C2^3.637C2^4 - GroupNames

G = C₂³.₆₃₇C₂⁴ order 128 = 2⁷

354^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₇₆C₂³, C₂³.₆₃₇C₂⁴, C₂².₄₁₀2₊¹⁺⁴, C₂².₃₀₉2_-¹⁺⁴, C₄²⋊₅C₄⋊₂₇C₂, C₂³⋊Q₈⋊₅₃C₂, C₂³.₇Q₈⋊₉₉C₂, C₂³.₁₈₁(C₄○D₄), C₂³.₃₄D₄⋊₅₁C₂, (C₂×C₄²).₆₈₆C₂², (C₂³×C₄).₄₇₉C₂², (C₂²×C₄).₂₀₁C₂³, C₂³.₁₁D₄⋊₁₀₄C₂, C₂³.₁₀D₄.₅₂C₂, C₂³.₂₃D₄.₆₀C₂, (C₂²×D₄).₂₆₀C₂², (C₂²×Q₈).₂₀₄C₂², C₂⁴.C₂²⋊₁₅₂C₂, C₂³.₇₈C₂³⋊₅₃C₂, C₂⁴.₃C₂².₆₇C₂, C₂³.₆₃C₂³⋊₁₅₃C₂, C₂.₂₂(C₂².₅₄C₂⁴), C₂.₈₉(C₂².₄₅C₂⁴), C₂.C₄².₃₄₁C₂², C₂.₃₈(C₂².₄₉C₂⁴), C₂.₈₁(C₂².₃₃C₂⁴), C₂.₈₈(C₂².₃₆C₂⁴), (C₂×C₄).₄₃₉(C₄○D₄), (C₂×C₄⋊C₄).₄₄₈C₂², C₂².₄₉₈(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₂₉₉C₂², SmallGroup(128,1469)

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

Derived series

C₁ — C₂³ — C₂³.₆₃₇C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂³.₃₄D₄ — C₂³.₆₃₇C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₃₇C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₃₇C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₃₇C₂⁴

Generators and relations for C₂³.₆₃₇C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=g²=1, d²=c, e²=b, f²=a, ab=ba, ac=ca, ede^-1=ad=da, geg=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, gdg=abd, fg=gf >

Subgroups: 468 in 221 conjugacy classes, 88 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂³×C₄, C₂²×D₄, C₂²×Q₈, C₂³.₇Q₈, C₂³.₃₄D₄, C₄²⋊₅C₄, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂⁴.₃C₂², C₂³⋊Q₈, C₂³.₁₀D₄, C₂³.₇₈C₂³, C₂³.₁₁D₄, C₂³.₆₃₇C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₃C₂⁴, C₂².₃₆C₂⁴, C₂².₄₅C₂⁴, C₂².₄₉C₂⁴, C₂².₅₄C₂⁴, C₂³.₆₃₇C₂⁴

Smallest permutation representation of C₂³.₆₃₇C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	4A	···	4P	4Q	···	4U
order	1	2	···	2	2	2	2	4	···	4	4	···	4
size	1	1	···	1	4	4	8	4	···	4	8	···	8

32 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂³.₆₃₇C₂⁴

C₂³.₇Q₈

C₂³.₃₄D₄

C₄²⋊₅C₄

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂⁴.₃C₂²

C₂³⋊Q₈

C₂³.₁₀D₄

C₂³.₇₈C₂³

C₂³.₁₁D₄

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₆₃₇C₂⁴ ►in GL₆(𝔽₅)

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

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

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

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

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	2	0
0	0	0	0	0	2

4	0	0	0	0	0
0	4	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	1	1

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

C₂³.₆₃₇C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{637}C_2^4

% in TeX

G:=Group("C2^3.637C2^4");

// GroupNames label

G:=SmallGroup(128,1469);

// by ID

G=gap.SmallGroup(128,1469);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,232,758,723,100,1571,346]);

// Polycyclic

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

// generators/relations

G = C23.637C24 order 128 = 27

354th central stem extension by C23 of C24

G = C₂³.₆₃₇C₂⁴ order 128 = 2⁷

354^th central stem extension by C₂³ of C₂⁴