C2^4.563C2^3 - GroupNames

G = C₂⁴.₅₆₃C₂³ order 128 = 2⁷

44^th non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂²=C₂

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

Aliases: C₂⁴.₅₆₃C₂³, C₂³.₃₁₉C₂⁴, C₂².₉₅2_-¹⁺⁴, C₂².₁₃₃2₊¹⁺⁴, (C₂×D₄).₂₈₁D₄, C₂³.₁₅₆(C₂×D₄), C₂.₁₄(D₄⋊₆D₄), C₂.₂₂(D₄⋊₅D₄), C₂³.₁₁D₄⋊₆C₂, (C₂²×C₄).₅₃C₂³, (C₂³×C₄).₇₂C₂², C₂³.₈Q₈⋊₃₁C₂, C₂³.₇Q₈⋊₃₉C₂, C₂³.₂₂₈(C₄○D₄), C₂³.₃₄D₄⋊₂₁C₂, (C₂×C₄²).₄₆₇C₂², C₂².₁₉₉(C₂²×D₄), C₂³.₈₃C₂³⋊₃C₂, C₂³.₈₁C₂³⋊₅C₂, C₂³.₂₃D₄.₁₇C₂, (C₂²×D₄).₅₀₅C₂², C₂³.₆₃C₂³⋊₂₈C₂, C₂.C₄².₈₂C₂², C₂.₁₃(C₂².₄₅C₂⁴), C₂².₁₅(C₂².D₄), C₂.₁₈(C₂³.₃₆C₂³), C₂.₁₀(C₂².₃₃C₂⁴), C₂.₁₅(C₂².₄₆C₂⁴), (C₂×C₄×D₄).₄₆C₂, (C₂×C₄).₃₁₀(C₂×D₄), (C₂×C₄).₇₂₈(C₄○D₄), (C₂×C₄⋊C₄).₂₀₈C₂², C₂².₁₉₈(C₂×C₄○D₄), (C₂×C₂.C₄²)⋊₂₈C₂, C₂.₁₆(C₂×C₂².D₄), (C₂×C₂²⋊C₄).₄₅₀C₂², (C₂×C₂².D₄).₁₁C₂, SmallGroup(128,1151)

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

Derived series

C₁ — C₂³ — C₂⁴.₅₆₃C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂×C₂.C₄² — C₂⁴.₅₆₃C₂³

Lower central

C₁ — C₂³ — C₂⁴.₅₆₃C₂³

Upper central

C₁ — C₂³ — C₂⁴.₅₆₃C₂³

Jennings

C₁ — C₂³ — C₂⁴.₅₆₃C₂³

Generators and relations for C₂⁴.₅₆₃C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=g²=1, e²=b, f²=a, ab=ba, ac=ca, ede^-1=gdg=ad=da, 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, eg=ge, fg=gf >

Subgroups: 532 in 280 conjugacy classes, 104 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊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, C₂³×C₄, C₂²×D₄, C₂×C₂.C₄², C₂³.₇Q₈, C₂³.₃₄D₄, C₂³.₈Q₈, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂³.₈₃C₂³, C₂×C₄×D₄, C₂×C₂².D₄, C₂⁴.₅₆₃C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂².D₄, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂×C₂².D₄, C₂³.₃₆C₂³, C₂².₃₃C₂⁴, D₄⋊₅D₄, D₄⋊₆D₄, C₂².₄₅C₂⁴, C₂².₄₆C₂⁴, C₂⁴.₅₆₃C₂³

Smallest permutation representation of C₂⁴.₅₆₃C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	2L	2M	4A	4B	4C	4D	4E	···	4T	4U	4V	4W	4X
order	1	2	···	2	2	2	2	2	2	2	4	4	4	4	4	···	4	4	4	4	4
size	1	1	···	1	2	2	2	2	4	4	2	2	2	2	4	···	4	8	8	8	8

38 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂⁴.₅₆₃C₂³

C₂×C₂.C₄²

C₂³.₇Q₈

C₂³.₃₄D₄

C₂³.₈Q₈

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂³.₈₃C₂³

C₂×C₄×D₄

C₂×C₂².D₄

C₂×D₄

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂⁴.₅₆₃C₂³ ►in GL₆(𝔽₅)

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

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

0	3	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	3	0
0	0	0	0	1	2

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

0	1	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	4	0
0	0	0	0	0	4

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

C₂⁴.₅₆₃C₂³ in GAP, Magma, Sage, TeX

C_2^4._{563}C_2^3

% in TeX

G:=Group("C2^4.563C2^3");

// GroupNames label

G:=SmallGroup(128,1151);

// by ID

G=gap.SmallGroup(128,1151);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,232,758,723,100,675]);

// Polycyclic

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

// generators/relations

G = C24.563C23 order 128 = 27

44th non-split extension by C24 of C23 acting via C23/C22=C2

G = C₂⁴.₅₆₃C₂³ order 128 = 2⁷

44^th non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂²=C₂