C2^3.413C2^4 - GroupNames

G = C₂³.₄₁₃C₂⁴ order 128 = 2⁷

130^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₁₈C₂³, C₂³.₄₁₃C₂⁴, C₂².₂₀₈2₊¹⁺⁴, C₄²⋊₈C₄⋊₃₄C₂, (C₂×C₄²).₅₃C₂², C₄.₄₂(C₄²⋊₂C₂), C₂³.₁₁D₄⋊₃₄C₂, (C₂²×C₄).₅₂₆C₂³, C₂⁴.C₂²⋊₇₀C₂, (C₂²×D₄).₁₅₄C₂², C₂³.₆₅C₂³⋊₇₆C₂, C₂⁴.₃C₂².₃₉C₂, C₂.C₄².₄₉₀C₂², C₂.₅₆(C₂³.₃₆C₂³), C₂.₁₃(C₂².₅₃C₂⁴), C₂.₁₉(C₂².₃₄C₂⁴), C₂.₄₂(C₂².₄₇C₂⁴), C₂.₁₈(C₂².₄₉C₂⁴), (C₄×C₄⋊C₄)⋊₇₆C₂, (C₂×C₄).₁₃₄(C₄○D₄), (C₂×C₄⋊C₄).₈₆₁C₂², C₂.₂₀(C₂×C₄²⋊₂C₂), C₂².₂₉₀(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₄₉C₂², SmallGroup(128,1245)

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

Derived series

C₁ — C₂³ — C₂³.₄₁₃C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₄×C₄⋊C₄ — C₂³.₄₁₃C₂⁴

Lower central

C₁ — C₂³ — C₂³.₄₁₃C₂⁴

Upper central

C₁ — C₂³ — C₂³.₄₁₃C₂⁴

Jennings

C₁ — C₂³ — C₂³.₄₁₃C₂⁴

Generators and relations for C₂³.₄₁₃C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=1, d²=cb=bc, e²=b, f²=ba=ab, g²=a, ac=ca, ede^-1=gdg^-1=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 >

Subgroups: 436 in 220 conjugacy classes, 96 normal (42 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂.C₄², C₂.C₄², C₂×C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂²×D₄, C₄×C₄⋊C₄, C₄²⋊₈C₄, C₂⁴.C₂², C₂³.₆₅C₂³, C₂⁴.₃C₂², C₂³.₁₁D₄, C₂³.₄₁₃C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₄²⋊₂C₂, C₂×C₄○D₄, 2₊¹⁺⁴, C₂×C₄²⋊₂C₂, C₂³.₃₆C₂³, C₂².₃₄C₂⁴, C₂².₄₇C₂⁴, C₂².₄₉C₂⁴, C₂².₅₃C₂⁴, C₂³.₄₁₃C₂⁴

Smallest permutation representation of C₂³.₄₁₃C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim	1	1	1	1	1	1	1	2	4
type	+	+	+	+	+	+	+		+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₄○D₄	2₊¹⁺⁴
kernel	C₂³.₄₁₃C₂⁴	C₄×C₄⋊C₄	C₄²⋊₈C₄	C₂⁴.C₂²	C₂³.₆₅C₂³	C₂⁴.₃C₂²	C₂³.₁₁D₄	C₂×C₄	C₂²
# reps	1	2	1	4	1	3	4	20	2

Matrix representation of C₂³.₄₁₃C₂⁴ ►in GL₆(𝔽₅)

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	4	0
0	0	0	0	0	4

4	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	1	0
0	0	0	0	0	1

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

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

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

4	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	0	1
0	0	0	0	4	0

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

C₂³.₄₁₃C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{413}C_2^4

% in TeX

G:=Group("C2^3.413C2^4");

// GroupNames label

G:=SmallGroup(128,1245);

// by ID

G=gap.SmallGroup(128,1245);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,672,253,232,758,723,268,675,80]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=c*b=b*c,e^2=b,f^2=b*a=a*b,g^2=a,a*c=c*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,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 = C23.413C24 order 128 = 27

130th central stem extension by C23 of C24

G = C₂³.₄₁₃C₂⁴ order 128 = 2⁷

130^th central stem extension by C₂³ of C₂⁴