C2^3.425C2^4

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

Aliases: C₂⁴.₂₂C₂³, C₂³.₄₂₅C₂⁴, C₂².₁₆₆2_-⁽¹⁺⁴⁾, C₂².₂₁₇2₊⁽¹⁺⁴⁾, C₄²⋊₅C₄⋊₁₆C₂, (C₂²×C₄).₈₇C₂³, (C₂×C₄²).₅₄₀C₂², C₂³.Q₈.₁₁C₂, C₂³.₁₁D₄.₁₄C₂, C₂³.₆₃C₂³⋊₇₇C₂, C₂³.₈₃C₂³⋊₃₁C₂, C₂³.₈₁C₂³⋊₃₁C₂, C₂³.₆₅C₂³⋊₈₀C₂, C₂⁴.C₂².₂₆C₂, C₂.C₄².₁₇₃C₂², C₂.₆₈(C₂³.₃₆C₂³), C₂.₃₁(C₂².₅₀C₂⁴), C₂.₅₃(C₂².₄₆C₂⁴), C₂.₄₇(C₂².₄₇C₂⁴), C₂.₃₉(C₂².₃₆C₂⁴), (C₄×C₄⋊C₄)⋊₈₂C₂, (C₂×C₄).₁₄₄(C₄○D₄), (C₂×C₄⋊C₄).₂₈₇C₂², C₂².₃₀₂(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₅₀C₂², SmallGroup(128,1257)

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

Derived series

C₁ — C₂³ — C₂³.₄₂₅C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₄×C₄⋊C₄ — C₂³.₄₂₅C₂⁴

Lower central

C₁ — C₂³ — C₂³.₄₂₅C₂⁴

Upper central

C₁ — C₂³ — C₂³.₄₂₅C₂⁴

Jennings

C₁ — C₂³ — C₂³.₄₂₅C₂⁴

Subgroups: 356 in 199 conjugacy classes, 92 normal (82 characteristic)
C₁, C₂^[×7], C₂, C₄^[×19], C₂²^[×7], C₂²^[×7], C₂×C₄^[×10], C₂×C₄^[×37], C₂³, C₂³^[×7], C₄²^[×8], C₂²⋊C₄^[×12], C₄⋊C₄^[×16], C₂²×C₄^[×14], C₂⁴, C₂.C₄²^[×14], C₂×C₄²^[×5], C₂×C₂²⋊C₄^[×7], C₂×C₄⋊C₄^[×9], C₄×C₄⋊C₄^[×2], C₄²⋊₅C₄, C₂³.₆₃C₂³^[×2], C₂⁴.C₂²^[×5], C₂³.₆₅C₂³, C₂³.Q₈, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂³.₈₃C₂³, C₂³.₄₂₅C₂⁴

Quotients:
C₁, C₂^[×15], C₂²^[×35], C₂³^[×15], C₄○D₄^[×10], C₂⁴, C₂×C₄○D₄^[×5], 2₊⁽¹⁺⁴⁾, 2_-⁽¹⁺⁴⁾, C₂³.₃₆C₂³^[×2], C₂².₃₆C₂⁴, C₂².₄₆C₂⁴, C₂².₄₇C₂⁴^[×2], C₂².₅₀C₂⁴, C₂³.₄₂₅C₂⁴

Generators and relations
G = < a,b,c,d,e,f,g | a²=b²=c²=1, d²=ca=ac, e²=f²=a, g²=ba=ab, 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 >

Smallest permutation representation
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

Matrix representation ►G ⊆ 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	1	0
0	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	0	0
0	0	0	0	4	0
0	0	0	0	0	4

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

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

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

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

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	2	0
0	0	0	0	0	2

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

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

C₄○D₄

2₊⁽¹⁺⁴⁾

2_-⁽¹⁺⁴⁾

kernel

C₂³.₄₂₅C₂⁴

C₄×C₄⋊C₄

C₄²⋊₅C₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂³.Q₈

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂³.₈₃C₂³

C₂×C₄

C₂²

# reps

In GAP, Magma, Sage, TeX

C_2^3._{425}C_2^4

% in TeX

G:=Group("C2^3.425C2^4");

// GroupNames label

G:=SmallGroup(128,1257);

// by ID

G=gap.SmallGroup(128,1257);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,120,758,723,100,675,192]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=c*a=a*c,e^2=f^2=a,g^2=b*a=a*b,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 = C₂³.₄₂₅C₂⁴ order 128 = 2⁷

142^nd central stem extension by C₂³ of C₂⁴

G = C23.425C24 order 128 = 27

142nd central stem extension by C23 of C24

G = C₂³.₄₂₅C₂⁴ order 128 = 2⁷

142^nd central stem extension by C₂³ of C₂⁴