C2^3.656C2^4

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

Aliases: C₂⁴.₈₂C₂³, C₂³.₆₅₆C₂⁴, C₂².₄₂₉2₊⁽¹⁺⁴⁾, C₄²⋊₅C₄⋊₃₀C₂, C₂³.₉₃(C₄○D₄), C₂³⋊₂D₄.₃₁C₂, (C₂×C₄²).₁₀₀C₂², (C₂³×C₄).₄₈₉C₂², (C₂²×C₄).₅₇₆C₂³, C₂³.₇Q₈⋊₁₀₆C₂, C₂³.₁₀D₄⋊₁₀₂C₂, C₂³.₁₁D₄⋊₁₁₃C₂, C₂³.₂₃D₄⋊₁₀₄C₂, C₂⁴.₃C₂²⋊₉₄C₂, (C₂²×D₄).₂₇₁C₂², C₂⁴.C₂²⋊₁₆₂C₂, C₂.₈₇(C₂².₃₂C₂⁴), C₂³.₈₁C₂³⋊₁₁₃C₂, C₂.₂₈(C₂².₅₄C₂⁴), C₂.C₄².₃₆₀C₂², C₂.₁₀₈(C₂².₄₅C₂⁴), C₂.₅₆(C₂².₃₄C₂⁴), C₂.₄₀(C₂².₄₉C₂⁴), C₂.₉₈(C₂².₄₇C₂⁴), (C₂×C₄).₄₅₄(C₄○D₄), (C₂×C₄⋊C₄).₄₆₇C₂², C₂².₅₁₇(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₃₀₆C₂², SmallGroup(128,1488)

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

Derived series

C₁ — C₂³ — C₂³.₆₅₆C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₂⁴.₃C₂² — C₂³.₆₅₆C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₅₆C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₅₆C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₅₆C₂⁴

Subgroups: 548 in 242 conjugacy classes, 88 normal (82 characteristic)
C₁, C₂^[×7], C₂^[×4], C₄^[×14], C₂²^[×7], C₂²^[×24], C₂×C₄^[×4], C₂×C₄^[×38], D₄^[×12], C₂³, C₂³^[×2], C₂³^[×20], C₄²^[×3], C₂²⋊C₄^[×18], C₄⋊C₄^[×5], C₂²×C₄^[×12], C₂²×C₄^[×4], C₂×D₄^[×12], C₂⁴^[×3], C₂.C₄²^[×10], C₂×C₄²^[×2], C₂×C₂²⋊C₄^[×14], C₂×C₄⋊C₄^[×5], C₂³×C₄, C₂²×D₄^[×3], C₂³.₇Q₈, C₄²⋊₅C₄, C₂³.₂₃D₄^[×2], C₂⁴.C₂²^[×4], C₂⁴.₃C₂², C₂³⋊₂D₄, C₂³.₁₀D₄^[×3], C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂³.₆₅₆C₂⁴

Quotients:
C₁, C₂^[×15], C₂²^[×35], C₂³^[×15], C₄○D₄^[×6], C₂⁴, C₂×C₄○D₄^[×3], 2₊⁽¹⁺⁴⁾^[×4], C₂².₃₂C₂⁴^[×2], C₂².₃₄C₂⁴, C₂².₄₅C₂⁴, C₂².₄₇C₂⁴, C₂².₄₉C₂⁴, C₂².₅₄C₂⁴, C₂³.₆₅₆C₂⁴

Generators and relations
G = < a,b,c,d,e,f,g | a²=b²=c²=e²=1, d²=c, f²=ba=ab, g²=a, ac=ca, ede=ad=da, geg^-1=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, gdg^-1=abd, fg=gf >

Smallest permutation representation
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

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

3	0	0	0	0	0
0	3	0	0	0	0
0	0	0	3	0	0
0	0	3	0	0	0
0	0	0	0	0	1
0	0	0	0	1	0

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	···	4P	4Q	4R	4S	4T
order	1	2	···	2	2	2	2	2	4	···	4	4	4	4	4
size	1	1	···	1	4	4	8	8	4	···	4	8	8	8	8

32 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊⁽¹⁺⁴⁾

kernel

C₂³.₆₅₆C₂⁴

C₂³.₇Q₈

C₄²⋊₅C₄

C₂³.₂₃D₄

C₂⁴.C₂²

C₂⁴.₃C₂²

C₂³⋊₂D₄

C₂³.₁₀D₄

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂×C₄

C₂³

C₂²

# reps

In GAP, Magma, Sage, TeX

C_2^3._{656}C_2^4

% in TeX

G:=Group("C2^3.656C2^4");

// GroupNames label

G:=SmallGroup(128,1488);

// by ID

G=gap.SmallGroup(128,1488);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,723,268,1571,346,80]);

// Polycyclic

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

// generators/relations

G = C₂³.₆₅₆C₂⁴ order 128 = 2⁷

373^rd central stem extension by C₂³ of C₂⁴

G = C23.656C24 order 128 = 27

373rd central stem extension by C23 of C24

G = C₂³.₆₅₆C₂⁴ order 128 = 2⁷

373^rd central stem extension by C₂³ of C₂⁴