C8:7D8 - GroupNames

G = C₈⋊₇D₈ order 128 = 2⁷

1^st semidirect product of C₈ and D₈ acting via D₈/D₄=C₂

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

Aliases: C₈⋊₇D₈, D₄⋊₁D₈, C₄².₂₁₈C₂³, (C₈×D₄)⋊₅C₂, C₈⋊₁C₈⋊₃C₂, C₈⋊₄D₄⋊₃C₂, C₄⋊D₈⋊₅C₂, C₄.₅₉(C₂×D₈), C₄⋊C₄.₁₉₃D₄, (C₂×C₈).₃₃₉D₄, C₄.D₈⋊₅C₂, (C₂×D₄).₁₈₇D₄, C₄.₈₄(C₄○D₈), C₂.₆(C₈⋊₇D₄), C₂.₇(C₄⋊D₈), C₄⋊C₈.₁₉C₂², (C₄×C₈).₄₉C₂², C₄.₁₁₆(C₈⋊C₂²), C₂.₆(D₄.₄D₄), (C₄×D₄).₂₇₅C₂², C₄⋊₁D₄.₂₆C₂², C₂².₁₇₉(C₄⋊D₄), (C₂×C₄).₃(C₄○D₄), (C₂×C₄).₁₂₅₃(C₂×D₄), SmallGroup(128,399)

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

Derived series

C₁ — C₄² — C₈⋊₇D₈

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄×D₄ — C₈×D₄ — C₈⋊₇D₈

Lower central

C₁ — C₂² — C₄² — C₈⋊₇D₈

Upper central

C₁ — C₂² — C₄² — C₈⋊₇D₈

Jennings

C₁ — C₂² — C₂² — C₄² — C₈⋊₇D₈

Generators and relations for C₈⋊₇D₈
G = < a,b,c | a⁸=b⁸=c²=1, bab^-1=cac=a^-1, cbc=b^-1 >

Subgroups: 304 in 102 conjugacy classes, 36 normal (24 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, D₄, D₄, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, D₈, C₂²×C₄, C₂×D₄, C₂×D₄, C₄×C₈, C₂²⋊C₈, D₄⋊C₄, C₄⋊C₈, C₄⋊C₈, C₄×D₄, C₄⋊₁D₄, C₂²×C₈, C₂×D₈, C₄.D₈, C₈⋊₁C₈, C₈×D₄, C₄⋊D₈, C₈⋊₄D₄, C₈⋊₇D₈
Quotients: C₁, C₂, C₂², D₄, C₂³, D₈, C₂×D₄, C₄○D₄, C₄⋊D₄, C₂×D₈, C₄○D₈, C₈⋊C₂², C₄⋊D₈, C₈⋊₇D₄, D₄.₄D₄, C₈⋊₇D₈

Character table of C₈⋊₇D₈

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

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

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

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

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

Matrix representation of C₈⋊₇D₈ ►in GL₄(𝔽₁₇) generated by

1	0	0	0
0	1	0	0
0	0	11	11
0	0	3	0

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

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

G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,11,3,0,0,11,0],[3,3,0,0,14,3,0,0,0,0,16,1,0,0,0,1],[1,0,0,0,0,16,0,0,0,0,1,16,0,0,0,16] >;

C₈⋊₇D₈ in GAP, Magma, Sage, TeX

C_8\rtimes_7D_8

% in TeX

G:=Group("C8:7D8");

// GroupNames label

G:=SmallGroup(128,399);

// by ID

G=gap.SmallGroup(128,399);

# by ID

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

// Polycyclic

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

// generators/relations

Export

Character table of C₈⋊₇D₈ in TeX

G = C8⋊7D8 order 128 = 27

1st semidirect product of C8 and D8 acting via D8/D4=C2

G = C₈⋊₇D₈ order 128 = 2⁷

1^st semidirect product of C₈ and D₈ acting via D₈/D₄=C₂