 27

3rd semidirect product of C8 and D8 acting via D8/C4=C22

G = C₈⋊₃D₈ order 128 = 2⁷

3^rd semidirect product of C₈ and D₈ acting via D₈/C₄=C₂²