C2^4.243C2^3 - GroupNames

G = C₂⁴.₂₄₃C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₂₄₃C₂³, C₂³.₃₀₆C₂⁴, C₂².₈₈2_-¹⁺⁴, (C₂×D₄).₂₇₉D₄, C₂.₉(D₄⋊₆D₄), (C₂²×C₄).₃₇₁D₄, C₂³.₁₄₉(C₂×D₄), C₂².₁₆C₂²≀C₂, C₂³.₁₀D₄⋊₄C₂, C₂³.₈Q₈⋊₂₃C₂, C₂³.₂₃D₄⋊₂₃C₂, C₂³.₃₄D₄⋊₁₈C₂, (C₂²×C₄).₇₈₇C₂³, (C₂³×C₄).₃₂₆C₂², C₂².₁₈₆(C₂²×D₄), C₂³.₇₈C₂³⋊₃C₂, (C₂²×D₄).₄₉₉C₂², (C₂²×Q₈).₄₁₃C₂², C₂.₁₇(C₂².₁₉C₂⁴), C₂.C₄².₇₄C₂², C₂.₈(C₂³.₃₈C₂³), (C₂×C₄)⋊₉(C₄○D₄), (C₂²×C₄⋊C₄)⋊₁₆C₂, (C₂×C₄).₃₀₂(C₂×D₄), C₂.₁₃(C₂×C₂²≀C₂), (C₂×C₄⋊C₄).₂₀₁C₂², (C₂²×C₄○D₄).₁₀C₂, C₂².₁₈₅(C₂×C₄○D₄), (C₂×C₂².D₄)⋊₃C₂, (C₂×C₂²⋊C₄).₁₀₅C₂², SmallGroup(128,1138)

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

Derived series

C₁ — C₂³ — C₂⁴.₂₄₃C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂²×C₄⋊C₄ — C₂⁴.₂₄₃C₂³

Lower central

C₁ — C₂³ — C₂⁴.₂₄₃C₂³

Upper central

C₁ — C₂³ — C₂⁴.₂₄₃C₂³

Jennings

C₁ — C₂³ — C₂⁴.₂₄₃C₂³

Generators and relations for C₂⁴.₂₄₃C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=e²=g²=1, f²=b, eae=gag=ab=ba, ac=ca, ad=da, faf^-1=acd, bc=cb, bd=db, geg=be=eb, bf=fb, bg=gb, cd=dc, fef^-1=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >

Subgroups: 788 in 432 conjugacy classes, 120 normal (18 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, C₂⁴, C₂.C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂².D₄, C₂³×C₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, C₂³.₃₄D₄, C₂³.₈Q₈, C₂³.₂₃D₄, C₂³.₁₀D₄, C₂³.₇₈C₂³, C₂²×C₄⋊C₄, C₂×C₂².D₄, C₂²×C₄○D₄, C₂⁴.₂₄₃C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²≀C₂, C₂²×D₄, C₂×C₄○D₄, 2_-¹⁺⁴, C₂×C₂²≀C₂, C₂².₁₉C₂⁴, C₂³.₃₈C₂³, D₄⋊₆D₄, C₂⁴.₂₄₃C₂³

Smallest permutation representation of C₂⁴.₂₄₃C₂³
►On 64 points

Generators in S₆₄

(2 35)(4 33)(5 48)(7 46)(10 30)(12 32)(13 15)(14 52)(16 50)(17 55)(18 20)(19 53)(21 57)(23 59)(25 27)(26 37)(28 39)(38 40)(41 43)(42 64)(44 62)(49 51)(54 56)(61 63)

(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 9)(2 10)(3 11)(4 12)(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 34)(30 35)(31 36)(32 33)(37 52)(38 49)(39 50)(40 51)(41 56)(42 53)(43 54)(44 55)(45 60)(46 57)(47 58)(48 59)

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

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	2L	2M	2N	2O	4A	4B	4C	4D	4E	···	4R	4S	4T	4U	4V
order	1	2	···	2	2	2	2	2	2	2	2	2	4	4	4	4	4	···	4	4	4	4	4
size	1	1	···	1	2	2	2	2	4	4	4	4	2	2	2	2	4	···	4	8	8	8	8

38 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2_-¹⁺⁴

kernel

C₂⁴.₂₄₃C₂³

C₂³.₃₄D₄

C₂³.₈Q₈

C₂³.₂₃D₄

C₂³.₁₀D₄

C₂³.₇₈C₂³

C₂²×C₄⋊C₄

C₂×C₂².D₄

C₂²×C₄○D₄

C₂²×C₄

C₂×D₄

C₂×C₄

C₂²

# reps

Matrix representation of C₂⁴.₂₄₃C₂³ ►in GL₆(𝔽₅)

1	0	0	0	0	0
1	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

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

2	1	0	0	0	0
2	3	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	4

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	0	1
0	0	0	0	1	0

1	3	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

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

C₂⁴.₂₄₃C₂³ in GAP, Magma, Sage, TeX

C_2^4._{243}C_2^3

% in TeX

G:=Group("C2^4.243C2^3");

// GroupNames label

G:=SmallGroup(128,1138);

// by ID

G=gap.SmallGroup(128,1138);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,120,758,723,100,675]);

// Polycyclic

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

// generators/relations

G = C24.243C23 order 128 = 27

83rd non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₂₄₃C₂³ order 128 = 2⁷

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