C2^3.360C2^4 - GroupNames

G = C₂³.₃₆₀C₂⁴ order 128 = 2⁷

77^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂³.₃₆₀C₂⁴, C₂⁴.₂₈₁C₂³, C₂².₁₆₇2₊¹⁺⁴, C₂².₁₂₂2_-¹⁺⁴, C₄⋊C₄.₃₂₉D₄, C₂.₄₆(D₄⋊₅D₄), C₂.₂₆(Q₈⋊₅D₄), C₂³.₃₀(C₄○D₄), (C₂³×C₄).₈₆C₂², C₂³.₇Q₈⋊₄₈C₂, C₂³.Q₈⋊₁₅C₂, C₂³.₈Q₈⋊₄₈C₂, C₂³.₁₁D₄⋊₁₅C₂, (C₂×C₄²).₅₀₃C₂², (C₂²×C₄).₈₁₃C₂³, C₂³.₁₀D₄.₅C₂, C₂².₂₄₀(C₂²×D₄), C₂⁴.C₂²⋊₄₅C₂, (C₂²×D₄).₅₁₈C₂², C₂³.₆₃C₂³⋊₄₂C₂, C₂.₃₂(C₂².₁₉C₂⁴), C₂.C₄².₁₁₇C₂², C₂.₃₃(C₂³.₃₆C₂³), C₂.₂₂(C₂².₄₆C₂⁴), C₂.₂₂(C₂².₄₇C₂⁴), C₂.₁₅(C₂².₃₃C₂⁴), (C₂×C₄×D₄).₅₂C₂, (C₄×C₂²⋊C₄)⋊₆₃C₂, (C₂×C₄).₃₃₇(C₂×D₄), (C₂×C₄².C₂)⋊₅C₂, (C₂×C₄).₁₁₃(C₄○D₄), (C₂×C₄⋊C₄).₂₄₁C₂², C₂².₂₃₇(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₄₅₅C₂², SmallGroup(128,1192)

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

Derived series

C₁ — C₂³ — C₂³.₃₆₀C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₄×C₂²⋊C₄ — C₂³.₃₆₀C₂⁴

Lower central

C₁ — C₂³ — C₂³.₃₆₀C₂⁴

Upper central

C₁ — C₂³ — C₂³.₃₆₀C₂⁴

Jennings

C₁ — C₂³ — C₂³.₃₆₀C₂⁴

Generators and relations for C₂³.₃₆₀C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=e²=1, d²=f²=a, g²=b, ab=ba, ac=ca, ede=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: 500 in 266 conjugacy classes, 100 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄×D₄, C₄².C₂, C₂³×C₄, C₂²×D₄, C₄×C₂²⋊C₄, C₂³.₇Q₈, C₂³.₈Q₈, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₁₀D₄, C₂³.Q₈, C₂³.₁₁D₄, C₂×C₄×D₄, C₂×C₄².C₂, C₂³.₃₆₀C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₁₉C₂⁴, C₂³.₃₆C₂³, C₂².₃₃C₂⁴, D₄⋊₅D₄, Q₈⋊₅D₄, C₂².₄₆C₂⁴, C₂².₄₇C₂⁴, C₂³.₃₆₀C₂⁴

Smallest permutation representation of C₂³.₃₆₀C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

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

D₄

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂³.₃₆₀C₂⁴

C₄×C₂²⋊C₄

C₂³.₇Q₈

C₂³.₈Q₈

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₁₀D₄

C₂³.Q₈

C₂³.₁₁D₄

C₂×C₄×D₄

C₂×C₄².C₂

C₄⋊C₄

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₃₆₀C₂⁴ ►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

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

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

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

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

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],[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],[0,1,0,0,0,0,4,0,0,0,0,0,0,0,1,2,0,0,0,0,0,4,0,0,0,0,0,0,0,3,0,0,0,0,2,0],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,2,0],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,4,0,0,0,0,3,3,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3] >;

C₂³.₃₆₀C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{360}C_2^4

% in TeX

G:=Group("C2^3.360C2^4");

// GroupNames label

G:=SmallGroup(128,1192);

// by ID

G=gap.SmallGroup(128,1192);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,758,723,100,675,136]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=1,d^2=f^2=a,g^2=b,a*b=b*a,a*c=c*a,e*d*e=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 = C23.360C24 order 128 = 27

77th central stem extension by C23 of C24

G = C₂³.₃₆₀C₂⁴ order 128 = 2⁷

77^th central stem extension by C₂³ of C₂⁴