C2^4.304C2^3 - GroupNames

G = C₂⁴.₃₀₄C₂³ order 128 = 2⁷

144^th non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²

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

Aliases: C₂⁴.₃₀₄C₂³, C₂³.₃₉₄C₂⁴, C₂².₁₉₄2₊¹⁺⁴, C₂².₁₄₆2_-¹⁺⁴, C₄²⋊₈C₄⋊₂₇C₂, C₂³.₄₂(C₄○D₄), (C₂×C₄²).₄₃C₂², (C₂²×C₄).₇₄C₂³, C₄.₂₆(C₄²⋊₂C₂), (C₂³×C₄).₃₇₈C₂², C₂³.₁₁D₄.₇C₂, C₂³.₇Q₈.₄₅C₂, C₂³.₈₄C₂³⋊₂C₂, C₂³.₆₃C₂³⋊₆₂C₂, C₂³.₆₅C₂³⋊₇₀C₂, C₂⁴.C₂².₂₀C₂, C₂.C₄².₁₄₆C₂², C₂.₃₀(C₂².₃₆C₂⁴), C₂.₃₈(C₂².₄₆C₂⁴), C₂.₅₀(C₂³.₃₆C₂³), C₂.₂₂(C₂².₅₀C₂⁴), C₂.₃₃(C₂².₄₇C₂⁴), (C₄×C₄⋊C₄)⋊₇₁C₂, (C₄×C₂²⋊C₄).₄₇C₂, (C₂×C₄).₈₅₇(C₄○D₄), (C₂×C₄⋊C₄).₂₆₄C₂², C₂.₁₃(C₂×C₄²⋊₂C₂), C₂².₂₇₁(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₁₅₈C₂², SmallGroup(128,1226)

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

Derived series

C₁ — C₂³ — C₂⁴.₃₀₄C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₄×C₂²⋊C₄ — C₂⁴.₃₀₄C₂³

Lower central

C₁ — C₂³ — C₂⁴.₃₀₄C₂³

Upper central

C₁ — C₂³ — C₂⁴.₃₀₄C₂³

Jennings

C₁ — C₂³ — C₂⁴.₃₀₄C₂³

Generators and relations for C₂⁴.₃₀₄C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=f²=1, d²=cb=bc, e²=b, g²=a, ab=ba, ac=ca, ede^-1=gdg^-1=ad=da, ae=ea, af=fa, ag=ga, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 372 in 206 conjugacy classes, 96 normal (42 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂.C₄², C₂.C₄², C₂×C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂³×C₄, C₄×C₂²⋊C₄, C₄×C₄⋊C₄, C₂³.₇Q₈, C₄²⋊₈C₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₅C₂³, C₂³.₁₁D₄, C₂³.₈₄C₂³, C₂⁴.₃₀₄C₂³
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₄²⋊₂C₂, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂×C₄²⋊₂C₂, C₂³.₃₆C₂³, C₂².₃₆C₂⁴, C₂².₄₆C₂⁴, C₂².₄₇C₂⁴, C₂².₅₀C₂⁴, C₂⁴.₃₀₄C₂³

Smallest permutation representation of C₂⁴.₃₀₄C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

(2 6)(4 8)(10 56)(12 54)(13 57)(14 16)(15 59)(17 61)(18 20)(19 63)(22 40)(24 38)(26 44)(28 42)(29 45)(30 32)(31 47)(33 49)(34 36)(35 51)(46 48)(50 52)(58 60)(62 64)

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂⁴.₃₀₄C₂³

C₄×C₂²⋊C₄

C₄×C₄⋊C₄

C₂³.₇Q₈

C₄²⋊₈C₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂³.₁₁D₄

C₂³.₈₄C₂³

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂⁴.₃₀₄C₂³ ►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	1	0
0	0	0	0	0	1

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

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

1	0	0	0	0	0
4	4	0	0	0	0
0	0	1	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	4	0	0	0
0	0	0	4	0	0
0	0	0	0	0	1
0	0	0	0	4	0

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

C₂⁴.₃₀₄C₂³ in GAP, Magma, Sage, TeX

C_2^4._{304}C_2^3

% in TeX

G:=Group("C2^4.304C2^3");

// GroupNames label

G:=SmallGroup(128,1226);

// by ID

G=gap.SmallGroup(128,1226);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,672,253,232,758,723,675,80]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=c*b=b*c,e^2=b,g^2=a,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,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 = C24.304C23 order 128 = 27

144th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₃₀₄C₂³ order 128 = 2⁷

144^th non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²