C2^3.473C2^4

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

Aliases: C₂³.₄₇₃C₂⁴, C₂⁴.₃₃₇C₂³, C₂².₁₉₃2_-⁽¹⁺⁴⁾, C₂².₂₅₆2₊⁽¹⁺⁴⁾, C₄²⋊₅C₄⋊₁₉C₂, (C₂×C₄²).₆₉C₂², C₂³.₁₅₈(C₄○D₄), (C₂²×C₄).₁₀₅C₂³, (C₂³×C₄).₁₂₀C₂², C₂³.₈Q₈.₃₂C₂, C₂³.Q₈.₁₄C₂, C₂³.₃₄D₄.₁₉C₂, C₂³.₁₁D₄.₁₉C₂, C₂³.₈₃C₂³⋊₄₃C₂, C₂³.₆₃C₂³⋊₉₀C₂, C₂³.₆₅C₂³⋊₉₀C₂, C₂⁴.C₂².₃₂C₂, C₂.₅₄(C₂².₄₅C₂⁴), C₂.C₄².₂₀₉C₂², C₂.₂₈(C₂².₃₃C₂⁴), C₂.₄₃(C₂².₅₀C₂⁴), C₂.₆₇(C₂².₄₆C₂⁴), C₂.₆₁(C₂².₄₇C₂⁴), C₂.₈₄(C₂³.₃₆C₂³), (C₄×C₄⋊C₄)⋊₉₉C₂, (C₄×C₂²⋊C₄).₆₄C₂, (C₂×C₄).₈₅₈(C₄○D₄), (C₂×C₄⋊C₄).₃₂₀C₂², C₂².₃₄₉(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₁₈₉C₂², SmallGroup(128,1305)

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

Derived series

C₁ — C₂³ — C₂³.₄₇₃C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₄×C₂²⋊C₄ — C₂³.₄₇₃C₂⁴

Lower central

C₁ — C₂³ — C₂³.₄₇₃C₂⁴

Upper central

C₁ — C₂³ — C₂³.₄₇₃C₂⁴

Jennings

C₁ — C₂³ — C₂³.₄₇₃C₂⁴

Subgroups: 372 in 206 conjugacy classes, 92 normal (82 characteristic)
C₁, C₂^[×7], C₂^[×2], C₄^[×18], C₂²^[×7], C₂²^[×10], C₂×C₄^[×8], C₂×C₄^[×42], C₂³, C₂³^[×2], C₂³^[×6], C₄²^[×6], C₂²⋊C₄^[×11], C₄⋊C₄^[×14], C₂²×C₄^[×14], C₂²×C₄^[×5], C₂⁴, C₂.C₄²^[×16], C₂×C₄²^[×4], C₂×C₂²⋊C₄^[×6], C₂×C₄⋊C₄^[×8], C₂³×C₄, C₄×C₂²⋊C₄, C₄×C₄⋊C₄, C₂³.₃₄D₄, C₄²⋊₅C₄, C₂³.₈Q₈, C₂³.₆₃C₂³^[×3], C₂⁴.C₂²^[×2], C₂³.₆₅C₂³, C₂³.Q₈, C₂³.₁₁D₄, C₂³.₈₃C₂³^[×2], C₂³.₄₇₃C₂⁴

Quotients:
C₁, C₂^[×15], C₂²^[×35], C₂³^[×15], C₄○D₄^[×10], C₂⁴, C₂×C₄○D₄^[×5], 2₊⁽¹⁺⁴⁾, 2_-⁽¹⁺⁴⁾, C₂³.₃₆C₂³^[×2], C₂².₃₃C₂⁴, C₂².₄₅C₂⁴, C₂².₄₆C₂⁴, C₂².₄₇C₂⁴, C₂².₅₀C₂⁴, C₂³.₄₇₃C₂⁴

Generators and relations
G = < a,b,c,d,e,f,g | a²=b²=c²=f²=1, d²=e²=ca=ac, g²=b, ab=ba, ede^-1=gdg^-1=ad=da, ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=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 37)(6 38)(7 39)(8 40)(13 52)(14 49)(15 50)(16 51)(17 46)(18 47)(19 48)(20 45)(21 43)(22 44)(23 41)(24 42)(25 54)(26 55)(27 56)(28 53)(29 60)(30 57)(31 58)(32 59)(33 64)(34 61)(35 62)(36 63)

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

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

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

(2 27)(4 25)(5 39)(6 44)(7 37)(8 42)(9 54)(11 56)(14 57)(16 59)(17 33)(18 45)(19 35)(20 47)(21 41)(22 38)(23 43)(24 40)(30 49)(32 51)(34 63)(36 61)(46 64)(48 62)

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

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

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

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

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	4	0
0	0	0	0	0	4

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	4A	···	4H	4I	···	4X	4Y	4Z	4AA	4AB
order	1	2	···	2	2	2	4	···	4	4	···	4	4	4	4	4
size	1	1	···	1	4	4	2	···	2	4	···	4	8	8	8	8

38 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊⁽¹⁺⁴⁾

2_-⁽¹⁺⁴⁾

kernel

C₂³.₄₇₃C₂⁴

C₄×C₂²⋊C₄

C₄×C₄⋊C₄

C₂³.₃₄D₄

C₄²⋊₅C₄

C₂³.₈Q₈

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂³.Q₈

C₂³.₁₁D₄

C₂³.₈₃C₂³

C₂×C₄

C₂³

C₂²

# reps

In GAP, Magma, Sage, TeX

C_2^3._{473}C_2^4

% in TeX

G:=Group("C2^3.473C2^4");

// GroupNames label

G:=SmallGroup(128,1305);

// by ID

G=gap.SmallGroup(128,1305);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,568,758,723,675,136]);

// Polycyclic

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

// generators/relations

G = C₂³.₄₇₃C₂⁴ order 128 = 2⁷

190^th central stem extension by C₂³ of C₂⁴

G = C23.473C24 order 128 = 27

190th central stem extension by C23 of C24

G = C₂³.₄₇₃C₂⁴ order 128 = 2⁷

190^th central stem extension by C₂³ of C₂⁴