C2^4.289C2^3 - GroupNames

G = C₂⁴.₂₈₉C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₂₈₉C₂³, C₂³.₃₇₀C₂⁴, C₂².₁₇₅2₊¹⁺⁴, C₂².₁₃₀2_-¹⁺⁴, C₂²⋊C₄.₁₃₁D₄, C₂³.₁₇₆(C₂×D₄), C₂.₅₂(D₄⋊₅D₄), C₂.₃₄(D₄⋊₆D₄), (C₂×C₄²).₃₁C₂², C₂³.₈Q₈⋊₅₄C₂, C₂³.Q₈⋊₁₈C₂, C₂³.₁₄₀(C₄○D₄), C₂³.₁₁D₄⋊₁₇C₂, (C₂²×C₄).₅₁₅C₂³, (C₂³×C₄).₃₅₉C₂², C₂³.₁₀D₄.₈C₂, C₂².₂₅₀(C₂²×D₄), C₂⁴.C₂²⋊₅₂C₂, C₂³.₂₃D₄.₂₀C₂, (C₂²×D₄).₁₃₇C₂², C₂³.₆₃C₂³⋊₄₉C₂, C₂³.₆₅C₂³⋊₆₂C₂, C₂³.₈₃C₂³⋊₁₁C₂, C₂.₄₂(C₂².₁₉C₂⁴), C₂.C₄².₁₂₇C₂², C₂.₂₄(C₂².₃₆C₂⁴), C₂.₃₈(C₂³.₃₆C₂³), C₂.₂₇(C₂².₄₇C₂⁴), C₂.₂₅(C₂².₄₆C₂⁴), (C₄×C₂²⋊C₄)⋊₆₇C₂, (C₂×C₄).₈₉₉(C₂×D₄), (C₂×C₄²⋊C₂)⋊₂₂C₂, (C₂×C₄).₃₆₇(C₄○D₄), (C₂×C₄⋊C₄).₂₅₀C₂², C₂².₂₄₇(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₄₅₆C₂², (C₂×C₂².D₄).₁₃C₂, SmallGroup(128,1202)

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

Derived series

C₁ — C₂³ — C₂⁴.₂₈₉C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₄×C₂²⋊C₄ — C₂⁴.₂₈₉C₂³

Lower central

C₁ — C₂³ — C₂⁴.₂₈₉C₂³

Upper central

C₁ — C₂³ — C₂⁴.₂₈₉C₂³

Jennings

C₁ — C₂³ — C₂⁴.₂₈₉C₂³

Generators and relations for C₂⁴.₂₈₉C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=f²=1, e²=ba=ab, g²=b, ac=ca, ede^-1=gdg^-1=ad=da, ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 500 in 266 conjugacy classes, 100 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₂².D₄, C₂³×C₄, C₂²×D₄, C₄×C₂²⋊C₄, C₂³.₈Q₈, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₅C₂³, C₂³.₁₀D₄, C₂³.Q₈, C₂³.₁₁D₄, C₂³.₈₃C₂³, C₂×C₄²⋊C₂, C₂×C₂².D₄, C₂⁴.₂₈₉C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₁₉C₂⁴, C₂³.₃₆C₂³, C₂².₃₆C₂⁴, D₄⋊₅D₄, D₄⋊₆D₄, C₂².₄₆C₂⁴, C₂².₄₇C₂⁴, C₂⁴.₂₈₉C₂³

Smallest permutation representation of C₂⁴.₂₈₉C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

(2 38)(4 40)(5 53)(6 24)(7 55)(8 22)(10 32)(12 30)(14 52)(16 50)(17 47)(18 63)(19 45)(20 61)(21 58)(23 60)(26 36)(28 34)(41 48)(42 64)(43 46)(44 62)(54 57)(56 59)

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂⁴.₂₈₉C₂³

C₄×C₂²⋊C₄

C₂³.₈Q₈

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂³.₁₀D₄

C₂³.Q₈

C₂³.₁₁D₄

C₂³.₈₃C₂³

C₂×C₄²⋊C₂

C₂×C₂².D₄

C₂²⋊C₄

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂⁴.₂₈₉C₂³ ►in GL₆(𝔽₅)

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

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	4	0
0	0	0	0	0	4

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

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

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

2	0	0	0	0	0
0	3	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	3

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

C₂⁴.₂₈₉C₂³ in GAP, Magma, Sage, TeX

C_2^4._{289}C_2^3

% in TeX

G:=Group("C2^4.289C2^3");

// GroupNames label

G:=SmallGroup(128,1202);

// by ID

G=gap.SmallGroup(128,1202);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,344,758,723,675,136]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=1,e^2=b*a=a*b,g^2=b,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=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 = C24.289C23 order 128 = 27

129th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₂₈₉C₂³ order 128 = 2⁷

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