C8:8D8 - GroupNames

G = C₈⋊₈D₈ order 128 = 2⁷

2^nd semidirect product of C₈ and D₈ acting via D₈/D₄=C₂

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

Aliases: C₈⋊₈D₈, D₄⋊₁SD₁₆, C₄².₂₁₆C₂³, C₈⋊₂C₈⋊₂C₂, (C₈×D₄)⋊₂₈C₂, C₄.₅₈(C₂×D₈), C₈⋊₅D₄⋊₁₆C₂, C₄⋊C₄.₁₉₁D₄, D₄⋊Q₈⋊₆C₂, (C₂×C₈).₃₆₃D₄, C₄.D₈⋊₄C₂, C₄⋊D₈.₄C₂, (C₂×D₄).₁₈₆D₄, C₄.₆₄(C₄○D₈), C₄.₁₀D₈⋊₄C₂, C₂.₆(C₄⋊D₈), C₂.₈(C₈⋊₈D₄), C₄⋊C₈.₁₈C₂², C₄.₃₈(C₂×SD₁₆), C₄⋊Q₈.₄₁C₂², (C₄×C₈).₂₄₉C₂², C₄.₁₁₄(C₈⋊C₂²), C₂.₈(D₄.₃D₄), (C₄×D₄).₂₇₄C₂², C₄⋊₁D₄.₂₄C₂², C₂².₁₇₇(C₄⋊D₄), (C₂×C₄).₁(C₄○D₄), (C₂×C₄).₁₂₅₁(C₂×D₄), SmallGroup(128,397)

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

Derived series

C₁ — C₄² — C₈⋊₈D₈

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄×D₄ — C₈×D₄ — C₈⋊₈D₈

Lower central

C₁ — C₂² — C₄² — C₈⋊₈D₈

Upper central

C₁ — C₂² — C₄² — C₈⋊₈D₈

Jennings

C₁ — C₂² — C₂² — C₄² — C₈⋊₈D₈

Generators and relations for C₈⋊₈D₈
G = < a,b,c | a⁸=b⁸=c²=1, bab^-1=cac=a³, cbc=b^-1 >

Subgroups: 248 in 94 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, D₈, SD₁₆, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄×C₈, C₂²⋊C₈, D₄⋊C₄, C₄⋊C₈, C₂.D₈, C₄×D₄, C₄⋊₁D₄, C₄⋊Q₈, C₂²×C₈, C₂×D₈, C₂×SD₁₆, C₄.D₈, C₄.₁₀D₈, C₈⋊₂C₈, C₈×D₄, C₄⋊D₈, D₄⋊Q₈, C₈⋊₅D₄, C₈⋊₈D₈
Quotients: C₁, C₂, C₂², D₄, C₂³, D₈, SD₁₆, C₂×D₄, C₄○D₄, C₄⋊D₄, C₂×D₈, C₂×SD₁₆, C₄○D₈, C₈⋊C₂², C₄⋊D₈, C₈⋊₈D₄, D₄.₃D₄, C₈⋊₈D₈

Character table of C₈⋊₈D₈

class	1	2A	2B	2C	2D	2E	2F	4A	4B	4C	4D	4E	4F	4G	4H	8A	8B	8C	8D	8E	8F	8G	8H	8I	8J	8K	8L	8M	8N
size	1	1	1	1	4	4	16	2	2	2	2	4	4	4	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	0	0	0	-2	2	-2	2	0	0	-2	0	2	2	2	2	0	0	0	-2	-2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	0	0	0	-2	2	-2	2	0	0	-2	0	-2	-2	-2	-2	0	0	0	2	2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	-2	-2	0	2	-2	2	-2	2	2	-2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	2	2	0	2	-2	2	-2	-2	-2	-2	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	0	0	2	0	-2	0	0	0	0	2	-2	-2	2	0	0	0	0	0	0	√2	√2	-√2	-√2	orthogonal lifted from D₈
ρ₁₄	2	-2	2	-2	0	0	0	0	2	0	-2	0	0	0	0	-2	2	2	-2	0	0	0	0	0	0	-√2	√2	√2	-√2	orthogonal lifted from D₈
ρ₁₅	2	-2	2	-2	0	0	0	0	2	0	-2	0	0	0	0	-2	2	2	-2	0	0	0	0	0	0	√2	-√2	-√2	√2	orthogonal lifted from D₈
ρ₁₆	2	-2	2	-2	0	0	0	0	2	0	-2	0	0	0	0	2	-2	-2	2	0	0	0	0	0	0	-√2	-√2	√2	√2	orthogonal lifted from D₈
ρ₁₇	2	2	-2	-2	0	0	0	-2	0	2	0	2i	-2i	0	0	-√-2	√-2	-√-2	√-2	-√2	√2	-√2	-√-2	√-2	√2	0	0	0	0	complex lifted from C₄○D₈
ρ₁₈	2	2	2	2	0	0	0	-2	-2	-2	-2	0	0	2	0	0	0	0	0	-2i	-2i	2i	0	0	2i	0	0	0	0	complex lifted from C₄○D₄
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	-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
ρ₂₁	2	2	2	2	0	0	0	-2	-2	-2	-2	0	0	2	0	0	0	0	0	2i	2i	-2i	0	0	-2i	0	0	0	0	complex lifted from C₄○D₄
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	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
ρ₂₄	2	2	-2	-2	0	0	0	-2	0	2	0	2i	-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	0	-2i	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	0	0	0	-2	0	2	0	-2i	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	0	0	-4	0	4	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	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⋊₈D₈
►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 57 42 20 55 40 10)(2 28 58 45 21 50 33 13)(3 31 59 48 22 53 34 16)(4 26 60 43 23 56 35 11)(5 29 61 46 24 51 36 14)(6 32 62 41 17 54 37 9)(7 27 63 44 18 49 38 12)(8 30 64 47 19 52 39 15)

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

Matrix representation of C₈⋊₈D₈ ►in GL₄(𝔽₁₇) generated by

0	5	0	0
7	10	0	0
0	0	16	0
0	0	0	16

1	0	0	0
2	16	0	0
0	0	3	14
0	0	3	3

1	0	0	0
2	16	0	0
0	0	1	0
0	0	0	16

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

C₈⋊₈D₈ in GAP, Magma, Sage, TeX

C_8\rtimes_8D_8

% in TeX

G:=Group("C8:8D8");

// GroupNames label

G:=SmallGroup(128,397);

// by ID

G=gap.SmallGroup(128,397);

# by ID

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

// Polycyclic

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

// generators/relations

Export

Character table of C₈⋊₈D₈ in TeX

G = C8⋊8D8 order 128 = 27

2nd semidirect product of C8 and D8 acting via D8/D4=C2

G = C₈⋊₈D₈ order 128 = 2⁷

2^nd semidirect product of C₈ and D₈ acting via D₈/D₄=C₂