C2^3.573C2^4

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

Aliases: C₂⁴.₅₈C₂³, C₂³.₅₇₃C₂⁴, C₂².₃₄₇2₊⁽¹⁺⁴⁾, C₄⋊C₄⋊₈D₄, (C₂×D₄)⋊₁₄D₄, C₂.₃₇(D₄²), C₂³.₆₀(C₂×D₄), C₂³⋊₂D₄⋊₃₅C₂, C₂.₈₅(D₄⋊₅D₄), C₂.₂₇(Q₈⋊₆D₄), C₂³.₁₀D₄⋊₇₃C₂, C₂³.₂₃D₄⋊₇₉C₂, C₂.₃₈(C₂³⋊₃D₄), (C₂³×C₄).₄₄₃C₂², (C₂²×C₄).₈₆₂C₂³, (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

Subgroups: 868 in 367 conjugacy classes, 104 normal (82 characteristic)
C₁, C₂^[×7], C₂^[×7], C₄^[×15], C₂²^[×7], C₂²^[×41], C₂×C₄^[×10], C₂×C₄^[×33], D₄^[×40], C₂³, C₂³^[×4], C₂³^[×33], C₄²^[×2], C₂²⋊C₄^[×21], C₄⋊C₄^[×4], C₄⋊C₄^[×4], C₂²×C₄^[×10], C₂²×C₄^[×9], C₂×D₄^[×4], C₂×D₄^[×42], C₂⁴^[×5], C₂.C₄²^[×6], C₂×C₄², C₂×C₂²⋊C₄^[×13], C₂×C₄⋊C₄^[×3], C₄⋊D₄^[×8], C₂².D₄^[×4], C₄⋊₁D₄^[×4], C₂³×C₄^[×2], C₂²×D₄^[×10], C₂³.₂₃D₄^[×3], C₂³.₆₃C₂³, C₂⁴.C₂², C₂³⋊₂D₄^[×4], C₂³.₁₀D₄^[×2], C₂×C₄⋊D₄^[×2], C₂×C₂².D₄, C₂×C₄⋊₁D₄, C₂³.₅₇₃C₂⁴

Quotients:
C₁, C₂^[×15], C₂²^[×35], D₄^[×8], C₂³^[×15], C₂×D₄^[×12], C₄○D₄^[×2], C₂⁴, C₂²×D₄^[×2], C₂×C₄○D₄, 2₊⁽¹⁺⁴⁾^[×4], C₂³⋊₃D₄, C₂².₂₉C₂⁴, C₂².₃₄C₂⁴, D₄², D₄⋊₅D₄, Q₈⋊₆D₄, C₂².₅₄C₂⁴, C₂³.₅₇₃C₂⁴

Generators and relations
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 >

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

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

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

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

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

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

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

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

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] >;

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

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

290^th central stem extension by C₂³ of C₂⁴

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