C2^3.50D8 - GroupNames

G = C₂³.₅₀D₈ order 128 = 2⁷

21^st non-split extension by C₂³ of D₈ acting via D₈/D₄=C₂

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C₂³.₅₀D₈, C₂².₆SD₃₂, (C₂×C₈).₇₅D₄, (C₂×C₄).₄₃D₈, C₁₆⋊₄C₄⋊₁₁C₂, C₈.₇₂(C₄○D₄), C₂²⋊C₁₆.₈C₂, C₂.₁₃(C₂×SD₃₂), C₂.Q₃₂⋊₁₂C₂, (C₂×C₈).₅₃₅C₂³, (C₂×C₁₆).₄₇C₂², C₈.₁₈D₄.₅C₂, C₂².₁₂₁(C₂×D₈), (C₂²×C₄).₃₅₅D₄, C₂.D₈.₂₀C₂², C₂.₁₈(Q₃₂⋊C₂), C₄.₁₇(C₈.C₂²), (C₂×Q₁₆).₁₀C₂², (C₂²×C₈).₁₃₁C₂², C₄.₄₂(C₂².D₄), C₂.₁₅(C₂².D₈), (C₂×C₄).₈₀₃(C₂×D₄), (C₂×C₂.D₈).₂₅C₂, SmallGroup(128,967)

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

Derived series

C₁ — C₂×C₈ — C₂³.₅₀D₈

Chief series

C₁ — C₂ — C₄ — C₈ — C₂×C₈ — C₂.D₈ — C₂×C₂.D₈ — C₂³.₅₀D₈

Lower central

C₁ — C₂ — C₄ — C₂×C₈ — C₂³.₅₀D₈

Upper central

C₁ — C₂² — C₂²×C₄ — C₂²×C₈ — C₂³.₅₀D₈

Jennings

C₁ — C₂ — C₂ — C₂ — C₂ — C₄ — C₄ — C₂×C₈ — C₂³.₅₀D₈

Generators and relations for C₂³.₅₀D₈
G = < a,b,c,d,e | a²=b²=c²=1, d⁸=e²=c, dad^-1=eae^-1=ab=ba, ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede^-1=bd⁷ >

Subgroups: 164 in 70 conjugacy classes, 32 normal (20 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, Q₈, C₂³, C₁₆, C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, Q₁₆, C₂²×C₄, C₂²×C₄, C₂×Q₈, Q₈⋊C₄, C₂.D₈, C₂.D₈, C₂.D₈, C₂×C₁₆, C₂×C₄⋊C₄, C₂²⋊Q₈, C₂²×C₈, C₂×Q₁₆, C₂²⋊C₁₆, C₂.Q₃₂, C₁₆⋊₄C₄, C₂×C₂.D₈, C₈.₁₈D₄, C₂³.₅₀D₈
Quotients: C₁, C₂, C₂², D₄, C₂³, D₈, C₂×D₄, C₄○D₄, SD₃₂, C₂².D₄, C₂×D₈, C₈.C₂², C₂².D₈, C₂×SD₃₂, Q₃₂⋊C₂, C₂³.₅₀D₈

