C2^3.410C2^4

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

Aliases: C₂⁴.₁₆C₂³, C₂³.₄₁₀C₂⁴, C₂².₂₀₅2₊⁽¹⁺⁴⁾, C₄²⋊₅C₄⋊₁₂C₂, (C₂×C₄²).₅₀C₂², (C₂²×C₄).₈₁C₂³, C₂³.₈Q₈⋊₆₄C₂, C₂³.₇Q₈⋊₅₉C₂, C₂³.₃₁₃(C₄○D₄), C₂³.₁₁D₄⋊₃₂C₂, (C₂³×C₄).₁₀₃C₂², C₂⁴.C₂²⋊₆₈C₂, C₂³.₈₄C₂³⋊₃C₂, C₂³.₂₃D₄.₂₈C₂, (C₂²×D₄).₁₅₂C₂², C₂³.₆₃C₂³⋊₆₈C₂, C₂³.₈₃C₂³⋊₂₈C₂, C₂.₂₀(C₂².₃₂C₂⁴), C₂².₁₉(C₄²⋊₂C₂), C₂.₃₁(C₂².₄₅C₂⁴), C₂.C₄².₄₈₈C₂², C₂.₅₅(C₂³.₃₆C₂³), C₂.₄₀(C₂².₄₇C₂⁴), (C₄×C₂²⋊C₄)⋊₁₅C₂, (C₂×C₄).₇₃₅(C₄○D₄), (C₂×C₄⋊C₄).₂₇₆C₂², C₂.₁₇(C₂×C₄²⋊₂C₂), C₂².₂₈₇(C₂×C₄○D₄), (C₂×C₂.C₄²)⋊₃₅C₂, (C₂×C₂²⋊C₄).₄₈C₂², SmallGroup(128,1242)

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

Derived series

C₁ — C₂³ — C₂³.₄₁₀C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂×C₂.C₄² — C₂³.₄₁₀C₂⁴

Lower central

C₁ — C₂³ — C₂³.₄₁₀C₂⁴

Upper central

C₁ — C₂³ — C₂³.₄₁₀C₂⁴

Jennings

C₁ — C₂³ — C₂³.₄₁₀C₂⁴

Subgroups: 484 in 241 conjugacy classes, 96 normal (82 characteristic)
C₁, C₂^[×7], C₂^[×5], C₄^[×15], C₂²^[×7], C₂²^[×4], C₂²^[×19], C₂×C₄^[×4], C₂×C₄^[×49], D₄^[×4], C₂³, C₂³^[×6], C₂³^[×11], C₄²^[×3], C₂²⋊C₄^[×14], C₄⋊C₄^[×5], C₂²×C₄^[×13], C₂²×C₄^[×14], C₂×D₄^[×6], C₂⁴^[×2], C₂.C₄²^[×16], C₂×C₄²^[×2], C₂×C₂²⋊C₄^[×9], C₂×C₄⋊C₄^[×4], C₂³×C₄^[×3], C₂²×D₄, C₂×C₂.C₄², C₄×C₂²⋊C₄, C₂³.₇Q₈, C₄²⋊₅C₄, C₂³.₈Q₈, C₂³.₂₃D₄^[×3], C₂³.₆₃C₂³, C₂⁴.C₂²^[×2], C₂³.₁₁D₄^[×2], C₂³.₈₃C₂³, C₂³.₈₄C₂³, C₂³.₄₁₀C₂⁴

Quotients:
C₁, C₂^[×15], C₂²^[×35], C₂³^[×15], C₄○D₄^[×10], C₂⁴, C₄²⋊₂C₂^[×4], C₂×C₄○D₄^[×5], 2₊⁽¹⁺⁴⁾^[×2], C₂×C₄²⋊₂C₂, C₂³.₃₆C₂³, C₂².₃₂C₂⁴, C₂².₄₅C₂⁴^[×2], C₂².₄₇C₂⁴^[×2], C₂³.₄₁₀C₂⁴

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

Smallest permutation representation
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

Matrix representation ►G ⊆ 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

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

0	2	0	0	0	0
2	0	0	0	0	0
0	0	3	0	0	0
0	0	0	3	0	0
0	0	0	0	3	3
0	0	0	0	0	2

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

0	1	0	0	0	0
1	0	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

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[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],[0,2,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,0,1,4],[0,2,0,0,0,0,2,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,3,2],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,2,1,0,0,0,0,0,3],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

38 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	2L	4A	4B	4C	4D	4E	···	4V	4W	4X	4Y
order	1	2	···	2	2	2	2	2	2	4	4	4	4	4	···	4	4	4	4
size	1	1	···	1	2	2	2	2	8	2	2	2	2	4	···	4	8	8	8

38 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊⁽¹⁺⁴⁾

kernel

C₂³.₄₁₀C₂⁴

C₂×C₂.C₄²

C₄×C₂²⋊C₄

C₂³.₇Q₈

C₄²⋊₅C₄

C₂³.₈Q₈

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₁₁D₄

C₂³.₈₃C₂³

C₂³.₈₄C₂³

C₂×C₄

C₂³

C₂²

# reps

In GAP, Magma, Sage, TeX

C_2^3._{410}C_2^4

% in TeX

G:=Group("C2^3.410C2^4");

// GroupNames label

G:=SmallGroup(128,1242);

// by ID

G=gap.SmallGroup(128,1242);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,672,253,344,758,723,184,675]);

// Polycyclic

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

// generators/relations

G = C₂³.₄₁₀C₂⁴ order 128 = 2⁷

127^th central stem extension by C₂³ of C₂⁴

G = C23.410C24 order 128 = 27

127th central stem extension by C23 of C24

G = C₂³.₄₁₀C₂⁴ order 128 = 2⁷

127^th central stem extension by C₂³ of C₂⁴