C2^3.493C2^4 - GroupNames

G = C₂³.₄₉₃C₂⁴ order 128 = 2⁷

210^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₃₄C₂³, C₂³.₄₉₃C₂⁴, C₂².₂₇₅2₊¹⁺⁴, (C₂×C₄²).₇₇C₂², (C₂²×C₄).₁₁₅C₂³, C₂⁴.C₂²⋊₉₄C₂, C₂³.₁₀D₄.₂₆C₂, (C₂²×D₄).₁₈₀C₂², C₂³.₈₃C₂³⋊₅₀C₂, C₂³.₈₁C₂³⋊₄₉C₂, C₂.₃₃(C₂².₃₂C₂⁴), C₂⁴.₃C₂².₅₂C₂, C₂.C₄².₂₂₇C₂², C₂.₂₆(C₂².₅₃C₂⁴), C₂.₉₅(C₂³.₃₆C₂³), C₂.₇₀(C₂².₄₇C₂⁴), C₂.₃₀(C₂².₄₉C₂⁴), (C₄×C₄⋊C₄)⋊₁₀₆C₂, (C₂×C₄).₈₅₉(C₄○D₄), (C₂×C₄⋊C₄).₈₈₂C₂², C₂².₃₆₉(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₅₇C₂², SmallGroup(128,1325)

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

Derived series

C₁ — C₂³ — C₂³.₄₉₃C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₄×C₄⋊C₄ — C₂³.₄₉₃C₂⁴

Lower central

C₁ — C₂³ — C₂³.₄₉₃C₂⁴

Upper central

C₁ — C₂³ — C₂³.₄₉₃C₂⁴

Jennings

C₁ — C₂³ — C₂³.₄₉₃C₂⁴

Generators and relations for C₂³.₄₉₃C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=1, d²=e²=ca=ac, f²=c, g²=b, ab=ba, ede^-1=gdg^-1=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 >

Subgroups: 436 in 220 conjugacy classes, 92 normal (22 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², 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, C₂⁴.C₂², C₂⁴.₃C₂², C₂³.₁₀D₄, C₂³.₈₁C₂³, C₂³.₈₃C₂³, C₂³.₄₉₃C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, C₂³.₃₆C₂³, C₂².₃₂C₂⁴, C₂².₄₇C₂⁴, C₂².₄₉C₂⁴, C₂².₅₃C₂⁴, C₂³.₄₉₃C₂⁴

Smallest permutation representation of C₂³.₄₉₃C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

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	C₂³.₄₉₃C₂⁴	C₄×C₄⋊C₄	C₂⁴.C₂²	C₂⁴.₃C₂²	C₂³.₁₀D₄	C₂³.₈₁C₂³	C₂³.₈₃C₂³	C₂×C₄	C₂²
# reps	1	2	8	1	2	1	1	20	2

Matrix representation of C₂³.₄₉₃C₂⁴ ►in 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

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

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

2	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	0	0	2
0	0	0	0	3	0

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

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

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

C₂³.₄₉₃C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{493}C_2^4

% in TeX

G:=Group("C2^3.493C2^4");

// GroupNames label

G:=SmallGroup(128,1325);

// by ID

G=gap.SmallGroup(128,1325);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,568,758,723,352,675,136]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=e^2=c*a=a*c,f^2=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^-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 = C23.493C24 order 128 = 27

210th central stem extension by C23 of C24

G = C₂³.₄₉₃C₂⁴ order 128 = 2⁷

210^th central stem extension by C₂³ of C₂⁴