C2^3.402C2^4 - GroupNames

G = C₂³.₄₀₂C₂⁴ order 128 = 2⁷

119^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂³.₄₀₂C₂⁴, C₂⁴.₅₇₈C₂³, C₂².₁₉₉2₊¹⁺⁴, C₂²⋊C₄⋊₁₆Q₈, C₄²⋊₉C₄⋊₂₄C₂, (C₂²×C₄).₃₈₉D₄, C₂³.₆₁₆(C₂×D₄), C₂².₁₆(C₄⋊Q₈), C₂³.₁₂₀(C₂×Q₈), C₂.₂₁(D₄⋊₃Q₈), (C₂²×C₄).₇₉C₂³, (C₂³×C₄).₉₉C₂², C₂².₈₇(C₂²×Q₈), (C₂×C₄²).₅₂₂C₂², C₂².₂₇₈(C₂²×D₄), C₂³.₈Q₈.₂₁C₂, C₄.₅₂(C₂².D₄), (C₂²×Q₈).₁₁₉C₂², C₂³.₈₁C₂³⋊₂₆C₂, C₂³.₆₇C₂³⋊₅₂C₂, C₂.₁₈(C₂².₂₉C₂⁴), C₂.C₄².₁₅₃C₂², C₂.₁₀(C₂×C₄⋊Q₈), (C₂×C₄).₄₀(C₂×Q₈), (C₂×C₄).₃₄₉(C₂×D₄), (C₂²×C₄⋊C₄).₃₇C₂, (C₄×C₂²⋊C₄).₅₂C₂, (C₂×C₂²⋊Q₈).₃₂C₂, (C₂×C₄).₈₁₄(C₄○D₄), (C₂×C₄⋊C₄).₈₅₉C₂², C₂².₂₇₉(C₂×C₄○D₄), C₂.₃₇(C₂×C₂².D₄), (C₂×C₂²⋊C₄).₅₀₁C₂², SmallGroup(128,1234)

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

Derived series

C₁ — C₂³ — C₂³.₄₀₂C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₄×C₂²⋊C₄ — C₂³.₄₀₂C₂⁴

Lower central

C₁ — C₂³ — C₂³.₄₀₂C₂⁴

Upper central

C₁ — C₂³ — C₂³.₄₀₂C₂⁴

Jennings

C₁ — C₂³ — C₂³.₄₀₂C₂⁴

Generators and relations for C₂³.₄₀₂C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=1, d²=ca=ac, e²=g²=a, f²=b, ab=ba, 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: 484 in 272 conjugacy classes, 124 normal (16 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂²⋊Q₈, C₂³×C₄, C₂³×C₄, C₂²×Q₈, C₄×C₂²⋊C₄, C₄²⋊₉C₄, C₂³.₈Q₈, C₂³.₆₇C₂³, C₂³.₈₁C₂³, C₂²×C₄⋊C₄, C₂×C₂²⋊Q₈, C₂³.₄₀₂C₂⁴
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, C₂².D₄, C₄⋊Q₈, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, 2₊¹⁺⁴, C₂×C₂².D₄, C₂×C₄⋊Q₈, C₂².₂₉C₂⁴, D₄⋊₃Q₈, C₂³.₄₀₂C₂⁴

Smallest permutation representation of C₂³.₄₀₂C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	4B	4C	4D	4E	···	4V	4W	4X	4Y	4Z
order	1	2	···	2	2	2	2	2	4	4	4	4	4	···	4	4	4	4	4
size	1	1	···	1	2	2	2	2	2	2	2	2	4	···	4	8	8	8	8

38 irreducible representations

dim

type

image

C₁

C₂

Q₈

D₄

C₄○D₄

2₊¹⁺⁴

kernel

C₂³.₄₀₂C₂⁴

C₄×C₂²⋊C₄

C₄²⋊₉C₄

C₂³.₈Q₈

C₂³.₆₇C₂³

C₂³.₈₁C₂³

C₂²×C₄⋊C₄

C₂×C₂²⋊Q₈

C₂²⋊C₄

C₂²×C₄

C₂×C₄

C₂²

# reps

Matrix representation of C₂³.₄₀₂C₂⁴ ►in GL₆(𝔽₅)

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

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

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

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

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

0	1	0	0	0	0
1	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

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	2	0
0	0	0	0	0	3

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

C₂³.₄₀₂C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{402}C_2^4

% in TeX

G:=Group("C2^3.402C2^4");

// GroupNames label

G:=SmallGroup(128,1234);

// by ID

G=gap.SmallGroup(128,1234);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,120,758,723,184,675,80]);

// 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=g^2=a,f^2=b,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^-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 = C23.402C24 order 128 = 27

119th central stem extension by C23 of C24

G = C₂³.₄₀₂C₂⁴ order 128 = 2⁷

119^th central stem extension by C₂³ of C₂⁴