C2^3.51D8 - GroupNames

G = C₂³.₅₁D₈ order 128 = 2⁷

22^nd non-split extension by C₂³ of D₈ acting via D₈/D₄=C₂

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

Aliases: C₂³.₅₁D₈, C₂².₃Q₃₂, C₁₆⋊₃C₄⋊₈C₂, (C₂×C₈).₇₆D₄, (C₂×C₄).₄₄D₈, C₂.₈(C₂×Q₃₂), C₂.Q₃₂⋊₇C₂, C₈.₇₃(C₄○D₄), C₂²⋊C₁₆.₆C₂, (C₂×C₈).₅₃₆C₂³, (C₂×C₁₆).₁₁C₂², C₈.₁₈D₄.₆C₂, C₂².₁₂₂(C₂×D₈), (C₂²×C₄).₃₅₆D₄, C₂.₁₉(C₁₆⋊C₂²), C₂.D₈.₂₁C₂², C₄.₁₈(C₈.C₂²), (C₂×Q₁₆).₁₁C₂², (C₂²×C₈).₁₃₂C₂², C₄.₄₃(C₂².D₄), C₂.₁₆(C₂².D₈), (C₂×C₄).₈₀₄(C₂×D₄), (C₂×C₂.D₈).₂₆C₂, SmallGroup(128,968)

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

Derived series

C₁ — C₂×C₈ — C₂³.₅₁D₈

Chief series

C₁ — C₂ — C₄ — C₈ — C₂×C₈ — C₂.D₈ — C₂×C₂.D₈ — C₂³.₅₁D₈

Lower central

C₁ — C₂ — C₄ — C₂×C₈ — C₂³.₅₁D₈

Upper central

C₁ — C₂² — C₂²×C₄ — C₂²×C₈ — C₂³.₅₁D₈

Jennings

C₁ — C₂ — C₂ — C₂ — C₂ — C₄ — C₄ — C₂×C₈ — C₂³.₅₁D₈

Generators and relations for C₂³.₅₁D₈
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=bcd⁷ >

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⋊₃C₄, C₂×C₂.D₈, C₈.₁₈D₄, C₂³.₅₁D₈
Quotients: C₁, C₂, C₂², D₄, C₂³, D₈, C₂×D₄, C₄○D₄, Q₃₂, C₂².D₄, C₂×D₈, C₈.C₂², C₂².D₈, C₂×Q₃₂, C₁₆⋊C₂², C₂³.₅₁D₈

Character table of C₂³.₅₁D₈

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	2	-2	0	0	0	0	0	0	0	0	0	√2	√2	-√2	-√2	-√2	√2	ζ₁₆⁷-ζ₁₆	-ζ₁₆⁷+ζ₁₆	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷-ζ₁₆	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆⁷+ζ₁₆	ζ₁₆⁵-ζ₁₆³	ζ₁₆⁵-ζ₁₆³	symplectic lifted from Q₃₂, Schur index 2
ρ₁₆	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	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁷+ζ₁₆	ζ₁₆⁵-ζ₁₆³	ζ₁₆⁷-ζ₁₆	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆⁵+ζ₁₆³	symplectic lifted from Q₃₂, Schur index 2
ρ₁₇	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	ζ₁₆⁵-ζ₁₆³	ζ₁₆⁷-ζ₁₆	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆⁷+ζ₁₆	-ζ₁₆⁷+ζ₁₆	symplectic lifted from Q₃₂, Schur index 2
ρ₁₈	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	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁷+ζ₁₆	ζ₁₆⁵-ζ₁₆³	ζ₁₆⁷-ζ₁₆	-ζ₁₆⁷+ζ₁₆	symplectic lifted from Q₃₂, Schur index 2
ρ₁₉	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	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆⁷+ζ₁₆	ζ₁₆⁵-ζ₁₆³	ζ₁₆⁷-ζ₁₆	ζ₁₆⁷-ζ₁₆	symplectic lifted from Q₃₂, Schur index 2
ρ₂₀	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	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷-ζ₁₆	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆⁷+ζ₁₆	ζ₁₆⁷-ζ₁₆	symplectic lifted from Q₃₂, Schur index 2
ρ₂₁	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	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷-ζ₁₆	ζ₁₆⁵-ζ₁₆³	ζ₁₆⁷-ζ₁₆	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁵-ζ₁₆³	symplectic lifted from Q₃₂, Schur index 2
ρ₂₂	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	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁷+ζ₁₆	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆⁷+ζ₁₆	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁵+ζ₁₆³	symplectic lifted from Q₃₂, Schur index 2
ρ₂₃	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
ρ₂₄	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
ρ₂₅	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
ρ₂₆	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
ρ₂₇	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 C₁₆⋊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 C₁₆⋊C₂²
ρ₂₉	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

Smallest permutation representation of C₂³.₅₁D₈
►On 64 points

Generators in S₆₄

(2 52)(4 54)(6 56)(8 58)(10 60)(12 62)(14 64)(16 50)(17 37)(19 39)(21 41)(23 43)(25 45)(27 47)(29 33)(31 35)

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

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

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

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

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

Matrix representation of C₂³.₅₁D₈ ►in GL₄(𝔽₁₇) generated by

1	0	0	0
0	1	0	0
0	0	1	0
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	13	0	0
4	6	0	0
0	0	0	13
0	0	13	0

10	1	0	0
1	7	0	0
0	0	0	1
0	0	1	0

G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,16],[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],[6,4,0,0,13,6,0,0,0,0,0,13,0,0,13,0],[10,1,0,0,1,7,0,0,0,0,0,1,0,0,1,0] >;

C₂³.₅₁D₈ in GAP, Magma, Sage, TeX

C_2^3._{51}D_8

% in TeX

G:=Group("C2^3.51D8");

// GroupNames label

G:=SmallGroup(128,968);

// by ID

G=gap.SmallGroup(128,968);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,422,394,1684,438,242,4037,1027,124]);

// 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*c*d^7>;

// generators/relations

Export

Character table of C₂³.₅₁D₈ in TeX

G = C23.51D8 order 128 = 27

22nd non-split extension by C23 of D8 acting via D8/D4=C2

G = C₂³.₅₁D₈ order 128 = 2⁷

22^nd non-split extension by C₂³ of D₈ acting via D₈/D₄=C₂