C2^4.430C2^3 - GroupNames

G = C₂⁴.₄₃₀C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₄₃₀C₂³, C₂³.₆₄₄C₂⁴, C₂².₃₁₆2_-¹⁺⁴, C₂².₄₁₇2₊¹⁺⁴, C₄²⋊₈C₄⋊₅₈C₂, (C₂×C₄²).₉₂C₂², C₂³.₁₈₆(C₄○D₄), (C₂³×C₄).₄₈₅C₂², (C₂²×C₄).₅₆₇C₂³, C₂³.₇Q₈.₇₀C₂, C₂³.₈Q₈.₅₅C₂, C₂³.Q₈.₃₄C₂, C₂³.₁₁D₄.₄₂C₂, C₂³.₃₄D₄.₃₀C₂, C₂³.₈₃C₂³⋊₉₆C₂, C₂⁴.C₂².₆₀C₂, C₂³.₆₃C₂³⋊₁₆₀C₂, C₂³.₈₁C₂³⋊₁₀₈C₂, C₂.₉₆(C₂².₄₅C₂⁴), C₂.C₄².₃₄₈C₂², C₂.₂₉(C₂².₅₆C₂⁴), C₂.₉₃(C₂².₄₆C₂⁴), C₂.₄₉(C₂².₃₅C₂⁴), C₂.₈₆(C₂².₃₃C₂⁴), C₂.₃₉(C₂².₄₉C₂⁴), (C₂×C₄).₄₄₅(C₄○D₄), (C₂×C₄⋊C₄).₄₅₅C₂², C₂².₅₀₅(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₆₂C₂², SmallGroup(128,1476)

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

Derived series

C₁ — C₂³ — C₂⁴.₄₃₀C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂³.₃₄D₄ — C₂⁴.₄₃₀C₂³

Lower central

C₁ — C₂³ — C₂⁴.₄₃₀C₂³

Upper central

C₁ — C₂³ — C₂⁴.₄₃₀C₂³

Jennings

C₁ — C₂³ — C₂⁴.₄₃₀C₂³

Generators and relations for C₂⁴.₄₃₀C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=bcd, f²=c, g²=b, faf^-1=ab=ba, ac=ca, ad=da, eae^-1=abc, ag=ga, bc=cb, 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, C₂³.₇Q₈, C₂³.₃₄D₄, C₄²⋊₈C₄, C₂³.₈Q₈, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.Q₈, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂³.₈₃C₂³, C₂⁴.₄₃₀C₂³
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₃C₂⁴, C₂².₃₅C₂⁴, C₂².₄₅C₂⁴, C₂².₄₆C₂⁴, C₂².₄₉C₂⁴, C₂².₅₆C₂⁴, C₂⁴.₄₃₀C₂³

Smallest permutation representation of C₂⁴.₄₃₀C₂³
►On 64 points

Generators in S₆₄

(2 11)(4 9)(5 61)(6 49)(7 63)(8 51)(14 25)(16 27)(17 62)(18 50)(19 64)(20 52)(21 58)(22 40)(23 60)(24 38)(30 48)(32 46)(33 59)(34 37)(35 57)(36 39)(41 56)(43 54)

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

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

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

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

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

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

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

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

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

C₂⁴.₄₃₀C₂³

C₂³.₇Q₈

C₂³.₃₄D₄

C₄²⋊₈C₄

C₂³.₈Q₈

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.Q₈

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂³.₈₃C₂³

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂⁴.₄₃₀C₂³ ►in GL₆(𝔽₅)

1	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	4	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	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

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

2	0	0	0	0	0
0	2	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	1	1

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

C₂⁴.₄₃₀C₂³ in GAP, Magma, Sage, TeX

C_2^4._{430}C_2^3

% in TeX

G:=Group("C2^4.430C2^3");

// GroupNames label

G:=SmallGroup(128,1476);

// by ID

G=gap.SmallGroup(128,1476);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,784,253,232,758,723,100,1571,346]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=b*c*d,f^2=c,g^2=b,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*c=c*b,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 = C24.430C23 order 128 = 27

270th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₄₃₀C₂³ order 128 = 2⁷

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