C2^3.622C2^4 - GroupNames

G = C₂³.₆₂₂C₂⁴ order 128 = 2⁷

339^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂³.₆₂₂C₂⁴, C₂⁴.₄₁₇C₂³, C₂².₂₉₇2_-¹⁺⁴, C₂².₃₉₅2₊¹⁺⁴, C₂²⋊C₄.₂₀D₄, C₂³.₇₄(C₂×D₄), C₂.₇₃(D₄⋊₆D₄), (C₂×C₄²).₆₇₃C₂², (C₂³×C₄).₄₇₃C₂², (C₂²×C₄).₅₆₁C₂³, C₂².₄₃₁(C₂²×D₄), C₂³.₇Q₈.₆₆C₂, C₂³.₈Q₈.₅₀C₂, C₂³.₁₁D₄.₃₈C₂, C₂³.₈₁C₂³⋊₉₈C₂, C₂³.₆₃C₂³⋊₁₄₇C₂, C₂³.₆₅C₂³⋊₁₃₂C₂, C₂.₈₃(C₂².₄₅C₂⁴), C₂.C₄².₃₂₈C₂², C₂.₂₄(C₂².₅₇C₂⁴), C₂.₄₆(C₂².₃₅C₂⁴), C₂.₅₂(C₂².₃₁C₂⁴), (C₂×C₄).₄₂₈(C₂×D₄), (C₂×C₄).₄₃₇(C₄○D₄), (C₂×C₄⋊C₄).₄₃₅C₂², C₂².₄₈₄(C₂×C₄○D₄), (C₂×C₄²⋊₂C₂).₁₆C₂, (C₂×C₂²⋊C₄).₂₈₆C₂², SmallGroup(128,1454)

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

Derived series

C₁ — C₂³ — C₂³.₆₂₂C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₂³.₆₅C₂³ — C₂³.₆₂₂C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₂₂C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₂₂C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₂₂C₂⁴

Generators and relations for C₂³.₆₂₂C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=f²=cb=bc, g²=b, eae^-1=ab=ba, faf^-1=ac=ca, ad=da, gag^-1=abc, bd=db, geg^-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef^-1=de=ed, df=fd, dg=gd, fg=gf >

Subgroups: 404 in 224 conjugacy classes, 96 normal (22 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², 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×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄²⋊₂C₂, C₂³×C₄, C₂³.₇Q₈, C₂³.₈Q₈, C₂³.₆₃C₂³, C₂³.₆₅C₂³, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂×C₄²⋊₂C₂, C₂³.₆₂₂C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₁C₂⁴, C₂².₃₅C₂⁴, D₄⋊₆D₄, C₂².₄₅C₂⁴, C₂².₅₇C₂⁴, C₂³.₆₂₂C₂⁴

Smallest permutation representation of C₂³.₆₂₂C₂⁴
►On 64 points

Generators in S₆₄

(2 10)(4 12)(6 38)(8 40)(13 15)(14 48)(16 46)(17 51)(18 20)(19 49)(21 43)(23 41)(26 56)(28 54)(29 31)(30 58)(32 60)(33 35)(34 63)(36 61)(45 47)(50 52)(57 59)(62 64)

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

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

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

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

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂³.₆₂₂C₂⁴

C₂³.₇Q₈

C₂³.₈Q₈

C₂³.₆₃C₂³

C₂³.₆₅C₂³

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂×C₄²⋊₂C₂

C₂²⋊C₄

C₂×C₄

C₂²

# reps

Matrix representation of C₂³.₆₂₂C₂⁴ ►in GL₆(𝔽₅)

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	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

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

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

0	4	0	0	0	0
1	0	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	0	4

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

C₂³.₆₂₂C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{622}C_2^4

% in TeX

G:=Group("C2^3.622C2^4");

// GroupNames label

G:=SmallGroup(128,1454);

// by ID

G=gap.SmallGroup(128,1454);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,344,758,723,268,1571,346,80]);

// Polycyclic

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

// generators/relations

G = C23.622C24 order 128 = 27

339th central stem extension by C23 of C24

G = C₂³.₆₂₂C₂⁴ order 128 = 2⁷

339^th central stem extension by C₂³ of C₂⁴