C2^3.344C2^4 - GroupNames

G = C₂³.₃₄₄C₂⁴ order 128 = 2⁷

61^st central stem extension by C₂³ of C₂⁴

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

Aliases: C₂³.₃₄₄C₂⁴, C₂⁴.₅₇₀C₂³, C₂².₁₅₃2₊¹⁺⁴, (C₂×D₄).₂₈₇D₄, C₂³.₁₆₅(C₂×D₄), C₂.₃₆(D₄⋊₅D₄), (C₂³×C₄).₇₆C₂², (C₂²×C₄).₆₁C₂³, C₂³.₈Q₈⋊₃₉C₂, C₂³.₂₃₂(C₄○D₄), C₂³.₁₀D₄⋊₂₄C₂, C₂³.₂₃D₄⋊₃₉C₂, C₂³.₁₁D₄⋊₁₂C₂, C₂³.₃₄D₄⋊₂₃C₂, (C₂×C₄²).₄₈₇C₂², C₂².₂₂₄(C₂²×D₄), C₂⁴.C₂²⋊₃₈C₂, C₂³.₈₃C₂³⋊₇C₂, (C₂²×D₄).₁₃₁C₂², C₂³.₆₃C₂³⋊₃₅C₂, C₂.₁₂(C₂².₃₂C₂⁴), C₂.C₄².₁₀₁C₂², C₂².₂₂(C₂².D₄), C₂.₁₉(C₂².₄₇C₂⁴), C₂.₂₄(C₂³.₃₆C₂³), (C₂×C₄×D₄)⋊₃₁C₂, (C₂×C₄).₃₂₆(C₂×D₄), (C₂×C₄⋊D₄).₂₉C₂, (C₂×C₄).₇₃₀(C₄○D₄), (C₂×C₄⋊C₄).₂₂₆C₂², C₂².₂₂₁(C₂×C₄○D₄), (C₂×C₂.C₄²)⋊₃₁C₂, C₂.₂₂(C₂×C₂².D₄), (C₂×C₂²⋊C₄).₄₅₃C₂², SmallGroup(128,1176)

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

Derived series

C₁ — C₂³ — C₂³.₃₄₄C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂×C₂.C₄² — C₂³.₃₄₄C₂⁴

Lower central

C₁ — C₂³ — C₂³.₃₄₄C₂⁴

Upper central

C₁ — C₂³ — C₂³.₃₄₄C₂⁴

Jennings

C₁ — C₂³ — C₂³.₃₄₄C₂⁴

Generators and relations for C₂³.₃₄₄C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=g²=1, e²=ba=ab, f²=b, ac=ca, ede^-1=gdg=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: 628 in 305 conjugacy classes, 104 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄×D₄, C₄⋊D₄, C₂³×C₄, C₂²×D₄, C₂×C₂.C₄², C₂³.₃₄D₄, C₂³.₈Q₈, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₁₀D₄, C₂³.₁₁D₄, C₂³.₈₃C₂³, C₂×C₄×D₄, C₂×C₄⋊D₄, C₂³.₃₄₄C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂².D₄, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, C₂×C₂².D₄, C₂³.₃₆C₂³, C₂².₃₂C₂⁴, D₄⋊₅D₄, C₂².₄₇C₂⁴, C₂³.₃₄₄C₂⁴

Smallest permutation representation of C₂³.₃₄₄C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	2L	2M	2N	4A	4B	4C	4D	4E	···	4T	4U	4V	4W
order	1	2	···	2	2	2	2	2	2	2	2	4	4	4	4	4	···	4	4	4	4
size	1	1	···	1	2	2	2	2	4	4	8	2	2	2	2	4	···	4	8	8	8

38 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

kernel

C₂³.₃₄₄C₂⁴

C₂×C₂.C₄²

C₂³.₃₄D₄

C₂³.₈Q₈

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₁₀D₄

C₂³.₁₁D₄

C₂³.₈₃C₂³

C₂×C₄×D₄

C₂×C₄⋊D₄

C₂×D₄

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₃₄₄C₂⁴ ►in GL₆(𝔽₅)

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

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

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

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

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

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

C₂³.₃₄₄C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{344}C_2^4

% in TeX

G:=Group("C2^3.344C2^4");

// GroupNames label

G:=SmallGroup(128,1176);

// by ID

G=gap.SmallGroup(128,1176);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,344,758,723,184,675]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=g^2=1,e^2=b*a=a*b,f^2=b,a*c=c*a,e*d*e^-1=g*d*g=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 = C23.344C24 order 128 = 27

61st central stem extension by C23 of C24

G = C₂³.₃₄₄C₂⁴ order 128 = 2⁷

61^st central stem extension by C₂³ of C₂⁴