C4^2.391C2^3 - GroupNames

G = C₄².₃₉₁C₂³ order 128 = 2⁷

252^nd non-split extension by C₄² of C₂³ acting via C₂³/C₂=C₂²

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

Aliases: C₄².₃₉₁C₂³, C₈⋊Q₈⋊₉C₂, (C₄×D₈)⋊₃₀C₂, C₈⋊D₄⋊₉C₂, C₈⋊₂D₄⋊₇C₂, C₈⋊₃D₄⋊₇C₂, C₈⋊₇D₄⋊₂₇C₂, C₈⋊₈D₄⋊₁₅C₂, C₄⋊C₄.₂₅₁D₄, D₈⋊C₄⋊₁₅C₂, (C₄×SD₁₆)⋊₁₈C₂, C₈.₃₀(C₄○D₄), C₂.₂₆(D₄○D₈), C₄.₄D₈⋊₄₂C₂, C₂²⋊C₄.₉₁D₄, C₂³.₈₈(C₂×D₄), C₄⋊C₄.₁₁₈C₂³, (C₂×C₄).₃₇₇C₂⁴, (C₄×C₈).₁₈₄C₂², (C₂×C₈).₂₇₉C₂³, (C₄×D₄).₉₇C₂², C₄⋊Q₈.₁₁₉C₂², SD₁₆⋊C₄⋊₂₂C₂, (C₄×Q₈).₉₄C₂², C₈○₂M₄(2)⋊₂₀C₂, C₄.Q₈.₂₉C₂², C₂.₄₀(D₄○SD₁₆), (C₂×D₈).₁₆₄C₂², (C₂×D₄).₁₃₁C₂³, C₄⋊D₄.₃₈C₂², C₄⋊₁D₄.₆₆C₂², (C₂×Q₈).₁₁₉C₂³, C₂.D₈.₂₂₁C₂², C₈⋊C₄.₁₃₄C₂², C₂²⋊Q₈.₃₈C₂², (C₂²×C₈).₂₇₉C₂², (C₂×SD₁₆).₂₅C₂², C₄.₄D₄.₃₇C₂², C₂².₆₃₇(C₂²×D₄), C₄².C₂.₂₃C₂², D₄⋊C₄.₁₄₉C₂², (C₂²×C₄).₁₀₅₇C₂³, C₂².₃₆C₂⁴⋊₆C₂, C₂².₃₄C₂⁴⋊₄C₂, Q₈⋊C₄.₁₄₁C₂², C₄².₇₈C₂²⋊₃₀C₂, C₄²⋊C₂.₃₃₄C₂², (C₂×M₄(2)).₂₈₇C₂², C₂.₇₄(C₂².₂₆C₂⁴), C₄.₆₂(C₂×C₄○D₄), (C₂×C₄).₁₄₉(C₂×D₄), SmallGroup(128,1911)

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

Derived series

C₁ — C₂×C₄ — C₄².₃₉₁C₂³

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₄² — C₄×C₈ — C₈○₂M₄(2) — C₄².₃₉₁C₂³

Lower central

C₁ — C₂ — C₂×C₄ — C₄².₃₉₁C₂³

Upper central

C₁ — C₂² — C₄²⋊C₂ — C₄².₃₉₁C₂³

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₄².₃₉₁C₂³

Generators and relations for C₄².₃₉₁C₂³
G = < a,b,c,d,e | a⁴=b⁴=d²=1, c²=b², e²=b, ab=ba, ac=ca, dad=ab², ae=ea, cbc^-1=b^-1, bd=db, be=eb, dcd=a²c, ece^-1=b^-1c, ede^-1=b²d >

Subgroups: 388 in 187 conjugacy classes, 88 normal (84 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₄², C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, M₄(2), D₈, SD₁₆, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄×C₈, C₈⋊C₄, D₄⋊C₄, Q₈⋊C₄, C₄.Q₈, C₂.D₈, C₄²⋊C₂, C₄×D₄, C₄×Q₈, C₄⋊D₄, C₄⋊D₄, C₂²⋊Q₈, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₄.₄D₄, C₄².C₂, C₄²⋊₂C₂, C₄⋊₁D₄, C₄⋊Q₈, C₂²×C₈, C₂×M₄(2), C₂×D₈, C₂×SD₁₆, C₈○₂M₄(2), C₄×D₈, C₄×SD₁₆, SD₁₆⋊C₄, D₈⋊C₄, C₈⋊₈D₄, C₈⋊₇D₄, C₈⋊D₄, C₈⋊₂D₄, C₄.₄D₈, C₄².₇₈C₂², C₈⋊₃D₄, C₈⋊Q₈, C₂².₃₄C₂⁴, C₂².₃₆C₂⁴, C₄².₃₉₁C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, C₂².₂₆C₂⁴, D₄○D₈, D₄○SD₁₆, C₄².₃₉₁C₂³

Smallest permutation representation of C₄².₃₉₁C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	4A	···	4F	4G	4H	4I	4J	···	4N	8A	8B	8C	8D	8E	···	8J
order	1	2	2	2	2	2	2	2	4	···	4	4	4	4	4	···	4	8	8	8	8	8	···	8
size	1	1	1	1	4	8	8	8	2	···	2	4	4	4	8	···	8	2	2	2	2	4	···	4

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

D₄○D₈

D₄○SD₁₆

kernel

C₄².₃₉₁C₂³

C₈○₂M₄(2)

C₄×D₈

C₄×SD₁₆

SD₁₆⋊C₄

D₈⋊C₄

C₈⋊₈D₄

C₈⋊₇D₄

C₈⋊D₄

C₈⋊₂D₄

C₄.₄D₈

C₄².₇₈C₂²

C₈⋊₃D₄

C₈⋊Q₈

C₂².₃₄C₂⁴

C₂².₃₆C₂⁴

C₂²⋊C₄

C₄⋊C₄

C₈

C₂

# reps

Matrix representation of C₄².₃₉₁C₂³ ►in GL₆(𝔽₁₇)

13	0	0	0	0	0
0	13	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	1	0	0	0
0	0	0	1	0	0

1	0	0	0	0	0
0	1	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

13	9	0	0	0	0
4	4	0	0	0	0
0	0	12	0	0	12
0	0	0	5	12	0
0	0	0	12	12	0
0	0	12	0	0	5

1	2	0	0	0	0
0	16	0	0	0	0
0	0	0	12	12	0
0	0	5	0	0	12
0	0	5	0	0	5
0	0	0	5	12	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	0	12	5
0	0	0	0	12	12
0	0	12	5	0	0
0	0	12	12	0	0

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

C₄².₃₉₁C₂³ in GAP, Magma, Sage, TeX

C_4^2._{391}C_2^3

% in TeX

G:=Group("C4^2.391C2^3");

// GroupNames label

G:=SmallGroup(128,1911);

// by ID

G=gap.SmallGroup(128,1911);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,723,184,1018,80,4037,124]);

// Polycyclic

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

// generators/relations

G = C42.391C23 order 128 = 27

252nd non-split extension by C42 of C23 acting via C23/C2=C22

G = C₄².₃₉₁C₂³ order 128 = 2⁷

252^nd non-split extension by C₄² of C₂³ acting via C₂³/C₂=C₂²