C2^3.396C2^4 - GroupNames

G = C₂³.₃₉₆C₂⁴ order 128 = 2⁷

113^rd central stem extension by C₂³ of C₂⁴

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

Aliases: C₂³.₃₉₆C₂⁴, C₂⁴.₃₀₆C₂³, C₂².₁₄₈2_-¹⁺⁴, C₂²⋊C₄.₇₃D₄, C₂³.₄₃(C₂×D₄), C₄²⋊₉C₄⋊₁₉C₂, C₄⋊₂(C₄²⋊₂C₂), C₂.₄₃(D₄⋊₆D₄), (C₂×C₄²).₄₅C₂², (C₂²×C₄).₇₆C₂³, (C₂³×C₄).₃₈₀C₂², C₂³.₁₁D₄.₉C₂, C₂².₂₇₃(C₂²×D₄), C₂³.₇Q₈.₄₇C₂, C₂³.₆₅C₂³⋊₇₂C₂, C₂³.₈₃C₂³⋊₂₃C₂, C₂³.₆₃C₂³⋊₆₃C₂, C₂.C₄².₁₄₈C₂², C₂.₂₂(C₂².₂₆C₂⁴), C₂.₁₇(C₂².₃₅C₂⁴), C₂.₄₀(C₂².₄₆C₂⁴), C₂.₂₃(C₂².₅₀C₂⁴), (C₄×C₄⋊C₄)⋊₇₃C₂, (C₂×C₄).₆₄(C₂×D₄), (C₄×C₂²⋊C₄).₄₉C₂, (C₂×C₄).₁₂₄(C₄○D₄), (C₂×C₄⋊C₄).₂₆₅C₂², (C₂×C₄²⋊₂C₂).₇C₂, C₂.₁₅(C₂×C₄²⋊₂C₂), C₂².₂₇₃(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₄₉₈C₂², SmallGroup(128,1228)

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

Derived series

C₁ — C₂³ — C₂³.₃₉₆C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₄×C₂²⋊C₄ — C₂³.₃₉₆C₂⁴

Lower central

C₁ — C₂³ — C₂³.₃₉₆C₂⁴

Upper central

C₁ — C₂³ — C₂³.₃₉₆C₂⁴

Jennings

C₁ — C₂³ — C₂³.₃₉₆C₂⁴

Generators and relations for C₂³.₃₉₆C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=dc=cd, f²=cb=bc, g²=b, ab=ba, eae^-1=ac=ca, faf^-1=ad=da, ag=ga, bd=db, fef^-1=geg^-1=be=eb, bf=fb, bg=gb, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >

Subgroups: 404 in 234 conjugacy classes, 104 normal (42 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂.C₄², C₂.C₄², C₂×C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄²⋊₂C₂, C₂³×C₄, C₄×C₂²⋊C₄, C₄×C₄⋊C₄, C₂³.₇Q₈, C₄²⋊₉C₄, C₂³.₆₃C₂³, C₂³.₆₅C₂³, C₂³.₁₁D₄, C₂³.₈₃C₂³, C₂×C₄²⋊₂C₂, C₂³.₃₉₆C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₄²⋊₂C₂, C₂²×D₄, C₂×C₄○D₄, 2_-¹⁺⁴, C₂×C₄²⋊₂C₂, C₂².₂₆C₂⁴, C₂².₃₅C₂⁴, D₄⋊₆D₄, C₂².₄₆C₂⁴, C₂².₅₀C₂⁴, C₂³.₃₉₆C₂⁴

Smallest permutation representation of C₂³.₃₉₆C₂⁴
►On 64 points

Generators in S₆₄

(2 6)(4 8)(10 56)(12 54)(13 57)(14 16)(15 59)(17 61)(18 20)(19 63)(22 40)(24 38)(26 44)(28 42)(29 45)(30 32)(31 47)(33 49)(34 36)(35 51)(46 48)(50 52)(58 60)(62 64)

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

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

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

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	4A	···	4H	4I	···	4X	4Y	4Z	4AA	4AB
order	1	2	···	2	2	2	4	···	4	4	···	4	4	4	4	4
size	1	1	···	1	4	4	2	···	2	4	···	4	8	8	8	8

38 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2_-¹⁺⁴

kernel

C₂³.₃₉₆C₂⁴

C₄×C₂²⋊C₄

C₄×C₄⋊C₄

C₂³.₇Q₈

C₄²⋊₉C₄

C₂³.₆₃C₂³

C₂³.₆₅C₂³

C₂³.₁₁D₄

C₂³.₈₃C₂³

C₂×C₄²⋊₂C₂

C₂²⋊C₄

C₂×C₄

C₂²

# reps

Matrix representation of C₂³.₃₉₆C₂⁴ ►in GL₆(𝔽₅)

1	0	0	0	0	0
0	4	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	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

4	0	0	0	0	0
0	4	0	0	0	0
0	0	4	0	0	0
0	0	0	4	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	1	0	0	0
0	0	0	1	0	0
0	0	0	0	4	0
0	0	0	0	0	4

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

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

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

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

C₂³.₃₉₆C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{396}C_2^4

% in TeX

G:=Group("C2^3.396C2^4");

// GroupNames label

G:=SmallGroup(128,1228);

// by ID

G=gap.SmallGroup(128,1228);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,672,253,344,758,723,675,80]);

// Polycyclic

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

// generators/relations

G = C23.396C24 order 128 = 27

113rd central stem extension by C23 of C24

G = C₂³.₃₉₆C₂⁴ order 128 = 2⁷

113^rd central stem extension by C₂³ of C₂⁴