C2^3.611C2^4 - GroupNames

G = C₂³.₆₁₁C₂⁴ order 128 = 2⁷

328^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₆₇C₂³, C₂³.₆₁₁C₂⁴, C₂².₂₈₇2_-¹⁺⁴, C₂².₃₈₅2₊¹⁺⁴, C₄⋊C₄.₁₂₁D₄, C₂.₄₉(Q₈⋊₅D₄), C₂³.₈₂(C₄○D₄), C₂.₁₁₆(D₄⋊₅D₄), C₂³.₄Q₈⋊₄₈C₂, C₂³.Q₈⋊₆₈C₂, C₂³.₁₀D₄⋊₉₄C₂, (C₂³×C₄).₄₆₈C₂², (C₂×C₄²).₆₆₂C₂², (C₂²×C₄).₁₈₉C₂³, C₂³.₈Q₈⋊₁₁₃C₂, C₂².₄₂₀(C₂²×D₄), C₂⁴.₃C₂²⋊₈₇C₂, (C₂²×D₄).₂₄₆C₂², C₂⁴.C₂²⋊₁₃₉C₂, C₂.₆₂(C₂².₂₉C₂⁴), C₂³.₆₃C₂³⋊₁₄₀C₂, C₂.C₄².₃₁₇C₂², C₂.₈₇(C₂².₄₇C₂⁴), C₂.₇₁(C₂².₃₃C₂⁴), C₂.₂₂(C₂².₅₆C₂⁴), C₂.₄₇(C₂².₃₄C₂⁴), (C₂×C₄).₄₂₄(C₂×D₄), (C₂×C₄⋊D₄).₄₈C₂, (C₂×C₄².C₂)⋊₂₄C₂, (C₂×C₄).₁₉₈(C₄○D₄), (C₂×C₄⋊C₄).₄₂₄C₂², C₂².₄₇₃(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₂₇₇C₂², SmallGroup(128,1443)

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

Derived series

C₁ — C₂³ — C₂³.₆₁₁C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₂⁴.₃C₂² — C₂³.₆₁₁C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₁₁C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₁₁C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₁₁C₂⁴

Generators and relations for C₂³.₆₁₁C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=e²=1, d²=g²=a, f²=b, ab=ba, ac=ca, ede=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: 580 in 270 conjugacy classes, 96 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄⋊D₄, C₄².C₂, C₂³×C₄, C₂²×D₄, C₂³.₈Q₈, C₂³.₆₃C₂³, C₂⁴.C₂², C₂⁴.₃C₂², C₂³.₁₀D₄, C₂³.Q₈, C₂³.₄Q₈, C₂×C₄⋊D₄, C₂×C₄².C₂, C₂³.₆₁₁C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₂₉C₂⁴, C₂².₃₃C₂⁴, C₂².₃₄C₂⁴, D₄⋊₅D₄, Q₈⋊₅D₄, C₂².₄₇C₂⁴, C₂².₅₆C₂⁴, C₂³.₆₁₁C₂⁴

Smallest permutation representation of C₂³.₆₁₁C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	···	4P	4Q	4R	4S	4T
order	1	2	···	2	2	2	2	2	4	···	4	4	4	4	4
size	1	1	···	1	4	4	8	8	4	···	4	8	8	8	8

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂³.₆₁₁C₂⁴

C₂³.₈Q₈

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂⁴.₃C₂²

C₂³.₁₀D₄

C₂³.Q₈

C₂³.₄Q₈

C₂×C₄⋊D₄

C₂×C₄².C₂

C₄⋊C₄

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₆₁₁C₂⁴ ►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

1	4	0	0	0	0
2	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	4

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

3	0	0	0	0	0
1	2	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	1	0

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

C₂³.₆₁₁C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{611}C_2^4

% in TeX

G:=Group("C2^3.611C2^4");

// GroupNames label

G:=SmallGroup(128,1443);

// by ID

G=gap.SmallGroup(128,1443);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,758,723,184,1571,346,80]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=1,d^2=g^2=a,f^2=b,a*b=b*a,a*c=c*a,e*d*e=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 = C23.611C24 order 128 = 27

328th central stem extension by C23 of C24

G = C₂³.₆₁₁C₂⁴ order 128 = 2⁷

328^th central stem extension by C₂³ of C₂⁴