C2^3.405C2^4

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

Aliases: C₂³.₄₀₅C₂⁴, C₂⁴.₅₈₀C₂³, C₂².₂₀₁2₊⁽¹⁺⁴⁾, C₂²⋊C₄.₈Q₈, C₄²⋊₈C₄⋊₃₃C₂, C₂³.₁₂₁(C₂×Q₈), C₂.₂₂(D₄⋊₃Q₈), (C₂²×C₄).₈₀C₂³, C₂³.₃₁₂(C₄○D₄), C₂².₈₈(C₂²×Q₈), (C₂×C₄²).₅₂₅C₂², (C₂³×C₄).₁₀₂C₂², C₂³.₇Q₈.₄₉C₂, C₂³.₈Q₈.₂₂C₂, C₂².₁₄(C₄².C₂), C₂³.₈₃C₂³⋊₂₇C₂, C₂³.₆₃C₂³⋊₆₇C₂, C₂³.₈₁C₂³⋊₂₇C₂, C₂³.₆₅C₂³⋊₇₄C₂, C₂.₁₉(C₂².₃₂C₂⁴), C₂.₃₀(C₂².₄₅C₂⁴), C₂.C₄².₁₅₆C₂², C₂.₅₂(C₂³.₃₆C₂³), (C₂×C₄).₄₁(C₂×Q₈), (C₄×C₂²⋊C₄).₅₄C₂, C₂.₁₀(C₂×C₄².C₂), (C₂×C₄).₃₇₇(C₄○D₄), (C₂×C₄⋊C₄).₂₇₁C₂², C₂².₂₈₂(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₅₀₂C₂², (C₂×C₂.C₄²).₂₇C₂, SmallGroup(128,1237)

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

Derived series

C₁ — C₂³ — C₂³.₄₀₅C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂×C₂.C₄² — C₂³.₄₀₅C₂⁴

Lower central

C₁ — C₂³ — C₂³.₄₀₅C₂⁴

Upper central

C₁ — C₂³ — C₂³.₄₀₅C₂⁴

Jennings

C₁ — C₂³ — C₂³.₄₀₅C₂⁴

Subgroups: 420 in 228 conjugacy classes, 104 normal (82 characteristic)
C₁, C₂^[×7], C₂^[×4], C₄^[×18], C₂²^[×7], C₂²^[×4], C₂²^[×12], C₂×C₄^[×8], C₂×C₄^[×50], C₂³, C₂³^[×6], C₂³^[×4], C₄²^[×3], C₂²⋊C₄^[×4], C₂²⋊C₄^[×6], C₄⋊C₄^[×13], C₂²×C₄^[×14], C₂²×C₄^[×14], C₂⁴, C₂.C₄²^[×16], C₂×C₄²^[×2], C₂×C₂²⋊C₄^[×4], C₂×C₄⋊C₄^[×10], C₂³×C₄^[×3], C₂×C₂.C₄², C₄×C₂²⋊C₄, C₂³.₇Q₈, C₄²⋊₈C₄, C₂³.₈Q₈^[×4], C₂³.₆₃C₂³^[×2], C₂³.₆₅C₂³, C₂³.₈₁C₂³, C₂³.₈₃C₂³^[×3], C₂³.₄₀₅C₂⁴

Quotients:
C₁, C₂^[×15], C₂²^[×35], Q₈^[×4], C₂³^[×15], C₂×Q₈^[×6], C₄○D₄^[×8], C₂⁴, C₄².C₂^[×4], C₂²×Q₈, C₂×C₄○D₄^[×4], 2₊⁽¹⁺⁴⁾^[×2], C₂×C₄².C₂, C₂³.₃₆C₂³, C₂².₃₂C₂⁴, C₂².₄₅C₂⁴^[×2], D₄⋊₃Q₈^[×2], C₂³.₄₀₅C₂⁴

Generators and relations
G = < a,b,c,d,e,f,g | a²=b²=c²=g²=1, d²=c, e²=ba=ab, f²=b, ac=ca, ede^-1=gdg=ad=da, ae=ea, af=fa, ag=ga, bc=cb, 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 11)(2 12)(3 9)(4 10)(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 36)(6 33)(7 34)(8 35)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(37 61)(38 62)(39 63)(40 64)(41 53)(42 54)(43 55)(44 56)(45 57)(46 58)(47 59)(48 60)

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

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

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

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

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

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

Matrix representation ►G ⊆ GL₆(𝔽₅)

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	4	0	0	0
0	0	0	4	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

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

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim

type

image

C₁

C₂

Q₈

C₄○D₄

2₊⁽¹⁺⁴⁾

kernel

C₂³.₄₀₅C₂⁴

C₂×C₂.C₄²

C₄×C₂²⋊C₄

C₂³.₇Q₈

C₄²⋊₈C₄

C₂³.₈Q₈

C₂³.₆₃C₂³

C₂³.₆₅C₂³

C₂³.₈₁C₂³

C₂³.₈₃C₂³

C₂²⋊C₄

C₂×C₄

C₂³

C₂²

# reps

In GAP, Magma, Sage, TeX

C_2^3._{405}C_2^4

% in TeX

G:=Group("C2^3.405C2^4");

// GroupNames label

G:=SmallGroup(128,1237);

// by ID

G=gap.SmallGroup(128,1237);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,344,758,723,184,675]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=g^2=1,d^2=c,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,b*c=c*b,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 = C₂³.₄₀₅C₂⁴ order 128 = 2⁷

122^nd central stem extension by C₂³ of C₂⁴

G = C23.405C24 order 128 = 27

122nd central stem extension by C23 of C24

G = C₂³.₄₀₅C₂⁴ order 128 = 2⁷

122^nd central stem extension by C₂³ of C₂⁴