C2^3.573C2^4 - GroupNames

G = C₂³.₅₇₃C₂⁴ order 128 = 2⁷

290^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₅₈C₂³, C₂³.₅₇₃C₂⁴, C₂².₃₄₇2₊¹⁺⁴, C₄⋊C₄⋊₈D₄, C₂.₃₇D₄², (C₂×D₄)⋊₁₄D₄, C₂³.₆₀(C₂×D₄), C₂³⋊₂D₄⋊₃₅C₂, C₂.₈₅(D₄⋊₅D₄), C₂.₂₇(Q₈⋊₆D₄), C₂³.₂₃D₄⋊₇₉C₂, C₂³.₁₀D₄⋊₇₃C₂, C₂.₃₈(C₂³⋊₃D₄), (C₂²×C₄).₈₆₂C₂³, (C₂³×C₄).₄₄₃C₂², (C₂×C₄²).₆₃₃C₂², C₂².₃₈₂(C₂²×D₄), (C₂²×D₄).₂₁₄C₂², C₂⁴.C₂²⋊₁₁₇C₂, C₂.₅₃(C₂².₂₉C₂⁴), C₂.₈(C₂².₅₄C₂⁴), C₂³.₆₃C₂³⋊₁₂₄C₂, C₂.C₄².₂₈₄C₂², C₂.₃₉(C₂².₃₄C₂⁴), (C₂×C₄⋊D₄)⋊₃₀C₂, (C₂×C₄⋊₁D₄)⋊₁₀C₂, (C₂×C₄).₄₁₃(C₂×D₄), (C₂×C₄).₁₈₈(C₄○D₄), (C₂×C₄⋊C₄).₃₉₁C₂², C₂².₄₃₉(C₂×C₄○D₄), (C₂×C₂².D₄)⋊₃₀C₂, (C₂×C₂²⋊C₄).₂₄₄C₂², SmallGroup(128,1405)

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

Derived series

C₁ — C₂³ — C₂³.₅₇₃C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂×C₂².D₄ — C₂³.₅₇₃C₂⁴

Lower central

C₁ — C₂³ — C₂³.₅₇₃C₂⁴

Upper central

C₁ — C₂³ — C₂³.₅₇₃C₂⁴

Jennings

C₁ — C₂³ — C₂³.₅₇₃C₂⁴

Generators and relations for C₂³.₅₇₃C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=g²=1, e²=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: 868 in 367 conjugacy classes, 104 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄⋊D₄, C₂².D₄, C₄⋊₁D₄, C₂³×C₄, C₂²×D₄, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³⋊₂D₄, C₂³.₁₀D₄, C₂×C₄⋊D₄, C₂×C₂².D₄, C₂×C₄⋊₁D₄, C₂³.₅₇₃C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, C₂³⋊₃D₄, C₂².₂₉C₂⁴, C₂².₃₄C₂⁴, D₄², D₄⋊₅D₄, Q₈⋊₆D₄, C₂².₅₄C₂⁴, C₂³.₅₇₃C₂⁴

Smallest permutation representation of C₂³.₅₇₃C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	2L	2M	2N	4A	···	4N	4O	4P	4Q
order	1	2	···	2	2	2	2	2	2	2	2	4	···	4	4	4	4
size	1	1	···	1	4	4	4	4	8	8	8	4	···	4	8	8	8

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

kernel

C₂³.₅₇₃C₂⁴

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³⋊₂D₄

C₂³.₁₀D₄

C₂×C₄⋊D₄

C₂×C₂².D₄

C₂×C₄⋊₁D₄

C₄⋊C₄

C₂×D₄

C₂×C₄

C₂²

# reps

Matrix representation of C₂³.₅₇₃C₂⁴ ►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

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

3	1	0	0	0	0
2	2	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	1

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

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

4	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	0	1
0	0	0	0	1	0

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

C₂³.₅₇₃C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{573}C_2^4

% in TeX

G:=Group("C2^3.573C2^4");

// GroupNames label

G:=SmallGroup(128,1405);

// by ID

G=gap.SmallGroup(128,1405);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,120,758,723,100,1571,346]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=g^2=1,e^2=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 = C23.573C24 order 128 = 27

290th central stem extension by C23 of C24

G = C₂³.₅₇₃C₂⁴ order 128 = 2⁷

290^th central stem extension by C₂³ of C₂⁴