C2^3.369C2^4 - GroupNames

G = C₂³.₃₆₉C₂⁴ order 128 = 2⁷

86^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₉C₂³, C₂³.₃₆₉C₂⁴, C₂².₁₂₉2_-¹⁺⁴, C₄⋊C₄.₃₃₂D₄, C₂.₃₃(D₄⋊₆D₄), C₂³⋊Q₈.₅C₂, C₂.₂₉(Q₈⋊₅D₄), (C₂²×C₄).₆₅C₂³, C₂³.Q₈.₅C₂, (C₂×C₄²).₅₁₂C₂², C₂².₂₄₉(C₂²×D₄), (C₂²×Q₈).₄₃₀C₂², C₂³.₈₃C₂³⋊₁₀C₂, C₂³.₆₅C₂³⋊₆₁C₂, C₂³.₇₈C₂³⋊₁₂C₂, C₂³.₆₃C₂³⋊₄₈C₂, C₂.₄₁(C₂².₁₉C₂⁴), C₂⁴.C₂².₁₅C₂, C₂.C₄².₁₂₆C₂², C₂.₁₃(C₂².₃₅C₂⁴), C₂.₃₇(C₂³.₃₆C₂³), C₂.₁₇(C₂².₅₀C₂⁴), (C₄×C₄⋊C₄)⋊₆₂C₂, (C₂×C₄×Q₈)⋊₂₀C₂, (C₂×C₄).₃₄₂(C₂×D₄), (C₂×C₄).₈₅₆(C₄○D₄), (C₂×C₄⋊C₄).₂₄₉C₂², (C₂×C₄²⋊₂C₂).₅C₂, C₂².₂₄₆(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₁₄₃C₂², SmallGroup(128,1201)

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

Derived series

C₁ — C₂³ — C₂³.₃₆₉C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₄×C₄⋊C₄ — C₂³.₃₆₉C₂⁴

Lower central

C₁ — C₂³ — C₂³.₃₆₉C₂⁴

Upper central

C₁ — C₂³ — C₂³.₃₆₉C₂⁴

Jennings

C₁ — C₂³ — C₂³.₃₆₉C₂⁴

Generators and relations for C₂³.₃₆₉C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=1, d²=e²=f²=a, g²=ba=ab, ac=ca, ede^-1=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: 404 in 235 conjugacy classes, 100 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄×Q₈, C₄²⋊₂C₂, C₂²×Q₈, C₄×C₄⋊C₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₅C₂³, C₂³⋊Q₈, C₂³.₇₈C₂³, C₂³.Q₈, C₂³.₈₃C₂³, C₂×C₄×Q₈, C₂×C₄²⋊₂C₂, C₂³.₃₆₉C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2_-¹⁺⁴, C₂².₁₉C₂⁴, C₂³.₃₆C₂³, C₂².₃₅C₂⁴, D₄⋊₆D₄, Q₈⋊₅D₄, C₂².₅₀C₂⁴, C₂³.₃₆₉C₂⁴

Smallest permutation representation of C₂³.₃₆₉C₂⁴
►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 43)(2 44)(3 41)(4 42)(5 39)(6 40)(7 37)(8 38)(9 50)(10 51)(11 52)(12 49)(13 22)(14 23)(15 24)(16 21)(17 45)(18 46)(19 47)(20 48)(25 54)(26 55)(27 56)(28 53)(29 60)(30 57)(31 58)(32 59)(33 62)(34 63)(35 64)(36 61)

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

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	4A	···	4H	4I	···	4Z	4AA	4AB	4AC
order	1	2	···	2	2	4	···	4	4	···	4	4	4	4
size	1	1	···	1	8	2	···	2	4	···	4	8	8	8

38 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2_-¹⁺⁴

kernel

C₂³.₃₆₉C₂⁴

C₄×C₄⋊C₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂³⋊Q₈

C₂³.₇₈C₂³

C₂³.Q₈

C₂³.₈₃C₂³

C₂×C₄×Q₈

C₂×C₄²⋊₂C₂

C₄⋊C₄

C₂×C₄

C₂²

# reps

Matrix representation of C₂³.₃₆₉C₂⁴ ►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	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

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

0	3	0	0	0	0
3	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

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

C₂³.₃₆₉C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{369}C_2^4

% in TeX

G:=Group("C2^3.369C2^4");

// GroupNames label

G:=SmallGroup(128,1201);

// by ID

G=gap.SmallGroup(128,1201);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,723,100,675,192]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=e^2=f^2=a,g^2=b*a=a*b,a*c=c*a,e*d*e^-1=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 = C23.369C24 order 128 = 27

86th central stem extension by C23 of C24

G = C₂³.₃₆₉C₂⁴ order 128 = 2⁷

86^th central stem extension by C₂³ of C₂⁴