C2^3.368C2^4 - GroupNames

G = C₂³.₃₆₈C₂⁴ order 128 = 2⁷

85^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂³.₃₆₈C₂⁴, C₂⁴.₂₈₈C₂³, C₂².₁₂₈2_-¹⁺⁴, C₂².₁₇₄2₊¹⁺⁴, C₂²⋊C₄.₁₃₀D₄, C₂³.₁₇₅(C₂×D₄), C₂.₅₁(D₄⋊₅D₄), C₂.₃₂(D₄⋊₆D₄), C₂³.₃₆(C₄○D₄), (C₂³×C₄).₉₂C₂², (C₂²×C₄).₆₄C₂³, C₂³.₈Q₈⋊₅₃C₂, C₂³.₇Q₈⋊₅₀C₂, C₂³.₃₄D₄⋊₂₈C₂, (C₂×C₄²).₅₁₁C₂², C₂³.₁₀D₄.₇C₂, C₂².₂₄₈(C₂²×D₄), C₂⁴.C₂²⋊₅₁C₂, (C₂²×D₄).₅₂₂C₂², C₂³.₆₃C₂³⋊₄₇C₂, C₂³.₈₁C₂³⋊₁₇C₂, C₂.₄₀(C₂².₁₉C₂⁴), C₂.C₄².₁₂₅C₂², C₂.₂₄(C₂².₄₆C₂⁴), C₂.₃₆(C₂³.₃₆C₂³), C₂.₂₃(C₂².₃₆C₂⁴), C₂.₂₆(C₂².₄₇C₂⁴), (C₄×C₄⋊C₄)⋊₆₁C₂, (C₂×C₄×D₄).₅₄C₂, (C₂×C₄).₈₉₈(C₂×D₄), (C₂×C₄²⋊₂C₂)⋊₃C₂, (C₂×C₄).₇₃₂(C₄○D₄), (C₂×C₄⋊C₄).₂₄₈C₂², C₂².₂₄₅(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₁₄₂C₂², SmallGroup(128,1200)

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

Derived series

C₁ — C₂³ — C₂³.₃₆₈C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₄×C₄⋊C₄ — C₂³.₃₆₈C₂⁴

Lower central

C₁ — C₂³ — C₂³.₃₆₈C₂⁴

Upper central

C₁ — C₂³ — C₂³.₃₆₈C₂⁴

Jennings

C₁ — C₂³ — C₂³.₃₆₈C₂⁴

Generators and relations for C₂³.₃₆₈C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=e²=1, f²=b, g²=cb=bc, eae=gag^-1=ab=ba, ac=ca, ad=da, faf^-1=acd, bd=db, geg^-1=be=eb, bf=fb, bg=gb, cd=dc, fef^-1=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, 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, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄×D₄, C₄²⋊₂C₂, C₂³×C₄, C₂²×D₄, C₄×C₄⋊C₄, C₂³.₇Q₈, C₂³.₃₄D₄, C₂³.₈Q₈, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₁₀D₄, C₂³.₈₁C₂³, C₂×C₄×D₄, C₂×C₄²⋊₂C₂, C₂³.₃₆₈C₂⁴
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, C₂².₃₆C₂⁴, D₄⋊₅D₄, D₄⋊₆D₄, C₂².₄₆C₂⁴, C₂².₄₇C₂⁴, C₂³.₃₆₈C₂⁴

Smallest permutation representation of C₂³.₃₆₈C₂⁴
►On 64 points

Generators in S₆₄

(2 35)(4 33)(5 48)(7 46)(10 30)(12 32)(13 15)(14 52)(16 50)(17 55)(18 20)(19 53)(21 57)(23 59)(25 27)(26 37)(28 39)(38 40)(41 43)(42 64)(44 62)(49 51)(54 56)(61 63)

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

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

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

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

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

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

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

C₂³.₃₆₈C₂⁴

C₄×C₄⋊C₄

C₂³.₇Q₈

C₂³.₃₄D₄

C₂³.₈Q₈

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₁₀D₄

C₂³.₈₁C₂³

C₂×C₄×D₄

C₂×C₄²⋊₂C₂

C₂²⋊C₄

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₃₆₈C₂⁴ ►in GL₆(𝔽₅)

1	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	2	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

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	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

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

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	3	0
0	0	0	0	0	3

3	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	2
0	0	0	0	0	2

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

C₂³.₃₆₈C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{368}C_2^4

% in TeX

G:=Group("C2^3.368C2^4");

// GroupNames label

G:=SmallGroup(128,1200);

// by ID

G=gap.SmallGroup(128,1200);

# by ID

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

// Polycyclic

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

// generators/relations

G = C23.368C24 order 128 = 27

85th central stem extension by C23 of C24

G = C₂³.₃₆₈C₂⁴ order 128 = 2⁷

85^th central stem extension by C₂³ of C₂⁴