C2^3.390C2^4 - GroupNames

G = C₂³.₃₉₀C₂⁴ order 128 = 2⁷

107^th central stem extension by C₂³ of C₂⁴

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C₂⁴.₁₁C₂³, C₂³.₃₉₀C₂⁴, C₂².₁₉₀2₊¹⁺⁴, C₄⋊C₄.₂₂₈D₄, C₂.₆₂(D₄⋊₅D₄), C₂.₂₀(Q₈⋊₆D₄), C₄.₄₃(C₄.₄D₄), C₂³.₄₁(C₄○D₄), C₂³.₇Q₈⋊₅₆C₂, C₂³.₁₀D₄⋊₃₅C₂, C₂³.₁₁D₄⋊₂₉C₂, (C₂³×C₄).₃₇₅C₂², (C₂×C₄²).₅₁₈C₂², C₂².₂₇₀(C₂²×D₄), C₂⁴.₃C₂²⋊₄₇C₂, (C₂²×C₄).₁₄₈₂C₂³, C₂⁴.C₂²⋊₆₄C₂, (C₂²×D₄).₁₄₆C₂², C₂.C₄².₁₄₃C₂², C₂.₄₈(C₂³.₃₆C₂³), C₂.₃₂(C₂².₄₇C₂⁴), C₂.₁₆(C₂².₃₄C₂⁴), (C₄×C₄⋊C₄)⋊₆₈C₂, (C₂×C₄).₆₁(C₂×D₄), (C₄×C₂²⋊C₄)⋊₇₃C₂, (C₂×C₄².C₂)⋊₉C₂, (C₂×C₄⋊D₄).₃₁C₂, C₂.₁₆(C₂×C₄.₄D₄), (C₂×C₄).₈₁₂(C₄○D₄), (C₂×C₄⋊C₄).₂₆₀C₂², C₂².₂₆₇(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₁₅₅C₂², SmallGroup(128,1222)

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

Derived series

C₁ — C₂³ — C₂³.₃₉₀C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₄×C₂²⋊C₄ — C₂³.₃₉₀C₂⁴

Lower central

C₁ — C₂³ — C₂³.₃₉₀C₂⁴

Upper central

C₁ — C₂³ — C₂³.₃₉₀C₂⁴

Jennings

C₁ — C₂³ — C₂³.₃₉₀C₂⁴

Generators and relations for C₂³.₃₉₀C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=f²=1, d²=ca=ac, e²=b, g²=a, ab=ba, ede^-1=gdg^-1=ad=da, ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 580 in 280 conjugacy classes, 104 normal (42 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂⁴, C₂.C₄², C₂.C₄², C₂×C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄⋊D₄, C₄².C₂, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₄×C₂²⋊C₄, C₄×C₄⋊C₄, C₂³.₇Q₈, C₂⁴.C₂², C₂⁴.₃C₂², C₂⁴.₃C₂², C₂³.₁₀D₄, C₂³.₁₁D₄, C₂×C₄⋊D₄, C₂×C₄².C₂, C₂³.₃₉₀C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₄.₄D₄, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, C₂×C₄.₄D₄, C₂³.₃₆C₂³, C₂².₃₄C₂⁴, D₄⋊₅D₄, Q₈⋊₆D₄, C₂².₄₇C₂⁴, C₂³.₃₉₀C₂⁴

Smallest permutation representation of C₂³.₃₉₀C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	···	4H	4I	···	4X	4Y	4Z
order	1	2	···	2	2	2	2	2	4	···	4	4	···	4	4	4
size	1	1	···	1	4	4	8	8	2	···	2	4	···	4	8	8

38 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

kernel

C₂³.₃₉₀C₂⁴

C₄×C₂²⋊C₄

C₄×C₄⋊C₄

C₂³.₇Q₈

C₂⁴.C₂²

C₂⁴.₃C₂²

C₂³.₁₀D₄

C₂³.₁₁D₄

C₂×C₄⋊D₄

C₂×C₄².C₂

C₄⋊C₄

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₃₉₀C₂⁴ ►in GL₆(𝔽₅)

4	0	0	0	0	0
0	4	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	4	0
0	0	0	0	0	4

4	0	0	0	0	0
0	4	0	0	0	0
0	0	4	0	0	0
0	0	0	4	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	4	0	0	0
0	0	0	4	0	0
0	0	0	0	1	0
0	0	0	0	0	1

2	0	0	0	0	0
2	3	0	0	0	0
0	0	4	2	0	0
0	0	4	1	0	0
0	0	0	0	2	1
0	0	0	0	0	3

1	3	0	0	0	0
1	4	0	0	0	0
0	0	4	2	0	0
0	0	4	1	0	0
0	0	0	0	2	1
0	0	0	0	2	3

2	1	0	0	0	0
2	3	0	0	0	0
0	0	4	0	0	0
0	0	4	1	0	0
0	0	0	0	4	0
0	0	0	0	0	4

1	3	0	0	0	0
1	4	0	0	0	0
0	0	4	0	0	0
0	0	0	4	0	0
0	0	0	0	4	2
0	0	0	0	4	1

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,2,0,0,0,0,0,3,0,0,0,0,0,0,4,4,0,0,0,0,2,1,0,0,0,0,0,0,2,0,0,0,0,0,1,3],[1,1,0,0,0,0,3,4,0,0,0,0,0,0,4,4,0,0,0,0,2,1,0,0,0,0,0,0,2,2,0,0,0,0,1,3],[2,2,0,0,0,0,1,3,0,0,0,0,0,0,4,4,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,1,0,0,0,0,3,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,4,0,0,0,0,2,1] >;

C₂³.₃₉₀C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{390}C_2^4

% in TeX

G:=Group("C2^3.390C2^4");

// GroupNames label

G:=SmallGroup(128,1222);

// by ID

G=gap.SmallGroup(128,1222);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,232,758,723,675,80]);

// Polycyclic

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

// generators/relations

G = C23.390C24 order 128 = 27

107th central stem extension by C23 of C24

G = C₂³.₃₉₀C₂⁴ order 128 = 2⁷

107^th central stem extension by C₂³ of C₂⁴