C2^3.491C2^4 - GroupNames

G = C₂³.₄₉₁C₂⁴ order 128 = 2⁷

208^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₃₂C₂³, C₂³.₄₉₁C₂⁴, C₂².₂₇₃2₊¹⁺⁴, C₂³.₆₂₂(C₂×D₄), (C₂²×C₄).₃₉₆D₄, (C₂×C₄²).₇₅C₂², C₂³.₃₃₃(C₄○D₄), C₂³.₂₃D₄⋊₆₀C₂, C₂³.₁₀D₄⋊₄₆C₂, C₂.₁₈(C₂³⋊₃D₄), (C₂²×C₄).₅₅₀C₂³, (C₂³×C₄).₄₁₀C₂², C₂².₃₂₆(C₂²×D₄), C₂⁴.C₂²⋊₉₂C₂, C₂².₄₀(C₄.₄D₄), (C₂²×D₄).₁₇₈C₂², C₂³.₈₃C₂³⋊₄₉C₂, C₂.₆₉(C₂².₁₉C₂⁴), C₂.C₄².₂₂₅C₂², C₂.₆₉(C₂².₄₇C₂⁴), (C₄×C₂²⋊C₄)⋊₉₃C₂, (C₂²×C₄⋊C₄)⋊₂₈C₂, (C₂×C₄).₃₆₆(C₂×D₄), (C₂×C₄⋊D₄).₃₇C₂, C₂.₂₇(C₂×C₄.₄D₄), (C₂×C₄).₄₀₃(C₄○D₄), (C₂×C₄⋊C₄).₈₈₀C₂², C₂².₃₆₇(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₁₉₆C₂², SmallGroup(128,1323)

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

Derived series

C₁ — C₂³ — C₂³.₄₉₁C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂²×C₄⋊C₄ — C₂³.₄₉₁C₂⁴

Lower central

C₁ — C₂³ — C₂³.₄₉₁C₂⁴

Upper central

C₁ — C₂³ — C₂³.₄₉₁C₂⁴

Jennings

C₁ — C₂³ — C₂³.₄₉₁C₂⁴

Generators and relations for C₂³.₄₉₁C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=g²=1, d²=cb=bc, e²=ca=ac, f²=c, ab=ba, ede^-1=gdg=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: 612 in 298 conjugacy classes, 104 normal (18 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄⋊D₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₄×C₂²⋊C₄, C₂³.₂₃D₄, C₂⁴.C₂², C₂³.₁₀D₄, C₂³.₈₃C₂³, C₂²×C₄⋊C₄, C₂×C₄⋊D₄, C₂³.₄₉₁C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₄.₄D₄, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, C₂².₁₉C₂⁴, C₂×C₄.₄D₄, C₂³⋊₃D₄, C₂².₄₇C₂⁴, C₂³.₄₉₁C₂⁴

Smallest permutation representation of C₂³.₄₉₁C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

kernel

C₂³.₄₉₁C₂⁴

C₄×C₂²⋊C₄

C₂³.₂₃D₄

C₂⁴.C₂²

C₂³.₁₀D₄

C₂³.₈₃C₂³

C₂²×C₄⋊C₄

C₂×C₄⋊D₄

C₂²×C₄

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₄₉₁C₂⁴ ►in 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	0	1	0	0
0	0	4	0	0	0
0	0	0	0	3	3
0	0	0	0	0	2

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

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

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

C₂³.₄₉₁C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{491}C_2^4

% in TeX

G:=Group("C2^3.491C2^4");

// GroupNames label

G:=SmallGroup(128,1323);

// by ID

G=gap.SmallGroup(128,1323);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,672,253,568,758,723,352,675]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=g^2=1,d^2=c*b=b*c,e^2=c*a=a*c,f^2=c,a*b=b*a,e*d*e^-1=g*d*g=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 = C23.491C24 order 128 = 27

208th central stem extension by C23 of C24

G = C₂³.₄₉₁C₂⁴ order 128 = 2⁷

208^th central stem extension by C₂³ of C₂⁴