C2^4.430C2^3

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), (C₂²×C₄).₅₆₇C₂³, (C₂³×C₄).₄₈₅C₂², C₂³.Q₈.₃₄C₂, C₂³.₈Q₈.₅₅C₂, C₂³.₇Q₈.₇₀C₂, 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), C₂.₈₆(C₂².₃₃C₂⁴), C₂.₄₉(C₂².₃₅C₂⁴), 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

Subgroups: 372 in 196 conjugacy classes, 88 normal (82 characteristic)
C₁, C₂^[×7], C₂^[×2], C₄^[×16], C₂²^[×7], C₂²^[×10], C₂×C₄^[×4], C₂×C₄^[×44], C₂³, C₂³^[×2], C₂³^[×6], C₄²^[×3], C₂²⋊C₄^[×9], C₄⋊C₄^[×13], C₂²×C₄^[×14], C₂²×C₄^[×5], C₂⁴, C₂.C₄²^[×16], C₂×C₄²^[×2], C₂×C₂²⋊C₄^[×6], C₂×C₄⋊C₄^[×10], C₂³×C₄, C₂³.₇Q₈, C₂³.₃₄D₄, C₄²⋊₈C₄, C₂³.₈Q₈, C₂³.₆₃C₂³^[×3], C₂⁴.C₂²^[×2], C₂³.Q₈, C₂³.₁₁D₄, C₂³.₈₁C₂³^[×2], C₂³.₈₃C₂³^[×2], C₂⁴.₄₃₀C₂³

Quotients:
C₁, C₂^[×15], C₂²^[×35], C₂³^[×15], C₄○D₄^[×6], C₂⁴, C₂×C₄○D₄^[×3], 2₊⁽¹⁺⁴⁾^[×2], 2_-⁽¹⁺⁴⁾^[×2], C₂².₃₃C₂⁴^[×2], C₂².₃₅C₂⁴, C₂².₄₅C₂⁴, C₂².₄₆C₂⁴, C₂².₄₉C₂⁴, C₂².₅₆C₂⁴, C₂⁴.₄₃₀C₂³

Generators and relations
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 >

Smallest permutation representation
►On 64 points

Generators in S₆₄

(2 43)(4 41)(5 37)(6 25)(7 39)(8 27)(9 56)(11 54)(14 57)(16 59)(17 35)(18 50)(19 33)(20 52)(21 26)(22 40)(23 28)(24 38)(30 48)(32 46)(34 61)(36 63)(49 62)(51 64)

(1 55)(2 56)(3 53)(4 54)(5 37)(6 38)(7 39)(8 40)(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 26)(22 27)(23 28)(24 25)(29 58)(30 59)(31 60)(32 57)(33 64)(34 61)(35 62)(36 63)

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

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

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

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

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

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

Matrix representation ►G ⊆ 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] >;

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

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

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

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