C2^3.401C2^4 - GroupNames

G = C₂³.₄₀₁C₂⁴ order 128 = 2⁷

118^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₁₄C₂³, C₂³.₄₀₁C₂⁴, C₂².₁₅₁2_-¹⁺⁴, C₂²⋊C₄⋊₂₆D₄, C₄²⋊₉C₄⋊₂₃C₂, (C₂²×C₄).₃₈₈D₄, C₂³.₆₁₅(C₂×D₄), C₂.₄₄(D₄⋊₆D₄), C₄⋊₂(C₂².D₄), (C₂³×C₄).₉₈C₂², C₂³.₄Q₈⋊₁₆C₂, C₂².₁₃(C₄⋊₁D₄), (C₂×C₄²).₅₂₁C₂², C₂².₂₇₇(C₂²×D₄), C₂⁴.₃C₂²⋊₅₀C₂, (C₂²×C₄).₁₄₈₄C₂³, (C₂²×D₄).₁₅₀C₂², C₂.C₄².₅₄₁C₂², C₂.₁₈(C₂³.₃₈C₂³), (C₂×C₄).₆₅(C₂×D₄), C₂.₈(C₂×C₄⋊₁D₄), (C₂²×C₄⋊C₄)⋊₂₅C₂, (C₄×C₂²⋊C₄)⋊₇₆C₂, (C₂×C₄⋊D₄).₃₄C₂, (C₂×C₄).₈₁₃(C₄○D₄), (C₂×C₄⋊C₄).₈₅₈C₂², C₂².₂₇₈(C₂×C₄○D₄), (C₂×C₂².D₄)⋊₁₈C₂, C₂.₃₆(C₂×C₂².D₄), (C₂×C₂²⋊C₄).₁₆₁C₂², SmallGroup(128,1233)

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

Derived series

C₁ — C₂³ — C₂³.₄₀₁C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₄×C₂²⋊C₄ — C₂³.₄₀₁C₂⁴

Lower central

C₁ — C₂³ — C₂³.₄₀₁C₂⁴

Upper central

C₁ — C₂³ — C₂³.₄₀₁C₂⁴

Jennings

C₁ — C₂³ — C₂³.₄₀₁C₂⁴

Generators and relations for C₂³.₄₀₁C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=1, d²=c, e²=g²=a, f²=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: 676 in 354 conjugacy classes, 124 normal (16 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄⋊D₄, C₂².D₄, C₂³×C₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₄×C₂²⋊C₄, C₄²⋊₉C₄, C₂⁴.₃C₂², C₂³.₄Q₈, C₂²×C₄⋊C₄, C₂×C₄⋊D₄, C₂×C₂².D₄, C₂³.₄₀₁C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂².D₄, C₄⋊₁D₄, C₂²×D₄, C₂×C₄○D₄, 2_-¹⁺⁴, C₂×C₂².D₄, C₂×C₄⋊₁D₄, C₂³.₃₈C₂³, D₄⋊₆D₄, C₂³.₄₀₁C₂⁴

Smallest permutation representation of C₂³.₄₀₁C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2_-¹⁺⁴

kernel

C₂³.₄₀₁C₂⁴

C₄×C₂²⋊C₄

C₄²⋊₉C₄

C₂⁴.₃C₂²

C₂³.₄Q₈

C₂²×C₄⋊C₄

C₂×C₄⋊D₄

C₂×C₂².D₄

C₂²⋊C₄

C₂²×C₄

C₂×C₄

C₂²

# reps

Matrix representation of C₂³.₄₀₁C₂⁴ ►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

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

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

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

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

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

C₂³.₄₀₁C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{401}C_2^4

% in TeX

G:=Group("C2^3.401C2^4");

// GroupNames label

G:=SmallGroup(128,1233);

// by ID

G=gap.SmallGroup(128,1233);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,723,184,675,80]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=c,e^2=g^2=a,f^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 = C23.401C24 order 128 = 27

118th central stem extension by C23 of C24

G = C₂³.₄₀₁C₂⁴ order 128 = 2⁷

118^th central stem extension by C₂³ of C₂⁴