Character table of C₂³.₅₀D₈

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	4I	8A	8B	8C	8D	8E	8F	16A	16B	16C	16D	16E	16F	16G	16H
size	1	1	1	1	2	2	2	2	4	8	8	8	8	16	16	2	2	2	2	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	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	0	0	0	0	0	0	-2	-2	-2	-2	2	2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	2	2	2	2	2	0	0	0	0	0	0	-2	-2	-2	-2	-2	-2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	2	2	-2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	0	√2	√2	-√2	√2	-√2	√2	-√2	-√2	orthogonal lifted from D₈
ρ₁₂	2	2	2	2	2	2	-2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	0	-√2	-√2	√2	-√2	√2	-√2	√2	√2	orthogonal lifted from D₈
ρ₁₃	2	2	2	2	-2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	0	-√2	√2	-√2	√2	√2	-√2	√2	-√2	orthogonal lifted from D₈
ρ₁₄	2	2	2	2	-2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	0	√2	-√2	√2	-√2	-√2	√2	-√2	√2	orthogonal lifted from D₈
ρ₁₅	2	-2	-2	2	0	0	2	-2	0	0	0	-2i	2i	0	0	-2	2	-2	2	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₁₆	2	-2	-2	2	0	0	2	-2	0	2i	-2i	0	0	0	0	2	-2	2	-2	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₁₇	2	-2	-2	2	0	0	2	-2	0	-2i	2i	0	0	0	0	2	-2	2	-2	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₁₈	2	-2	-2	2	0	0	2	-2	0	0	0	2i	-2i	0	0	-2	2	-2	2	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₁₉	2	-2	2	-2	2	-2	0	0	0	0	0	0	0	0	0	√2	√2	-√2	-√2	-√2	√2	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷+ζ₁₆	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁷+ζ₁₆	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹⁵+ζ₁₆⁹	complex lifted from SD₃₂
ρ₂₀	2	-2	2	-2	2	-2	0	0	0	0	0	0	0	0	0	√2	√2	-√2	-√2	-√2	√2	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁷+ζ₁₆	ζ₁₆⁷+ζ₁₆	complex lifted from SD₃₂
ρ₂₁	2	-2	2	-2	-2	2	0	0	0	0	0	0	0	0	0	√2	√2	-√2	-√2	√2	-√2	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷+ζ₁₆	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁷+ζ₁₆	complex lifted from SD₃₂
ρ₂₂	2	-2	2	-2	-2	2	0	0	0	0	0	0	0	0	0	√2	√2	-√2	-√2	√2	-√2	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷+ζ₁₆	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁷+ζ₁₆	ζ₁₆¹⁵+ζ₁₆⁹	complex lifted from SD₃₂
ρ₂₃	2	-2	2	-2	2	-2	0	0	0	0	0	0	0	0	0	-√2	-√2	√2	√2	√2	-√2	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁷+ζ₁₆	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁷+ζ₁₆	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁵+ζ₁₆³	complex lifted from SD₃₂
ρ₂₄	2	-2	2	-2	2	-2	0	0	0	0	0	0	0	0	0	-√2	-√2	√2	√2	√2	-√2	ζ₁₆⁷+ζ₁₆	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷+ζ₁₆	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹³+ζ₁₆¹¹	complex lifted from SD₃₂
ρ₂₅	2	-2	2	-2	-2	2	0	0	0	0	0	0	0	0	0	-√2	-√2	√2	√2	-√2	√2	ζ₁₆⁷+ζ₁₆	ζ₁₆⁷+ζ₁₆	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁵+ζ₁₆³	complex lifted from SD₃₂
ρ₂₆	2	-2	2	-2	-2	2	0	0	0	0	0	0	0	0	0	-√2	-√2	√2	√2	-√2	√2	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷+ζ₁₆	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁷+ζ₁₆	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹³+ζ₁₆¹¹	complex lifted from SD₃₂
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	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	symplectic lifted from Q₃₂⋊C₂, Schur index 2
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	symplectic lifted from Q₃₂⋊C₂, Schur index 2

Smallest permutation representation of C₂³.₅₀D₈
►On 64 points

Generators in S₆₄

(2 34)(4 36)(6 38)(8 40)(10 42)(12 44)(14 46)(16 48)(18 51)(20 53)(22 55)(24 57)(26 59)(28 61)(30 63)(32 49)

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

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

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

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

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

Matrix representation of C₂³.₅₀D₈ ►in GL₄(𝔽₁₇) generated by

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

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

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

6	0	0	0
0	14	0	0
0	0	4	9
0	0	4	13

0	14	0	0
6	0	0	0
0	0	1	15
0	0	0	16

G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[6,0,0,0,0,14,0,0,0,0,4,4,0,0,9,13],[0,6,0,0,14,0,0,0,0,0,1,0,0,0,15,16] >;

C₂³.₅₀D₈ in GAP, Magma, Sage, TeX

C_2^3._{50}D_8

% in TeX

G:=Group("C2^3.50D8");

// GroupNames label

G:=SmallGroup(128,967);

// by ID

G=gap.SmallGroup(128,967);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,422,58,1684,438,242,4037,1027,124]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^8=e^2=c,d*a*d^-1=e*a*e^-1=a*b=b*a,a*c=c*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^7>;

// generators/relations

Export

Character table of C₂³.₅₀D₈ in TeX

G = C23.50D8 order 128 = 27

21st non-split extension by C23 of D8 acting via D8/D4=C2

G = C₂³.₅₀D₈ order 128 = 2⁷

21^st non-split extension by C₂³ of D₈ acting via D₈/D₄=C₂