C2^3.419C2^4 - GroupNames

G = C₂³.₄₁₉C₂⁴ order 128 = 2⁷

136^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₂₁C₂³, C₂³.₄₁₉C₂⁴, C₂².₁₆₁2_-¹⁺⁴, C₄⋊C₄.₂₃₂D₄, C₄²⋊₈C₄⋊₃₈C₂, C₂.₄₇(D₄⋊₆D₄), C₂.₃₅(Q₈⋊₅D₄), C₂³.Q₈.₉C₂, C₂³.₄Q₈.₆C₂, (C₂×C₄²).₅₃₄C₂², (C₂²×C₄).₅₂₉C₂³, C₂².₂₈₃(C₂²×D₄), C₂³.₆₃C₂³⋊₇₃C₂, C₂³.₈₁C₂³⋊₂₉C₂, C₂³.₆₅C₂³⋊₇₈C₂, C₂⁴.C₂².₂₄C₂, C₂.C₄².₁₆₇C₂², C₂.₂₈(C₂².₂₆C₂⁴), C₂.₂₀(C₂².₃₅C₂⁴), C₂.₅₀(C₂².₄₆C₂⁴), C₂.₆₂(C₂³.₃₆C₂³), (C₄×C₄⋊C₄)⋊₇₉C₂, (C₂×C₄).₆₈(C₂×D₄), (C₂×C₄².C₂)⋊₁₀C₂, (C₂×C₄).₃₇₈(C₄○D₄), (C₂×C₄⋊C₄).₂₈₂C₂², (C₂×C₄²⋊₂C₂).₈C₂, C₂².₂₉₆(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₁₆₅C₂², SmallGroup(128,1251)

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

Derived series

C₁ — C₂³ — C₂³.₄₁₉C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₄×C₄⋊C₄ — C₂³.₄₁₉C₂⁴

Lower central

C₁ — C₂³ — C₂³.₄₁₉C₂⁴

Upper central

C₁ — C₂³ — C₂³.₄₁₉C₂⁴

Jennings

C₁ — C₂³ — C₂³.₄₁₉C₂⁴

Generators and relations for C₂³.₄₁₉C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=1, d²=c, e²=f²=a, g²=b, ab=ba, ac=ca, ede^-1=gdg^-1=ad=da, ae=ea, af=fa, ag=ga, bc=cb, fdf^-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef^-1=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 388 in 227 conjugacy classes, 100 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄².C₂, C₄²⋊₂C₂, C₄×C₄⋊C₄, C₄²⋊₈C₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₅C₂³, C₂³.Q₈, C₂³.₈₁C₂³, C₂³.₄Q₈, C₂×C₄².C₂, C₂×C₄²⋊₂C₂, C₂³.₄₁₉C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2_-¹⁺⁴, C₂³.₃₆C₂³, C₂².₂₆C₂⁴, C₂².₃₅C₂⁴, D₄⋊₆D₄, Q₈⋊₅D₄, C₂².₄₆C₂⁴, C₂³.₄₁₉C₂⁴

Smallest permutation representation of C₂³.₄₁₉C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	4A	···	4H	4I	···	4Z	4AA	4AB	4AC
order	1	2	···	2	2	4	···	4	4	···	4	4	4	4
size	1	1	···	1	8	2	···	2	4	···	4	8	8	8

38 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2_-¹⁺⁴

kernel

C₂³.₄₁₉C₂⁴

C₄×C₄⋊C₄

C₄²⋊₈C₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂³.Q₈

C₂³.₈₁C₂³

C₂³.₄Q₈

C₂×C₄².C₂

C₂×C₄²⋊₂C₂

C₄⋊C₄

C₂×C₄

C₂²

# reps

Matrix representation of C₂³.₄₁₉C₂⁴ ►in GL₆(𝔽₅)

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

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

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

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

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

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

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

C₂³.₄₁₉C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{419}C_2^4

% in TeX

G:=Group("C2^3.419C2^4");

// GroupNames label

G:=SmallGroup(128,1251);

// by ID

G=gap.SmallGroup(128,1251);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,723,100,675,136]);

// Polycyclic

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

// generators/relations

G = C23.419C24 order 128 = 27

136th central stem extension by C23 of C24

G = C₂³.₄₁₉C₂⁴ order 128 = 2⁷

136^th central stem extension by C₂³ of C₂⁴