C2^3.431C2^4

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

Aliases: C₂⁴.₂₄C₂³, C₂³.₄₃₁C₂⁴, C₂².₂₂₂2₊⁽¹⁺⁴⁾, C₄²⋊₄C₄⋊₂₄C₂, C₂³.₄₈(C₄○D₄), (C₂×C₄²).₆₀C₂², (C₂²×C₄).₉₁C₂³, C₂³.₇Q₈⋊₆₁C₂, C₂³.₁₁D₄⋊₃₉C₂, (C₂³×C₄).₃₈₈C₂², C₂⁴.C₂²⋊₇₅C₂, C₂³.₁₀D₄.₁₇C₂, C₂³.₂₃D₄.₃₃C₂, (C₂²×D₄).₁₅₉C₂², C₂³.₆₃C₂³⋊₈₂C₂, C₂³.₈₁C₂³⋊₃₃C₂, C₂⁴.₃C₂².₄₂C₂, C₂.₄₃(C₂².₄₅C₂⁴), C₂.C₄².₅₄₄C₂², C₂.₅₁(C₂².₄₇C₂⁴), C₂.₂₂(C₂².₄₉C₂⁴), C₂.₂₃(C₂².₃₄C₂⁴), C₂.₇₄(C₂³.₃₆C₂³), (C₄×C₂²⋊C₄)⋊₈₂C₂, (C₂×C₄).₃₈₃(C₄○D₄), (C₂×C₄⋊C₄).₂₉₃C₂², C₂².₃₀₈(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₁₇₀C₂², SmallGroup(128,1263)

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

Derived series

C₁ — C₂³ — C₂³.₄₃₁C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₄×C₂²⋊C₄ — C₂³.₄₃₁C₂⁴

Lower central

C₁ — C₂³ — C₂³.₄₃₁C₂⁴

Upper central

C₁ — C₂³ — C₂³.₄₃₁C₂⁴

Jennings

C₁ — C₂³ — C₂³.₄₃₁C₂⁴

Subgroups: 452 in 227 conjugacy classes, 92 normal (82 characteristic)
C₁, C₂^[×7], C₂^[×3], C₄^[×17], C₂²^[×7], C₂²^[×17], C₂×C₄^[×8], C₂×C₄^[×39], D₄^[×4], C₂³, C₂³^[×2], C₂³^[×13], C₄²^[×7], C₂²⋊C₄^[×17], C₄⋊C₄^[×8], C₂²×C₄^[×13], C₂²×C₄^[×5], C₂×D₄^[×5], C₂⁴^[×2], C₂.C₄²^[×12], C₂×C₄²^[×4], C₂×C₂²⋊C₄^[×11], C₂×C₄⋊C₄^[×6], C₂³×C₄, C₂²×D₄, C₄²⋊₄C₄, C₄×C₂²⋊C₄, C₂³.₇Q₈, C₂³.₂₃D₄, C₂³.₆₃C₂³^[×2], C₂⁴.C₂²^[×4], C₂⁴.₃C₂², C₂³.₁₀D₄, C₂³.₁₁D₄^[×2], C₂³.₈₁C₂³, C₂³.₄₃₁C₂⁴

Quotients:
C₁, C₂^[×15], C₂²^[×35], C₂³^[×15], C₄○D₄^[×10], C₂⁴, C₂×C₄○D₄^[×5], 2₊⁽¹⁺⁴⁾^[×2], C₂³.₃₆C₂³^[×2], C₂².₃₄C₂⁴, C₂².₄₅C₂⁴^[×2], C₂².₄₇C₂⁴, C₂².₄₉C₂⁴, C₂³.₄₃₁C₂⁴

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

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

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

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

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

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

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

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

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[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],[0,3,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,4,0,0,0,0,0,0,0,3,0,0,0,0,0,0,2],[0,3,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,2,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

38 conjugacy classes

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

38 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊⁽¹⁺⁴⁾

kernel

C₂³.₄₃₁C₂⁴

C₄²⋊₄C₄

C₄×C₂²⋊C₄

C₂³.₇Q₈

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂⁴.₃C₂²

C₂³.₁₀D₄

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂×C₄

C₂³

C₂²

# reps

In GAP, Magma, Sage, TeX

C_2^3._{431}C_2^4

% in TeX

G:=Group("C2^3.431C2^4");

// GroupNames label

G:=SmallGroup(128,1263);

// by ID

G=gap.SmallGroup(128,1263);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,232,758,723,675,192]);

// 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=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=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 = C₂³.₄₃₁C₂⁴ order 128 = 2⁷

148^th central stem extension by C₂³ of C₂⁴

G = C23.431C24 order 128 = 27

148th central stem extension by C23 of C24

G = C₂³.₄₃₁C₂⁴ order 128 = 2⁷

148^th central stem extension by C₂³ of C₂⁴