C2^3.660C2^4 - GroupNames

G = C₂³.₆₆₀C₂⁴ order 128 = 2⁷

377^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂³.₆₆₀C₂⁴, C₂⁴.₄₃₉C₂³, C₂².₄₃₃2₊¹⁺⁴, C₂³⋊Q₈⋊₅₅C₂, C₂³⋊₂D₄.₃₂C₂, C₂³.₁₉₀(C₄○D₄), C₂³.₃₄D₄⋊₅₉C₂, (C₂²×C₄).₅₈₀C₂³, (C₂³×C₄).₁₆₆C₂², C₂³.₂₃D₄⋊₁₀₅C₂, (C₂²×D₄).₂₇₃C₂², (C₂²×Q₈).₂₁₂C₂², C₂³.₈₄C₂³⋊₁₂C₂, C₂.₈₉(C₂².₃₂C₂⁴), C₂.₃₀(C₂².₅₄C₂⁴), C₂.C₄².₃₆₄C₂², C₂.₁₁₂(C₂².₄₅C₂⁴), C₂².₅₂₁(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₃₀₉C₂², SmallGroup(128,1492)

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

Derived series

C₁ — C₂³ — C₂³.₆₆₀C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂³.₂₃D₄ — C₂³.₆₆₀C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₆₀C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₆₀C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₆₀C₂⁴

Generators and relations for C₂³.₆₆₀C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=f²=1, e²=dc=cd, g²=cb=bc, faf=ab=ba, ac=ca, ad=da, eae^-1=abc, ag=ga, bd=db, fef=be=eb, bf=fb, bg=gb, geg^-1=ce=ec, cf=fc, cg=gc, de=ed, gfg^-1=df=fd, dg=gd >

Subgroups: 596 in 260 conjugacy classes, 88 normal (9 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₂²⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂.C₄², C₂×C₂²⋊C₄, C₂³×C₄, C₂²×D₄, C₂²×Q₈, C₂³.₃₄D₄, C₂³.₂₃D₄, C₂³⋊₂D₄, C₂³⋊Q₈, C₂³.₈₄C₂³, C₂³.₆₆₀C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, C₂².₃₂C₂⁴, C₂².₄₅C₂⁴, C₂².₅₄C₂⁴, C₂³.₆₆₀C₂⁴

Smallest permutation representation of C₂³.₆₆₀C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	···	2M	4A	···	4L	4M	···	4R
order	1	2	···	2	2	···	2	4	···	4	4	···	4
size	1	1	···	1	4	···	4	4	···	4	8	···	8

32 irreducible representations

dim	1	1	1	1	1	1	2	4
type	+	+	+	+	+	+		+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄○D₄	2₊¹⁺⁴
kernel	C₂³.₆₆₀C₂⁴	C₂³.₃₄D₄	C₂³.₂₃D₄	C₂³⋊₂D₄	C₂³⋊Q₈	C₂³.₈₄C₂³	C₂³	C₂²
# reps	1	3	6	1	3	2	12	4

Matrix representation of C₂³.₆₆₀C₂⁴ ►in GL₆(𝔽₅)

0	1	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

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

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

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

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

C₂³.₆₆₀C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{660}C_2^4

% in TeX

G:=Group("C2^3.660C2^4");

// GroupNames label

G:=SmallGroup(128,1492);

// by ID

G=gap.SmallGroup(128,1492);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,672,253,758,723,268,1571,346]);

// Polycyclic

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

// generators/relations

G = C23.660C24 order 128 = 27

377th central stem extension by C23 of C24

G = C₂³.₆₆₀C₂⁴ order 128 = 2⁷

377^th central stem extension by C₂³ of C₂⁴