C2^3.628C2^4

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

Aliases: C₂³.₆₂₈C₂⁴, C₂⁴.₄₂₀C₂³, C₂².₄₀₁2₊⁽¹⁺⁴⁾, C₂².₃₀₃2_-⁽¹⁺⁴⁾, (C₂×Q₈)⋊₁₂D₄, C₂.₅₄(Q₈⋊₅D₄), C₂³.₈₅(C₄○D₄), C₂.₅₂(C₂³⋊₃D₄), (C₂×C₄²).₆₇₉C₂², (C₂²×C₄).₁₉₇C₂³, (C₂³×C₄).₄₇₅C₂², C₂³.₈Q₈⋊₁₂₁C₂, C₂².₄₃₇(C₂²×D₄), C₂³.₁₁D₄⋊₁₀₀C₂, C₂³.₂₃D₄.₅₈C₂, C₂³.₁₀D₄.₅₀C₂, (C₂²×D₄).₂₅₅C₂², (C₂²×Q₈).₁₉₇C₂², C₂⁴.C₂²⋊₁₄₆C₂, C₂³.₆₇C₂³⋊₈₉C₂, C₂³.₈₃C₂³⋊₈₉C₂, C₂³.₈₁C₂³⋊₁₀₁C₂, C₂.₈₅(C₂².₄₅C₂⁴), C₂.C₄².₃₃₄C₂², C₂.₂₉(C₂².₅₇C₂⁴), C₂.₇₉(C₂².₃₃C₂⁴), (C₂×C₄).₄₂₉(C₂×D₄), (C₂×C₂²⋊Q₈)⋊₄₄C₂, (C₂×C₄⋊C₄).₄₄₁C₂², C₂².₄₉₀(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₂₉₁C₂², SmallGroup(128,1460)

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

Derived series

C₁ — C₂³ — C₂³.₆₂₈C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂³.₂₃D₄ — C₂³.₆₂₈C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₂₈C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₂₈C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₂₈C₂⁴

Subgroups: 516 in 256 conjugacy classes, 96 normal (22 characteristic)
C₁, C₂^[×3], C₂^[×4], C₂^[×4], C₄^[×16], C₂²^[×3], C₂²^[×4], C₂²^[×20], C₂×C₄^[×6], C₂×C₄^[×44], D₄^[×4], Q₈^[×4], C₂³, C₂³^[×4], C₂³^[×12], C₄², C₂²⋊C₄^[×18], C₄⋊C₄^[×14], C₂²×C₄^[×3], C₂²×C₄^[×10], C₂²×C₄^[×8], C₂×D₄^[×5], C₂×Q₈^[×4], C₂×Q₈^[×2], C₂⁴^[×2], C₂.C₄²^[×2], C₂.C₄²^[×10], C₂×C₄², C₂×C₂²⋊C₄^[×10], C₂×C₄⋊C₄^[×2], C₂×C₄⋊C₄^[×6], C₂²⋊Q₈^[×8], C₂³×C₄^[×2], C₂²×D₄, C₂²×Q₈, C₂³.₈Q₈^[×2], C₂³.₂₃D₄^[×2], C₂⁴.C₂²^[×2], C₂³.₆₇C₂³, C₂³.₁₀D₄, C₂³.₁₁D₄^[×2], C₂³.₈₁C₂³, C₂³.₈₃C₂³^[×2], C₂×C₂²⋊Q₈^[×2], C₂³.₆₂₈C₂⁴

Quotients:
C₁, C₂^[×15], C₂²^[×35], D₄^[×4], C₂³^[×15], C₂×D₄^[×6], C₄○D₄^[×4], C₂⁴, C₂²×D₄, C₂×C₄○D₄^[×2], 2₊⁽¹⁺⁴⁾^[×2], 2_-⁽¹⁺⁴⁾^[×2], C₂³⋊₃D₄, C₂².₃₃C₂⁴^[×2], Q₈⋊₅D₄^[×2], C₂².₄₅C₂⁴, C₂².₅₇C₂⁴, C₂³.₆₂₈C₂⁴

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

Smallest permutation representation
►On 64 points

Generators in S₆₄

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

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

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

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

(2 12)(4 10)(5 42)(6 23)(7 44)(8 21)(14 48)(16 46)(17 63)(18 35)(19 61)(20 33)(22 39)(24 37)(26 54)(28 56)(30 58)(32 60)(34 51)(36 49)(38 41)(40 43)(50 62)(52 64)

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

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

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

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

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

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	4	0	0	0
0	0	0	4	0	0
0	0	0	0	1	0
0	0	0	0	0	1

0	1	0	0	0	0
4	0	0	0	0	0
0	0	4	0	0	0
0	0	0	4	0	0
0	0	0	0	0	1
0	0	0	0	4	0

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	0	1
0	0	0	0	1	0

1	0	0	0	0	0
0	4	0	0	0	0
0	0	1	0	0	0
0	0	0	4	0	0
0	0	0	0	1	0
0	0	0	0	0	1

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

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],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,1,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,0,0,0,0,0,0,3,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,0,3] >;

32 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	···	4N	4O	···	4T
order	1	2	···	2	2	2	2	2	4	···	4	4	···	4
size	1	1	···	1	4	4	4	4	4	···	4	8	···	8

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊⁽¹⁺⁴⁾

2_-⁽¹⁺⁴⁾

kernel

C₂³.₆₂₈C₂⁴

C₂³.₈Q₈

C₂³.₂₃D₄

C₂⁴.C₂²

C₂³.₆₇C₂³

C₂³.₁₀D₄

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂³.₈₃C₂³

C₂×C₂²⋊Q₈

C₂×Q₈

C₂³

C₂²

# reps

In GAP, Magma, Sage, TeX

C_2^3._{628}C_2^4

% in TeX

G:=Group("C2^3.628C2^4");

// GroupNames label

G:=SmallGroup(128,1460);

// by ID

G=gap.SmallGroup(128,1460);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,758,723,1571,346,192]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=f^2=1,d^2=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,b*c=c*b,f*d*f=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f=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 = C₂³.₆₂₈C₂⁴ order 128 = 2⁷

345^th central stem extension by C₂³ of C₂⁴

G = C23.628C24 order 128 = 27

345th central stem extension by C23 of C24

G = C₂³.₆₂₈C₂⁴ order 128 = 2⁷

345^th central stem extension by C₂³ of C₂⁴