C2^4.401C2^3 - GroupNames

G = C₂⁴.₄₀₁C₂³ order 128 = 2⁷

241^st non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²

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

Aliases: C₂⁴.₄₀₁C₂³, C₂³.₅₉₄C₂⁴, C₂².₂₇₄2_-¹⁺⁴, C₂².₃₆₈2₊¹⁺⁴, C₄⋊C₄.₁₁₈D₄, C₂.₉₉(D₄⋊₅D₄), C₂.₂₉(Q₈⋊₆D₄), C₂³.₇Q₈⋊₈₈C₂, C₂³.₄Q₈⋊₄₄C₂, C₂³.Q₈⋊₅₉C₂, C₂³.₁₇₂(C₄○D₄), C₂³.₁₁D₄⋊₈₃C₂, (C₂×C₄²).₆₄₇C₂², (C₂²×C₄).₈₇₃C₂³, (C₂³×C₄).₄₅₇C₂², C₂³.₈Q₈⋊₁₀₆C₂, C₂².₄₀₃(C₂²×D₄), C₂³.₁₀D₄.₄₁C₂, (C₂²×D₄).₂₃₁C₂², C₂⁴.C₂²⋊₁₂₆C₂, C₂³.₈₁C₂³⋊₈₄C₂, C₂⁴.₃C₂².₆₁C₂, C₂³.₆₅C₂³⋊₁₁₉C₂, C₂³.₆₃C₂³⋊₁₃₃C₂, C₂.C₄².₃₀₁C₂², C₂.₁₄(C₂².₅₆C₂⁴), C₂.₆₄(C₂².₃₃C₂⁴), C₂.₈₂(C₂².₄₆C₂⁴), C₂.₅₉(C₂³.₃₈C₂³), (C₂×C₄).₉₇(C₂×D₄), (C₂×C₄².C₂)⋊₂₁C₂, (C₂×C₄).₁₉₂(C₄○D₄), (C₂×C₄⋊C₄).₄₀₈C₂², C₂².₄₅₆(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₂₆₁C₂², (C₂×C₂².D₄).₂₅C₂, SmallGroup(128,1426)

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

Derived series

C₁ — C₂³ — C₂⁴.₄₀₁C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂³.₇Q₈ — C₂⁴.₄₀₁C₂³

Lower central

C₁ — C₂³ — C₂⁴.₄₀₁C₂³

Upper central

C₁ — C₂³ — C₂⁴.₄₀₁C₂³

Jennings

C₁ — C₂³ — C₂⁴.₄₀₁C₂³

Generators and relations for C₂⁴.₄₀₁C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=g²=b, f²=c, eae^-1=ab=ba, ac=ca, ad=da, faf^-1=abd, ag=ga, bc=cb, bd=db, fef^-1=be=eb, bf=fb, bg=gb, cd=dc, geg^-1=ce=ec, cf=fc, cg=gc, de=ed, gfg^-1=df=fd, dg=gd >

Subgroups: 484 in 245 conjugacy classes, 96 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂².D₄, C₄².C₂, C₂³×C₄, C₂²×D₄, C₂³.₇Q₈, C₂³.₈Q₈, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₅C₂³, C₂⁴.₃C₂², C₂³.₁₀D₄, C₂³.Q₈, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂³.₄Q₈, C₂×C₂².D₄, C₂×C₄².C₂, C₂⁴.₄₀₁C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂³.₃₈C₂³, C₂².₃₃C₂⁴, D₄⋊₅D₄, Q₈⋊₆D₄, C₂².₄₆C₂⁴, C₂².₅₆C₂⁴, C₂⁴.₄₀₁C₂³

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

Generators in S₆₄

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	4A	···	4P	4Q	···	4U
order	1	2	···	2	2	2	2	4	···	4	4	···	4
size	1	1	···	1	4	4	8	4	···	4	8	···	8

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂⁴.₄₀₁C₂³

C₂³.₇Q₈

C₂³.₈Q₈

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂⁴.₃C₂²

C₂³.₁₀D₄

C₂³.Q₈

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂³.₄Q₈

C₂×C₂².D₄

C₂×C₄².C₂

C₄⋊C₄

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂⁴.₄₀₁C₂³ ►in GL₆(𝔽₅)

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

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	1	0
0	0	0	0	0	1

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	4	0	0	0	0
0	0	4	0	0	0
0	0	0	4	0	0
0	0	0	0	0	3
0	0	0	0	3	0

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

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

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

C₂⁴.₄₀₁C₂³ in GAP, Magma, Sage, TeX

C_2^4._{401}C_2^3

% in TeX

G:=Group("C2^4.401C2^3");

// GroupNames label

G:=SmallGroup(128,1426);

// by ID

G=gap.SmallGroup(128,1426);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,232,758,723,100,1571,346,80]);

// Polycyclic

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

// generators/relations

G = C24.401C23 order 128 = 27

241st non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₄₀₁C₂³ order 128 = 2⁷

241^st non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²