C2^3.608C2^4 - GroupNames

G = C₂³.₆₀₈C₂⁴ order 128 = 2⁷

325^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂³.₆₀₈C₂⁴, C₂⁴.₄₁₀C₂³, C₂².₃₈₂2₊¹⁺⁴, C₂².₂₈₅2_-¹⁺⁴, (C₂×D₄).₁₄₂D₄, C₂³.₆₉(C₂×D₄), C₂.₆₅(D₄⋊₆D₄), C₂³.₈₁(C₄○D₄), C₂.₁₁₃(D₄⋊₅D₄), C₂³.₇Q₈⋊₉₆C₂, C₂³.₂₃D₄⋊₉₅C₂, C₂³.₁₁D₄⋊₉₂C₂, C₂³.₁₀D₄⋊₉₁C₂, C₂.₄₈(C₂³⋊₃D₄), (C₂²×C₄).₁₈₆C₂³, (C₂³×C₄).₄₆₇C₂², C₂³.₈Q₈⋊₁₁₁C₂, C₂².₄₁₇(C₂²×D₄), (C₂²×D₄).₂₄₃C₂², C₂³.₈₁C₂³⋊₉₃C₂, C₂³.₈₃C₂³⋊₈₄C₂, C₂.₇₀(C₂².₃₂C₂⁴), C₂.C₄².₃₁₄C₂², C₂.₇₀(C₂².₃₃C₂⁴), C₂.₂₀(C₂².₅₆C₂⁴), C₂.₈₆(C₂².₄₇C₂⁴), (C₂×C₄).₄₂₂(C₂×D₄), (C₂×C₄⋊D₄).₄₇C₂, (C₂×C₄).₄₃₂(C₄○D₄), (C₂×C₄⋊C₄).₄₂₁C₂², C₂².₄₇₀(C₂×C₄○D₄), (C₂×C₂².D₄)⋊₄₁C₂, (C₂×C₂²⋊C₄).₂₇₄C₂², SmallGroup(128,1440)

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

Derived series

C₁ — C₂³ — C₂³.₆₀₈C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂³.₂₃D₄ — C₂³.₆₀₈C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₀₈C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₀₈C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₀₈C₂⁴

Generators and relations for C₂³.₆₀₈C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=e²=g²=1, f²=ba=ab, ac=ca, ede=ad=da, geg=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=abd, fg=gf >

Subgroups: 612 in 284 conjugacy classes, 96 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂.C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄⋊D₄, C₂².D₄, C₂³×C₄, C₂²×D₄, C₂³.₇Q₈, C₂³.₈Q₈, C₂³.₂₃D₄, C₂³.₁₀D₄, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂³.₈₃C₂³, C₂×C₄⋊D₄, C₂×C₂².D₄, C₂³.₆₀₈C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂³⋊₃D₄, C₂².₃₂C₂⁴, C₂².₃₃C₂⁴, D₄⋊₅D₄, D₄⋊₆D₄, C₂².₄₇C₂⁴, C₂².₅₆C₂⁴, C₂³.₆₀₈C₂⁴

Smallest permutation representation of C₂³.₆₀₈C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

(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)

(5 15)(6 16)(7 13)(8 14)(9 62)(10 63)(11 64)(12 61)(29 33)(30 34)(31 35)(32 36)(37 41)(38 42)(39 43)(40 44)(45 47)(46 48)(49 51)(50 52)(53 55)(54 56)(57 59)(58 60)

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	···	2M	4A	···	4L	4M	···	4R
order	1	2	···	2	2	···	2	4	···	4	4	···	4
size	1	1	···	1	4	···	4	4	···	4	8	···	8

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂³.₆₀₈C₂⁴

C₂³.₇Q₈

C₂³.₈Q₈

C₂³.₂₃D₄

C₂³.₁₀D₄

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂³.₈₃C₂³

C₂×C₄⋊D₄

C₂×C₂².D₄

C₂×D₄

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₆₀₈C₂⁴ ►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

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

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

4	2	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	0	1
0	0	0	0	1	0

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

1	0	0	0	0	0
1	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	1

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

C₂³.₆₀₈C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{608}C_2^4

% in TeX

G:=Group("C2^3.608C2^4");

// GroupNames label

G:=SmallGroup(128,1440);

// by ID

G=gap.SmallGroup(128,1440);

# by ID

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

// Polycyclic

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

// generators/relations

G = C23.608C24 order 128 = 27

325th central stem extension by C23 of C24

G = C₂³.₆₀₈C₂⁴ order 128 = 2⁷

325^th central stem extension by C₂³ of C₂⁴