Q8.(C2xD4) - GroupNames

G = Q₈.(C₂×D₄) order 128 = 2⁷

10^th non-split extension by Q₈ of C₂×D₄ acting via C₂×D₄/C₂³=C₂

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

Aliases: C₄○D₄.₁₄D₄, D₄.₄₆(C₂×D₄), Q₈.₄₆(C₂×D₄), C₂.₆(Q₈○D₈), C₄.₄₂C₂²≀C₂, (C₂×D₄).₂₉₅D₄, C₄⋊C₄.₁₃C₂³, (C₂×Q₈).₁₂₃D₄, (C₂²×Q₁₆)⋊₇C₂, C₄.₄₈(C₂²×D₄), D₄.₇D₄⋊₁₆C₂, (C₂×C₈).₁₃₀C₂³, (C₂×C₄).₂₃₀C₂⁴, C₂²⋊Q₁₆⋊₁₃C₂, C₂³.₂₃₄(C₂×D₄), (C₂×Q₈).₂₃C₂³, C₂².₂₂C₂²≀C₂, (C₂×D₄).₃₈₃C₂³, C₂³.₂₄D₄⋊₇C₂, C₂³.₃₈D₄⋊₅C₂, C₂²⋊C₈.₁₂C₂², (C₂×SD₁₆).₁C₂², C₂²⋊Q₈.₁₇C₂², (C₂²×C₄).₂₇₈C₂³, (C₂²×C₈).₁₃₅C₂², Q₈⋊C₄.₂₀C₂², (C₂×Q₁₆).₁₁₃C₂², C₂².₄₉₀(C₂²×D₄), (C₂×2_-¹⁺⁴).₆C₂, D₄⋊C₄.₁₅₃C₂², C₄²⋊C₂.₉₈C₂², C₂³.₃₈C₂³⋊₄C₂, (C₂×M₄(2)).₄₂C₂², (C₂²×Q₈).₂₆₂C₂², (C₂×C₄).₄₅₇(C₂×D₄), (C₂×C₈.C₂²)⋊₉C₂, (C₂²×C₈)⋊C₂⋊₆C₂, C₂.₄₈(C₂×C₂²≀C₂), (C₂×C₄○D₄).₁₀₂C₂², SmallGroup(128,1743)

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

Derived series

C₁ — C₂×C₄ — Q₈.(C₂×D₄)

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂×C₄○D₄ — C₂×2_-¹⁺⁴ — Q₈.(C₂×D₄)

Lower central

C₁ — C₂ — C₂×C₄ — Q₈.(C₂×D₄)

Upper central

C₁ — C₂² — C₂×C₄○D₄ — Q₈.(C₂×D₄)

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — Q₈.(C₂×D₄)

Generators and relations for Q₈.(C₂×D₄)
G = < a,b,c,d,e | a⁴=1, b²=c²=d⁴=e²=a², bab^-1=eae^-1=a^-1, ac=ca, ad=da, bc=cb, dbd^-1=a^-1b, ebe^-1=ab, cd=dc, ece^-1=a²c, ede^-1=a²d³ >

Subgroups: 620 in 350 conjugacy classes, 108 normal (28 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, M₄(2), SD₁₆, Q₁₆, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂×Q₈, C₄○D₄, C₄○D₄, C₂²⋊C₈, D₄⋊C₄, Q₈⋊C₄, C₄²⋊C₂, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₄⋊Q₈, C₂²×C₈, C₂×M₄(2), C₂×SD₁₆, C₂×Q₁₆, C₂×Q₁₆, C₈.C₂², C₂²×Q₈, C₂²×Q₈, C₂²×Q₈, C₂×C₄○D₄, C₂×C₄○D₄, C₂×C₄○D₄, 2_-¹⁺⁴, (C₂²×C₈)⋊C₂, C₂³.₂₄D₄, C₂³.₃₈D₄, C₂²⋊Q₁₆, D₄.₇D₄, C₂³.₃₈C₂³, C₂²×Q₁₆, C₂×C₈.C₂², C₂×2_-¹⁺⁴, Q₈.(C₂×D₄)
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₂⁴, C₂²≀C₂, C₂²×D₄, C₂×C₂²≀C₂, Q₈○D₈, Q₈.(C₂×D₄)

Smallest permutation representation of Q₈.(C₂×D₄)
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	4A	4B	4C	4D	4E	···	4L	4M	4N	4O	4P	8A	8B	8C	8D	8E	8F
order	1	2	2	2	2	2	2	2	2	2	4	4	4	4	4	···	4	4	4	4	4	8	8	8	8	8	8
size	1	1	1	1	2	2	4	4	4	4	2	2	2	2	4	···	4	8	8	8	8	4	4	4	4	8	8

32 irreducible representations

dim

type

image

C₁

C₂

D₄

Q₈○D₈

kernel

Q₈.(C₂×D₄)

(C₂²×C₈)⋊C₂

C₂³.₂₄D₄

C₂³.₃₈D₄

C₂²⋊Q₁₆

D₄.₇D₄

C₂³.₃₈C₂³

C₂²×Q₁₆

C₂×C₈.C₂²

C₂×2_-¹⁺⁴

C₂×D₄

C₂×Q₈

C₄○D₄

C₂

# reps

Matrix representation of Q₈.(C₂×D₄) ►in GL₆(𝔽₁₇)

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

16	0	0	0	0	0
6	1	0	0	0	0
0	0	12	12	0	0
0	0	12	5	0	0
0	0	5	5	5	5
0	0	5	12	5	12

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	1	0	2
0	0	16	0	15	0
0	0	0	0	0	16
0	0	0	0	1	0

11	15	0	0	0	0
9	6	0	0	0	0
0	0	14	14	11	11
0	0	3	14	6	11
0	0	0	0	3	3
0	0	0	0	14	3

6	2	0	0	0	0
8	11	0	0	0	0
0	0	13	0	9	0
0	0	0	4	0	8
0	0	0	0	4	0
0	0	0	0	0	13

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[16,6,0,0,0,0,0,1,0,0,0,0,0,0,12,12,5,5,0,0,12,5,5,12,0,0,0,0,5,5,0,0,0,0,5,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,15,0,1,0,0,2,0,16,0],[11,9,0,0,0,0,15,6,0,0,0,0,0,0,14,3,0,0,0,0,14,14,0,0,0,0,11,6,3,14,0,0,11,11,3,3],[6,8,0,0,0,0,2,11,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,9,0,4,0,0,0,0,8,0,13] >;

Q₈.(C₂×D₄) in GAP, Magma, Sage, TeX

Q_8.(C_2\times D_4)

% in TeX

G:=Group("Q8.(C2xD4)");

// GroupNames label

G:=SmallGroup(128,1743);

// by ID

G=gap.SmallGroup(128,1743);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,521,2804,1411,172]);

// Polycyclic

G:=Group<a,b,c,d,e|a^4=1,b^2=c^2=d^4=e^2=a^2,b*a*b^-1=e*a*e^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^-1*b,e*b*e^-1=a*b,c*d=d*c,e*c*e^-1=a^2*c,e*d*e^-1=a^2*d^3>;

// generators/relations

G = Q8.(C2×D4) order 128 = 27

10th non-split extension by Q8 of C2×D4 acting via C2×D4/C23=C2

G = Q₈.(C₂×D₄) order 128 = 2⁷

10^th non-split extension by Q₈ of C₂×D₄ acting via C₂×D₄/C₂³=C₂