C2^3.681C2^4

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

Aliases: C₂⁴.₈₉C₂³, C₂³.₆₈₁C₂⁴, C₂².₄₅₄2₊⁽¹⁺⁴⁾, C₂².₃₄₅2_-⁽¹⁺⁴⁾, C₄²⋊₈C₄⋊₆₈C₂, C₂³.₉₈(C₄○D₄), C₂³.Q₈⋊₈₅C₂, (C₂³×C₄).₄₉₃C₂², (C₂²×C₄).₅₉₅C₂³, (C₂×C₄²).₁₀₇C₂², C₂³.₈Q₈⋊₁₃₅C₂, C₂³.₇Q₈⋊₁₁₀C₂, C₂³.₁₁D₄⋊₁₁₈C₂, C₂³.₁₀D₄.₆₂C₂, C₂³.₂₃D₄.₇₂C₂, (C₂²×D₄).₂₇₈C₂², C₂⁴.C₂²⋊₁₆₇C₂, C₂³.₈₁C₂³⋊₁₂₄C₂, C₂³.₈₃C₂³⋊₁₁₇C₂, C₂³.₆₃C₂³⋊₁₈₂C₂, C₂.₁₀₁(C₂².₃₂C₂⁴), C₂.C₄².₃₈₅C₂², C₂.₄₃(C₂².₄₉C₂⁴), C₂.₃₉(C₂².₅₆C₂⁴), C₂.₁₁₇(C₂².₄₆C₂⁴), C₂.₁₁₀(C₂².₄₇C₂⁴), C₂.₁₀₁(C₂².₃₃C₂⁴), (C₂×C₄).₄₆₈(C₄○D₄), (C₂×C₄⋊C₄).₄₉₁C₂², C₂².₅₄₂(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₃₁₇C₂², SmallGroup(128,1513)

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

Derived series

C₁ — C₂³ — C₂³.₆₈₁C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₂⁴.C₂² — C₂³.₆₈₁C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₈₁C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₈₁C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₈₁C₂⁴

Subgroups: 452 in 217 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], C₂³, C₂³^[×2], C₂³^[×13], C₄²^[×3], C₂²⋊C₄^[×16], C₄⋊C₄^[×9], C₂²×C₄^[×13], C₂²×C₄^[×4], C₂×D₄^[×4], C₂⁴^[×2], C₂.C₄²^[×12], C₂×C₄²^[×2], C₂×C₂²⋊C₄^[×11], C₂×C₄⋊C₄^[×8], C₂³×C₄, C₂²×D₄, C₂³.₇Q₈, C₄²⋊₈C₄, C₂³.₈Q₈, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂⁴.C₂²^[×4], C₂³.₁₀D₄^[×2], C₂³.Q₈, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂³.₈₃C₂³, C₂³.₆₈₁C₂⁴

Quotients:
C₁, C₂^[×15], C₂²^[×35], C₂³^[×15], C₄○D₄^[×6], C₂⁴, C₂×C₄○D₄^[×3], 2₊⁽¹⁺⁴⁾^[×3], 2_-⁽¹⁺⁴⁾, C₂².₃₂C₂⁴^[×2], C₂².₃₃C₂⁴, C₂².₄₆C₂⁴, C₂².₄₇C₂⁴, C₂².₄₉C₂⁴, C₂².₅₆C₂⁴, C₂³.₆₈₁C₂⁴

Generators and relations
G = < a,b,c,d,e,f,g | a²=b²=c²=e²=1, d²=cb=bc, f²=a, g²=ba=ab, ac=ca, ede=ad=da, geg^-1=ae=ea, af=fa, ag=ga, fdf^-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef^-1=ce=ec, cf=fc, cg=gc, gdg^-1=abd, fg=gf >

Smallest permutation representation
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

Matrix representation ►G ⊆ GL₆(𝔽₅)

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

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

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

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

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

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

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

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

C₂³.₆₈₁C₂⁴

C₂³.₇Q₈

C₄²⋊₈C₄

C₂³.₈Q₈

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₁₀D₄

C₂³.Q₈

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂³.₈₃C₂³

C₂×C₄

C₂³

C₂²

# reps

In GAP, Magma, Sage, TeX

C_2^3._{681}C_2^4

% in TeX

G:=Group("C2^3.681C2^4");

// GroupNames label

G:=SmallGroup(128,1513);

// by ID

G=gap.SmallGroup(128,1513);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,672,253,758,723,100,1571,346,192]);

// Polycyclic

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

// generators/relations

G = C₂³.₆₈₁C₂⁴ order 128 = 2⁷

398^th central stem extension by C₂³ of C₂⁴

G = C23.681C24 order 128 = 27

398th central stem extension by C23 of C24

G = C₂³.₆₈₁C₂⁴ order 128 = 2⁷

398^th central stem extension by C₂³ of C₂⁴