C2^3.664C2^4 - GroupNames

G = C₂³.₆₆₄C₂⁴ order 128 = 2⁷

381^st central stem extension by C₂³ of C₂⁴

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

Aliases: C₂³.₆₆₄C₂⁴, C₂⁴.₄₄₂C₂³, C₂².₄₃₇2₊¹⁺⁴, C₂².₃₃₀2_-¹⁺⁴, C₄²⋊₅C₄⋊₃₃C₂, C₂³⋊Q₈.₂₇C₂, C₂³.₁₉₂(C₄○D₄), (C₂×C₄²).₆₉₆C₂², (C₂³×C₄).₁₆₉C₂², (C₂²×C₄).₅₈₃C₂³, C₂³.₈Q₈.₅₉C₂, C₂³.₁₁D₄.₄₇C₂, C₂³.₃₄D₄.₃₂C₂, (C₂²×Q₈).₂₁₅C₂², C₂.₉₁(C₂².₃₂C₂⁴), C₂⁴.C₂².₆₆C₂, C₂³.₈₁C₂³⋊₁₁₆C₂, C₂³.₈₃C₂³⋊₁₀₆C₂, C₂³.₆₇C₂³⋊₁₀₀C₂, C₂.C₄².₃₆₈C₂², C₂.₁₁₆(C₂².₄₅C₂⁴), C₂.₃₈(C₂².₅₇C₂⁴), C₂.₁₀₂(C₂².₄₆C₂⁴), C₂.₁₀₅(C₂².₃₆C₂⁴), (C₂×C₄).₄₅₈(C₄○D₄), (C₂×C₄⋊C₄).₄₇₄C₂², C₂².₅₂₅(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₃₁₁C₂², SmallGroup(128,1496)

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

Derived series

C₁ — C₂³ — C₂³.₆₆₄C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₂³.₆₇C₂³ — C₂³.₆₆₄C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₆₄C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₆₄C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₆₄C₂⁴

Generators and relations for C₂³.₆₆₄C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=b, f²=g²=cb=bc, ab=ba, gag^-1=ac=ca, eae^-1=ad=da, faf^-1=acd, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef^-1=de=ed, df=fd, dg=gd, geg^-1=bde, gfg^-1=cdf >

Subgroups: 388 in 200 conjugacy classes, 88 normal (22 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂³×C₄, C₂²×Q₈, C₂³.₃₄D₄, C₄²⋊₅C₄, C₂³.₈Q₈, C₂⁴.C₂², C₂³.₆₇C₂³, C₂³⋊Q₈, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂³.₈₃C₂³, C₂³.₆₆₄C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₂C₂⁴, C₂².₃₆C₂⁴, C₂².₄₅C₂⁴, C₂².₄₆C₂⁴, C₂².₅₇C₂⁴, C₂³.₆₆₄C₂⁴

Smallest permutation representation of C₂³.₆₆₄C₂⁴
►On 64 points

Generators in S₆₄

(2 52)(4 50)(6 32)(8 30)(9 42)(10 25)(11 44)(12 27)(14 19)(16 17)(21 39)(22 47)(23 37)(24 45)(26 59)(28 57)(33 53)(35 55)(38 64)(40 62)(41 60)(43 58)(46 61)(48 63)

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	4A	···	4P	4Q	···	4V
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

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂³.₆₆₄C₂⁴

C₂³.₃₄D₄

C₄²⋊₅C₄

C₂³.₈Q₈

C₂⁴.C₂²

C₂³.₆₇C₂³

C₂³⋊Q₈

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂³.₈₃C₂³

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₆₆₄C₂⁴ ►in GL₆(𝔽₅)

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

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

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,4,0,0,0,0,0,4],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[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],[0,3,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,1,2],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,4,0,0,0,0,2,4] >;

C₂³.₆₆₄C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{664}C_2^4

% in TeX

G:=Group("C2^3.664C2^4");

// GroupNames label

G:=SmallGroup(128,1496);

// by ID

G=gap.SmallGroup(128,1496);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,672,253,120,758,723,268,1571,346,80]);

// Polycyclic

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

// generators/relations

G = C23.664C24 order 128 = 27

381st central stem extension by C23 of C24

G = C₂³.₆₆₄C₂⁴ order 128 = 2⁷

381^st central stem extension by C₂³ of C₂⁴