C2^3.678C2^4 - GroupNames

G = C₂³.₆₇₈C₂⁴ order 128 = 2⁷

395^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂³.₆₇₈C₂⁴, C₂⁴.₄₄₇C₂³, C₂².₄₅₁2₊¹⁺⁴, C₂³.₉₆(C₄○D₄), C₂³⋊₂D₄.₃₃C₂, (C₂²×C₄).₅₉₃C₂³, (C₂³×C₄).₁₇₂C₂², (C₂×C₄²).₁₀₅C₂², C₂³.₈Q₈⋊₁₃₂C₂, C₂³.₇Q₈⋊₁₀₇C₂, C₂³.₁₁D₄⋊₁₁₆C₂, C₂³.₂₃D₄⋊₁₀₆C₂, C₂³.₁₀D₄⋊₁₀₃C₂, (C₂²×D₄).₂₇₅C₂², C₂⁴.C₂²⋊₁₆₅C₂, C₂³.₈₄C₂³⋊₁₄C₂, C₂.₉₈(C₂².₃₂C₂⁴), C₂³.₈₃C₂³⋊₁₁₄C₂, C₂.₃₁(C₂².₅₄C₂⁴), C₂.C₄².₃₈₂C₂², C₂.₁₁₇(C₂².₄₅C₂⁴), C₂.₁₀₇(C₂².₄₇C₂⁴), (C₂×C₄).₄₆₆(C₄○D₄), (C₂×C₄⋊C₄).₄₈₈C₂², C₂².₅₃₉(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₇₅C₂², SmallGroup(128,1510)

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

Derived series

C₁ — C₂³ — C₂³.₆₇₈C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂³.₂₃D₄ — C₂³.₆₇₈C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₇₈C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₇₈C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₇₈C₂⁴

Generators and relations for C₂³.₆₇₈C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=f²=1, e²=bcd, g²=cb=bc, eae^-1=gag^-1=ab=ba, ac=ca, faf=ad=da, bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, geg^-1=bce, fg=gf >

Subgroups: 564 in 249 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, C₂⁴.C₂², C₂³⋊₂D₄, C₂³.₁₀D₄, C₂³.₁₁D₄, C₂³.₈₃C₂³, C₂³.₈₄C₂³, C₂³.₆₇₈C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, C₂².₃₂C₂⁴, C₂².₄₅C₂⁴, C₂².₄₇C₂⁴, C₂².₅₄C₂⁴, C₂³.₆₇₈C₂⁴

Smallest permutation representation of C₂³.₆₇₈C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	2L	4A	···	4N	4O	···	4S
order	1	2	···	2	2	2	2	2	2	4	···	4	4	···	4
size	1	1	···	1	4	4	4	4	8	4	···	4	8	···	8

32 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊¹⁺⁴

kernel

C₂³.₆₇₈C₂⁴

C₂³.₇Q₈

C₂³.₈Q₈

C₂³.₂₃D₄

C₂⁴.C₂²

C₂³⋊₂D₄

C₂³.₁₀D₄

C₂³.₁₁D₄

C₂³.₈₃C₂³

C₂³.₈₄C₂³

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₆₇₈C₂⁴ ►in GL₆(𝔽₅)

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	0	2
0	0	0	0	3	0

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

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

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

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

C₂³.₆₇₈C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{678}C_2^4

% in TeX

G:=Group("C2^3.678C2^4");

// GroupNames label

G:=SmallGroup(128,1510);

// by ID

G=gap.SmallGroup(128,1510);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,784,253,758,723,1571,346,192]);

// Polycyclic

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

// generators/relations

G = C23.678C24 order 128 = 27

395th central stem extension by C23 of C24

G = C₂³.₆₇₈C₂⁴ order 128 = 2⁷

395^th central stem extension by C₂³ of C₂⁴