C2^3.625C2^4 - GroupNames

G = C₂³.₆₂₅C₂⁴ order 128 = 2⁷

342^nd central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₇₂C₂³, C₂³.₆₂₅C₂⁴, C₂².₃₉₈2₊¹⁺⁴, C₂².₃₀₀2_-¹⁺⁴, C₄⋊C₄.₁₂₇D₄, C₂.₇₆(D₄⋊₆D₄), C₂.₅₃(Q₈⋊₅D₄), (C₂²×C₄).₁₉₆C₂³, (C₂×C₄²).₆₇₆C₂², C₂².₄₃₄(C₂²×D₄), C₂³.Q₈.₃₁C₂, C₂³.₁₁D₄.₃₉C₂, C₂³.₈₃C₂³⋊₈₈C₂, C₂⁴.C₂².₅₅C₂, C₂³.₆₃C₂³⋊₁₄₉C₂, C₂³.₈₁C₂³⋊₁₀₀C₂, C₂³.₆₅C₂³⋊₁₃₅C₂, C₂.C₄².₃₃₁C₂², C₂.₂₇(C₂².₅₇C₂⁴), C₂.₄₇(C₂².₃₅C₂⁴), C₂.₉₀(C₂².₄₇C₂⁴), C₂.₅₃(C₂².₃₁C₂⁴), (C₂×C₄).₁₂₀(C₂×D₄), (C₂×C₄².C₂)⋊₂₇C₂, (C₂×C₄).₂₀₆(C₄○D₄), (C₂×C₄⋊C₄).₄₃₈C₂², C₂².₄₈₇(C₂×C₄○D₄), (C₂×C₄²⋊₂C₂).₁₇C₂, (C₂×C₂²⋊C₄).₂₈₉C₂², SmallGroup(128,1457)

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

Derived series

C₁ — C₂³ — C₂³.₆₂₅C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₂³.₆₅C₂³ — C₂³.₆₂₅C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₂₅C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₂₅C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₂₅C₂⁴

Generators and relations for C₂³.₆₂₅C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=1, d²=ba=ab, e²=f²=b, g²=a, ac=ca, ede^-1=ad=da, geg^-1=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^-1=abd, fg=gf >

Subgroups: 388 in 217 conjugacy classes, 96 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄².C₂, C₄²⋊₂C₂, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₅C₂³, C₂³.Q₈, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂³.₈₃C₂³, C₂×C₄².C₂, C₂×C₄²⋊₂C₂, C₂³.₆₂₅C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₁C₂⁴, C₂².₃₅C₂⁴, D₄⋊₆D₄, Q₈⋊₅D₄, C₂².₄₇C₂⁴, C₂².₅₇C₂⁴, C₂³.₆₂₅C₂⁴

Smallest permutation representation of C₂³.₆₂₅C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	4A	···	4R	4S	···	4W
order	1	2	···	2	2	4	···	4	4	···	4
size	1	1	···	1	8	4	···	4	8	···	8

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂³.₆₂₅C₂⁴

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂³.Q₈

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂³.₈₃C₂³

C₂×C₄².C₂

C₂×C₄²⋊₂C₂

C₄⋊C₄

C₂×C₄

C₂²

# reps

Matrix representation of C₂³.₆₂₅C₂⁴ ►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	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

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	3	3

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

2	0	0	0	0	0
0	3	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	4

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

C₂³.₆₂₅C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{625}C_2^4

% in TeX

G:=Group("C2^3.625C2^4");

// GroupNames label

G:=SmallGroup(128,1457);

// by ID

G=gap.SmallGroup(128,1457);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,232,758,723,184,1571,346,80]);

// Polycyclic

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

// generators/relations

G = C23.625C24 order 128 = 27

342nd central stem extension by C23 of C24

G = C₂³.₆₂₅C₂⁴ order 128 = 2⁷

342^nd central stem extension by C₂³ of C₂⁴