C2^3.637C2^4

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

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

Subgroups: 468 in 221 conjugacy classes, 88 normal (82 characteristic)
C₁, C₂^[×7], C₂^[×3], C₄^[×15], C₂²^[×7], C₂²^[×17], C₂×C₄^[×4], C₂×C₄^[×41], D₄^[×4], Q₈^[×4], C₂³, C₂³^[×2], C₂³^[×13], C₄²^[×3], C₂²⋊C₄^[×13], C₄⋊C₄^[×7], C₂²×C₄^[×13], C₂²×C₄^[×5], C₂×D₄^[×5], C₂×Q₈^[×3], C₂⁴^[×2], C₂.C₄²^[×14], C₂×C₄²^[×2], C₂×C₂²⋊C₄^[×11], C₂×C₄⋊C₄^[×5], C₂³×C₄, C₂²×D₄, C₂²×Q₈, C₂³.₇Q₈, C₂³.₃₄D₄, C₄²⋊₅C₄, C₂³.₂₃D₄, C₂³.₆₃C₂³^[×2], C₂⁴.C₂²^[×2], C₂⁴.₃C₂², C₂³⋊Q₈^[×2], C₂³.₁₀D₄, C₂³.₇₈C₂³, C₂³.₁₁D₄^[×2], C₂³.₆₃₇C₂⁴

Quotients:
C₁, C₂^[×15], C₂²^[×35], C₂³^[×15], C₄○D₄^[×6], C₂⁴, C₂×C₄○D₄^[×3], 2₊⁽¹⁺⁴⁾^[×3], 2_-⁽¹⁺⁴⁾, C₂².₃₃C₂⁴, C₂².₃₆C₂⁴^[×2], C₂².₄₅C₂⁴^[×2], C₂².₄₉C₂⁴, C₂².₅₄C₂⁴, C₂³.₆₃₇C₂⁴

Generators and relations
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 >

Smallest permutation representation
►On 64 points

Generators in S₆₄

(1 21)(2 22)(3 23)(4 24)(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 40)(30 37)(31 38)(32 39)(49 64)(50 61)(51 62)(52 63)

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

(1 47 21 6)(2 28 22 44)(3 45 23 8)(4 26 24 42)(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 43 57 27)(16 41 59 25)(17 37 35 30)(19 39 33 32)(29 50 40 61)(31 52 38 63)

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

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

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

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

Matrix representation ►G ⊆ 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] >;

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

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

354^th central stem extension by C₂³ of C₂⁴

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