C2^4.443C2^3 - GroupNames

G = C₂⁴.₄₄₃C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₄₄₃C₂³, C₂³.₆₆₅C₂⁴, C₂².₃₃₁2_-¹⁺⁴, C₂².₄₃₈2₊¹⁺⁴, C₄²⋊₈C₄⋊₆₃C₂, C₂³.₁₉₃(C₄○D₄), (C₂²×C₄).₅₈₄C₂³, (C₂×C₄²).₆₉₇C₂², (C₂³×C₄).₁₇₀C₂², C₂³.Q₈.₃₅C₂, C₂³.₈Q₈.₆₀C₂, C₂³.₁₁D₄.₄₈C₂, C₂³.₃₄D₄.₃₃C₂, C₂³.₈₄C₂³⋊₁₃C₂, C₂.₉₂(C₂².₃₂C₂⁴), C₂⁴.C₂².₆₇C₂, C₂³.₆₅C₂³⋊₁₄₃C₂, C₂³.₈₁C₂³⋊₁₁₇C₂, C₂³.₆₃C₂³⋊₁₇₂C₂, C₂³.₈₃C₂³⋊₁₀₇C₂, C₂.C₄².₃₆₉C₂², C₂.₉₂(C₂².₃₃C₂⁴), C₂.₃₉(C₂².₅₇C₂⁴), C₂.₁₀₁(C₂².₄₇C₂⁴), C₂.₁₀₆(C₂².₃₆C₂⁴), C₂.₁₀₃(C₂².₄₆C₂⁴), (C₂×C₄).₄₅₉(C₄○D₄), (C₂×C₄⋊C₄).₄₇₅C₂², C₂².₅₂₆(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₇₁C₂², SmallGroup(128,1497)

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

Derived series

C₁ — C₂³ — C₂⁴.₄₄₃C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂³.₈Q₈ — C₂⁴.₄₄₃C₂³

Lower central

C₁ — C₂³ — C₂⁴.₄₄₃C₂³

Upper central

C₁ — C₂³ — C₂⁴.₄₄₃C₂³

Jennings

C₁ — C₂³ — C₂⁴.₄₄₃C₂³

Generators and relations for C₂⁴.₄₄₃C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=d, f²=g²=cb=bc, faf^-1=ab=ba, ac=ca, ad=da, eae^-1=abc, ag=ga, 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: 372 in 196 conjugacy classes, 88 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂³×C₄, C₂³.₃₄D₄, C₄²⋊₈C₄, C₂³.₈Q₈, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₅C₂³, C₂³.Q₈, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂³.₈₃C₂³, C₂³.₈₄C₂³, C₂⁴.₄₄₃C₂³
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₂C₂⁴, C₂².₃₃C₂⁴, C₂².₃₆C₂⁴, C₂².₄₆C₂⁴, C₂².₄₇C₂⁴, C₂².₅₇C₂⁴, C₂⁴.₄₄₃C₂³

Smallest permutation representation of C₂⁴.₄₄₃C₂³
►On 64 points

Generators in S₆₄

(2 24)(4 22)(5 62)(6 39)(7 64)(8 37)(10 52)(12 50)(14 56)(16 54)(17 45)(18 32)(19 47)(20 30)(26 44)(28 42)(29 57)(31 59)(33 63)(34 40)(35 61)(36 38)(46 60)(48 58)

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

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

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

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂⁴.₄₄₃C₂³

C₂³.₃₄D₄

C₄²⋊₈C₄

C₂³.₈Q₈

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂³.Q₈

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂³.₈₃C₂³

C₂³.₈₄C₂³

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂⁴.₄₄₃C₂³ ►in GL₆(𝔽₅)

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

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

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

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

0	1	0	0	0	0
1	0	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	2

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

C₂⁴.₄₄₃C₂³ in GAP, Magma, Sage, TeX

C_2^4._{443}C_2^3

% in TeX

G:=Group("C2^4.443C2^3");

// GroupNames label

G:=SmallGroup(128,1497);

// by ID

G=gap.SmallGroup(128,1497);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,344,758,723,268,1571,346]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=d,f^2=g^2=c*b=b*c,f*a*f^-1=a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b*c,a*g=g*a,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 = C24.443C23 order 128 = 27

283rd non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₄₄₃C₂³ order 128 = 2⁷

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