C2^3.478C2^4 - GroupNames

G = C₂³.₄₇₈C₂⁴ order 128 = 2⁷

195^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂³.₄₇₈C₂⁴, C₂⁴.₃₄₂C₂³, C₂².₂₆₁2₊¹⁺⁴, C₄²⋊₅C₄⋊₂₁C₂, C₂³.₅₇(C₄○D₄), (C₂×C₄²).₇₂C₂², C₂³.₈Q₈⋊₇₁C₂, C₂³.₇Q₈⋊₇₂C₂, C₂³.₁₁D₄⋊₄₈C₂, (C₂²×C₄).₁₀₉C₂³, (C₂³×C₄).₁₂₃C₂², C₂⁴.C₂²⋊₉₁C₂, C₂³.₂₃D₄.₃₉C₂, C₂³.₁₀D₄.₂₅C₂, (C₂²×D₄).₁₇₇C₂², C₂³.₈₃C₂³⋊₄₆C₂, C₂³.₆₃C₂³⋊₉₅C₂, C₂.₂₉(C₂².₃₂C₂⁴), C₂.₅₉(C₂².₄₅C₂⁴), C₂.C₄².₄₉₂C₂², C₂.₆₄(C₂².₄₇C₂⁴), C₂.₈₉(C₂³.₃₆C₂³), (C₄×C₂²⋊C₄)⋊₁₇C₂, (C₂×C₄).₃₉₆(C₄○D₄), (C₂×C₄⋊C₄).₃₂₅C₂², C₂².₃₅₄(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₁₉₃C₂², SmallGroup(128,1310)

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

Derived series

C₁ — C₂³ — C₂³.₄₇₈C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₄×C₂²⋊C₄ — C₂³.₄₇₈C₂⁴

Lower central

C₁ — C₂³ — C₂³.₄₇₈C₂⁴

Upper central

C₁ — C₂³ — C₂³.₄₇₈C₂⁴

Jennings

C₁ — C₂³ — C₂³.₄₇₈C₂⁴

Generators and relations for C₂³.₄₇₈C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=f²=1, d²=ca=ac, e²=cb=bc, g²=ba=ab, ede^-1=gdg^-1=ad=da, ae=ea, af=fa, ag=ga, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 468 in 234 conjugacy classes, 92 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, C₄×C₂²⋊C₄, C₂³.₇Q₈, C₄²⋊₅C₄, C₂³.₈Q₈, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₁₀D₄, C₂³.₁₁D₄, C₂³.₈₃C₂³, C₂³.₄₇₈C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, C₂³.₃₆C₂³, C₂².₃₂C₂⁴, C₂².₄₅C₂⁴, C₂².₄₇C₂⁴, C₂³.₄₇₈C₂⁴

Smallest permutation representation of C₂³.₄₇₈C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	···	4H	4I	···	4V	4W	4X	4Y	4Z
order	1	2	···	2	2	2	2	2	4	···	4	4	···	4	4	4	4	4
size	1	1	···	1	4	4	4	4	2	···	2	4	···	4	8	8	8	8

38 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊¹⁺⁴

kernel

C₂³.₄₇₈C₂⁴

C₄×C₂²⋊C₄

C₂³.₇Q₈

C₄²⋊₅C₄

C₂³.₈Q₈

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₁₀D₄

C₂³.₁₁D₄

C₂³.₈₃C₂³

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₄₇₈C₂⁴ ►in GL₆(𝔽₅)

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

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	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	4	0
0	0	0	0	0	4

3	0	0	0	0	0
0	2	0	0	0	0
0	0	2	0	0	0
0	0	0	2	0	0
0	0	0	0	2	0
0	0	0	0	0	3

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

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

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

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

C₂³.₄₇₈C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{478}C_2^4

% in TeX

G:=Group("C2^3.478C2^4");

// GroupNames label

G:=SmallGroup(128,1310);

// by ID

G=gap.SmallGroup(128,1310);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,680,758,723,675,192]);

// Polycyclic

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

// generators/relations

G = C23.478C24 order 128 = 27

195th central stem extension by C23 of C24

G = C₂³.₄₇₈C₂⁴ order 128 = 2⁷

195^th central stem extension by C₂³ of C₂⁴