C2^4.403C2^3 - GroupNames

G = C₂⁴.₄₀₃C₂³ order 128 = 2⁷

243^rd non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²

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

Aliases: C₂⁴.₄₀₃C₂³, C₂³.₅₉₆C₂⁴, C₂².₃₇₀2₊¹⁺⁴, C₂².₂₇₆2_-¹⁺⁴, C₄⋊C₄⋊₁₄D₄, C₂³⋊Q₈⋊₄₅C₂, C₂.₄₆(Q₈⋊₅D₄), C₂³⋊₂D₄.₂₂C₂, C₂.₁₀₁(D₄⋊₅D₄), C₂³.₇Q₈⋊₉₀C₂, C₂³.₁₇₄(C₄○D₄), C₂³.₂₃D₄⋊₈₈C₂, C₂³.₁₀D₄⋊₈₄C₂, C₂³.₁₁D₄⋊₈₅C₂, (C₂²×C₄).₈₇₄C₂³, (C₂×C₄²).₆₄₉C₂², (C₂³×C₄).₄₅₉C₂², C₂².₄₀₅(C₂²×D₄), C₂⁴.₃C₂²⋊₇₉C₂, (C₂²×D₄).₂₃₃C₂², (C₂²×Q₈).₁₈₄C₂², C₂³.₈₃C₂³⋊₈₀C₂, C₂⁴.C₂²⋊₁₂₈C₂, C₂.₆₆(C₂².₃₂C₂⁴), C₂.₅₈(C₂².₂₉C₂⁴), C₂³.₆₃C₂³⋊₁₃₄C₂, C₂.₇₈(C₂².₄₅C₂⁴), C₂.C₄².₃₀₃C₂², C₂.₆₆(C₂².₃₃C₂⁴), C₂.₁₅(C₂².₅₆C₂⁴), (C₂×C₄).₉₈(C₂×D₄), (C₂×C₂²⋊Q₈)⋊₄₀C₂, (C₂×C₄⋊C₄).₄₁₀C₂², C₂².₄₅₈(C₂×C₄○D₄), (C₂×C₂².D₄)⋊₃₅C₂, (C₂×C₂²⋊C₄).₂₆₃C₂², SmallGroup(128,1428)

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

Derived series

C₁ — C₂³ — C₂⁴.₄₀₃C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂³.₇Q₈ — C₂⁴.₄₀₃C₂³

Lower central

C₁ — C₂³ — C₂⁴.₄₀₃C₂³

Upper central

C₁ — C₂³ — C₂⁴.₄₀₃C₂³

Jennings

C₁ — C₂³ — C₂⁴.₄₀₃C₂³

Generators and relations for C₂⁴.₄₀₃C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=g²=cb=bc, f²=b, ab=ba, gag^-1=ac=ca, ad=da, ae=ea, faf^-1=acd, bd=db, geg^-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef^-1=de=ed, df=fd, dg=gd, gfg^-1=cdf >

Subgroups: 612 in 281 conjugacy classes, 96 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂²⋊Q₈, C₂².D₄, C₂³×C₄, C₂²×D₄, C₂²×Q₈, C₂³.₇Q₈, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂⁴.₃C₂², C₂³⋊₂D₄, C₂³⋊Q₈, C₂³.₁₀D₄, C₂³.₁₁D₄, C₂³.₈₃C₂³, C₂×C₂²⋊Q₈, C₂×C₂².D₄, C₂⁴.₄₀₃C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₂₉C₂⁴, C₂².₃₂C₂⁴, C₂².₃₃C₂⁴, D₄⋊₅D₄, Q₈⋊₅D₄, C₂².₄₅C₂⁴, C₂².₅₆C₂⁴, C₂⁴.₄₀₃C₂³

Smallest permutation representation of C₂⁴.₄₀₃C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	2L	4A	···	4N	4O	···	4S
order	1	2	···	2	2	2	2	2	2	4	···	4	4	···	4
size	1	1	···	1	4	4	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

C₂⁴.₄₀₃C₂³

C₂³.₇Q₈

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂⁴.₃C₂²

C₂³⋊₂D₄

C₂³⋊Q₈

C₂³.₁₀D₄

C₂³.₁₁D₄

C₂³.₈₃C₂³

C₂×C₂²⋊Q₈

C₂×C₂².D₄

C₄⋊C₄

C₂³

C₂²

# reps

Matrix representation of C₂⁴.₄₀₃C₂³ ►in GL₆(𝔽₅)

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

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	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

3	0	0	0	0	0
0	2	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

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

0	1	0	0	0	0
4	0	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	1	0

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

C₂⁴.₄₀₃C₂³ in GAP, Magma, Sage, TeX

C_2^4._{403}C_2^3

% in TeX

G:=Group("C2^4.403C2^3");

// GroupNames label

G:=SmallGroup(128,1428);

// by ID

G=gap.SmallGroup(128,1428);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,344,758,723,100,1571,346,80]);

// Polycyclic

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

// generators/relations

G = C24.403C23 order 128 = 27

243rd non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₄₀₃C₂³ order 128 = 2⁷

243^rd non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²