C2^3.679C2^4 - GroupNames

G = C₂³.₆₇₉C₂⁴ order 128 = 2⁷

396^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₈₈C₂³, C₂³.₆₇₉C₂⁴, C₂².₃₄₄2_-¹⁺⁴, C₂².₄₅₂2₊¹⁺⁴, C₂³.₉₇(C₄○D₄), C₂³.Q₈⋊₈₃C₂, (C₂²×C₄).₅₉₄C₂³, (C₂×C₄²).₁₀₆C₂², (C₂³×C₄).₄₉₂C₂², C₂³.₇Q₈⋊₁₀₈C₂, C₂³.₈Q₈⋊₁₃₃C₂, C₂³.₁₁D₄⋊₁₁₇C₂, C₂³.₁₀D₄.₆₁C₂, C₂³.₂₃D₄.₇₀C₂, (C₂²×D₄).₂₇₆C₂², C₂⁴.C₂²⋊₁₆₆C₂, C₂.₉₉(C₂².₃₂C₂⁴), C₂³.₆₅C₂³⋊₁₅₁C₂, C₂³.₆₃C₂³⋊₁₈₀C₂, C₂³.₈₃C₂³⋊₁₁₅C₂, C₂.C₄².₃₈₃C₂², C₂.₃₈(C₂².₅₆C₂⁴), C₂.₁₁₆(C₂².₄₆C₂⁴), C₂.₁₁₅(C₂².₃₆C₂⁴), C₂.₁₀₈(C₂².₄₇C₂⁴), (C₂×C₄).₄₆₇(C₄○D₄), (C₂×C₄⋊C₄).₄₈₉C₂², C₂².₅₄₀(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₇₆C₂², SmallGroup(128,1511)

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

Derived series

C₁ — C₂³ — C₂³.₆₇₉C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂³.₈Q₈ — C₂³.₆₇₉C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₇₉C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₇₉C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₇₉C₂⁴

Generators and relations for C₂³.₆₇₉C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=f²=1, d²=cb=bc, e²=a, g²=ba=ab, ac=ca, ede^-1=ad=da, geg^-1=ae=ea, af=fa, ag=ga, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, gdg^-1=abd, fg=gf >

Subgroups: 452 in 217 conjugacy classes, 88 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂³×C₄, C₂²×D₄, C₂³.₇Q₈, C₂³.₈Q₈, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₅C₂³, C₂³.₁₀D₄, C₂³.Q₈, C₂³.₁₁D₄, C₂³.₈₃C₂³, C₂³.₆₇₉C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₂C₂⁴, C₂².₃₆C₂⁴, C₂².₄₆C₂⁴, C₂².₄₇C₂⁴, C₂².₅₆C₂⁴, C₂³.₆₇₉C₂⁴

Smallest permutation representation of C₂³.₆₇₉C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	4A	···	4P	4Q	···	4U
order	1	2	···	2	2	2	2	4	···	4	4	···	4
size	1	1	···	1	4	4	8	4	···	4	8	···	8

32 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂³.₆₇₉C₂⁴

C₂³.₇Q₈

C₂³.₈Q₈

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂³.₁₀D₄

C₂³.Q₈

C₂³.₁₁D₄

C₂³.₈₃C₂³

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₆₇₉C₂⁴ ►in GL₆(𝔽₅)

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

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

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

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

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

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

C₂³.₆₇₉C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{679}C_2^4

% in TeX

G:=Group("C2^3.679C2^4");

// GroupNames label

G:=SmallGroup(128,1511);

// by ID

G=gap.SmallGroup(128,1511);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,672,253,120,758,723,1571,346,192]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=c*b=b*c,e^2=a,g^2=b*a=a*b,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,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 = C23.679C24 order 128 = 27

396th central stem extension by C23 of C24

G = C₂³.₆₇₉C₂⁴ order 128 = 2⁷

396^th central stem extension by C₂³ of C₂⁴