C2^3.576C2^4 - GroupNames

G = C₂³.₅₇₆C₂⁴ order 128 = 2⁷

293^rd central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₆₀C₂³, C₂³.₅₇₆C₂⁴, C₂².₂₆₁2_-¹⁺⁴, C₂².₃₅₀2₊¹⁺⁴, C₄⋊C₄⋊₉D₄, C₂.₄₀D₄², C₂²⋊C₄⋊₁₃D₄, C₂³.₆₂(C₂×D₄), C₂³⋊₂D₄⋊₃₇C₂, C₂³⋊Q₈⋊₃₉C₂, C₂.₈₆(D₄⋊₅D₄), C₂.₄₃(Q₈⋊₅D₄), C₂³.₇₄(C₄○D₄), C₂³.₈Q₈⋊₉₆C₂, C₂³.Q₈⋊₅₀C₂, C₂³.₂₃D₄⋊₈₀C₂, C₂³.₁₀D₄⋊₇₄C₂, (C₂×C₄²).₆₃₅C₂², (C₂³×C₄).₄₄₆C₂², (C₂²×C₄).₈₆₅C₂³, C₂².₃₈₅(C₂²×D₄), C₂⁴.₃C₂²⋊₇₂C₂, (C₂²×D₄).₂₁₇C₂², (C₂²×Q₈).₁₇₄C₂², C₂.₅₈(C₂².₃₂C₂⁴), C₂.₅₄(C₂².₂₉C₂⁴), C₂³.₆₃C₂³⋊₁₂₆C₂, C₂.C₄².₂₈₇C₂², C₂.₈(C₂².₅₆C₂⁴), C₂.₃₉(C₂².₃₁C₂⁴), (C₂×C₄⋊D₄)⋊₃₂C₂, (C₂×C₄).₄₁₅(C₂×D₄), (C₂×C₂²⋊Q₈)⋊₃₄C₂, (C₂×C₄.₄D₄)⋊₂₆C₂, (C₂×C₄⋊C₄).₃₉₄C₂², C₂².₄₄₁(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₂₄₇C₂², SmallGroup(128,1408)

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

Derived series

C₁ — C₂³ — C₂³.₅₇₆C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂×C₄⋊D₄ — C₂³.₅₇₆C₂⁴

Lower central

C₁ — C₂³ — C₂³.₅₇₆C₂⁴

Upper central

C₁ — C₂³ — C₂³.₅₇₆C₂⁴

Jennings

C₁ — C₂³ — C₂³.₅₇₆C₂⁴

Generators and relations for C₂³.₅₇₆C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=e²=f²=1, d²=g²=a, ab=ba, ac=ca, ede=ad=da, geg^-1=ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, gdg^-1=abd, fg=gf >

Subgroups: 756 in 338 conjugacy classes, 104 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄⋊D₄, C₂²⋊Q₈, C₄.₄D₄, C₂³×C₄, C₂²×D₄, C₂²×Q₈, C₂³.₈Q₈, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂⁴.₃C₂², C₂³⋊₂D₄, C₂³⋊Q₈, C₂³.₁₀D₄, C₂³.Q₈, C₂×C₄⋊D₄, C₂×C₂²⋊Q₈, C₂×C₄.₄D₄, C₂³.₅₇₆C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₂₉C₂⁴, C₂².₃₁C₂⁴, C₂².₃₂C₂⁴, D₄², D₄⋊₅D₄, Q₈⋊₅D₄, C₂².₅₆C₂⁴, C₂³.₅₇₆C₂⁴

Smallest permutation representation of C₂³.₅₇₆C₂⁴
►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 38)(2 64)(3 40)(4 62)(5 43)(6 16)(7 41)(8 14)(9 61)(10 37)(11 63)(12 39)(13 33)(15 35)(17 57)(18 27)(19 59)(20 25)(21 30)(22 54)(23 32)(24 56)(26 50)(28 52)(29 45)(31 47)(34 42)(36 44)(46 53)(48 55)(49 60)(51 58)

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

C₂³.₅₇₆C₂⁴

C₂³.₈Q₈

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.₃C₂²

C₂³⋊₂D₄

C₂³⋊Q₈

C₂³.₁₀D₄

C₂³.Q₈

C₂×C₄⋊D₄

C₂×C₂²⋊Q₈

C₂×C₄.₄D₄

C₂²⋊C₄

C₄⋊C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₅₇₆C₂⁴ ►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	1	0	0	0
0	0	0	1	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

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

1	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

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

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

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

C₂³.₅₇₆C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{576}C_2^4

% in TeX

G:=Group("C2^3.576C2^4");

// GroupNames label

G:=SmallGroup(128,1408);

// by ID

G=gap.SmallGroup(128,1408);

# by ID

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

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=f^2=1,d^2=g^2=a,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=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f=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 = C23.576C24 order 128 = 27

293rd central stem extension by C23 of C24

G = C₂³.₅₇₆C₂⁴ order 128 = 2⁷

293^rd central stem extension by C₂³ of C₂⁴