C2^3.412C2^4 - GroupNames

G = C₂³.₄₁₂C₂⁴ order 128 = 2⁷

129^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₁₇C₂³, C₂³.₄₁₂C₂⁴, C₂².₁₅₇2_-¹⁺⁴, C₂².₂₀₇2₊¹⁺⁴, C₄⋊C₄⋊₂₄D₄, C₄²⋊₉C₄⋊₂₅C₂, C₄⋊₃(C₄²⋊₂C₂), C₂.₄₆(D₄⋊₆D₄), C₂.₂₁(Q₈⋊₆D₄), (C₂×C₄²).₅₂C₂², C₂³.Q₈⋊₂₃C₂, C₂³.₁₁D₄⋊₃₃C₂, C₂².₂₈₂(C₂²×D₄), (C₂²×C₄).₁₄₈₅C₂³, C₂⁴.C₂²⋊₆₉C₂, (C₂²×D₄).₁₅₃C₂², C₂³.₆₅C₂³⋊₇₅C₂, C₂⁴.₃C₂².₃₈C₂, C₂.C₄².₄₈₉C₂², C₂.₂₇(C₂².₂₆C₂⁴), C₂.₂₆(C₂².₅₀C₂⁴), C₂.₃₅(C₂².₃₆C₂⁴), C₂.₄₁(C₂².₄₇C₂⁴), (C₄×C₄⋊C₄)⋊₇₅C₂, (C₂×C₄).₆₇(C₂×D₄), (C₂×C₄²⋊₂C₂)⋊₇C₂, (C₂×C₄).₁₃₃(C₄○D₄), (C₂×C₄⋊C₄).₂₇₈C₂², C₂.₁₉(C₂×C₄²⋊₂C₂), C₂².₂₈₉(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₁₆₂C₂², SmallGroup(128,1244)

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

Derived series

C₁ — C₂³ — C₂³.₄₁₂C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₄×C₄⋊C₄ — C₂³.₄₁₂C₂⁴

Lower central

C₁ — C₂³ — C₂³.₄₁₂C₂⁴

Upper central

C₁ — C₂³ — C₂³.₄₁₂C₂⁴

Jennings

C₁ — C₂³ — C₂³.₄₁₂C₂⁴

Generators and relations for C₂³.₄₁₂C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=1, d²=abc, e²=f²=b, g²=a, ab=ba, ac=ca, ede^-1=gdg^-1=ad=da, 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, eg=ge, fg=gf >

Subgroups: 468 in 248 conjugacy classes, 104 normal (42 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂.C₄², C₂.C₄², C₂×C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄²⋊₂C₂, C₂²×D₄, C₄×C₄⋊C₄, C₄²⋊₉C₄, C₂⁴.C₂², C₂³.₆₅C₂³, C₂⁴.₃C₂², C₂³.Q₈, C₂³.₁₁D₄, C₂×C₄²⋊₂C₂, C₂³.₄₁₂C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₄²⋊₂C₂, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂×C₄²⋊₂C₂, C₂².₂₆C₂⁴, C₂².₃₆C₂⁴, D₄⋊₆D₄, Q₈⋊₆D₄, C₂².₄₇C₂⁴, C₂².₅₀C₂⁴, C₂³.₄₁₂C₂⁴

Smallest permutation representation of C₂³.₄₁₂C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂³.₄₁₂C₂⁴

C₄×C₄⋊C₄

C₄²⋊₉C₄

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂⁴.₃C₂²

C₂³.Q₈

C₂³.₁₁D₄

C₂×C₄²⋊₂C₂

C₄⋊C₄

C₂×C₄

C₂²

# reps

Matrix representation of C₂³.₄₁₂C₂⁴ ►in GL₆(𝔽₅)

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	1	0
0	0	0	0	0	1

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	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

3	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	4	4
0	0	0	0	0	1

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

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	3	0
0	0	0	0	4	2

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	4	0
0	0	0	0	0	4

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

C₂³.₄₁₂C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{412}C_2^4

% in TeX

G:=Group("C2^3.412C2^4");

// GroupNames label

G:=SmallGroup(128,1244);

// by ID

G=gap.SmallGroup(128,1244);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,784,253,232,758,723,184,675,80]);

// Polycyclic

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

// generators/relations

G = C23.412C24 order 128 = 27

129th central stem extension by C23 of C24

G = C₂³.₄₁₂C₂⁴ order 128 = 2⁷

129^th central stem extension by C₂³ of C₂⁴