C2^4.282C2^3 - GroupNames

G = C₂⁴.₂₈₂C₂³ order 128 = 2⁷

122^nd non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²

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

Aliases: C₂⁴.₂₈₂C₂³, C₂³.₃₆₁C₂⁴, C₂².₁₂₃2_-¹⁺⁴, C₂².₁₆₈2₊¹⁺⁴, C₂.₂₃D₄², C₄⋊C₄⋊₂₃D₄, C₂²⋊C₄⋊₄₁D₄, C₂³⋊₂D₄.₆C₂, C₂³.₁₇₂(C₂×D₄), C₂.₂₉(D₄⋊₆D₄), C₂.₂₇(Q₈⋊₅D₄), C₂³.₃₁(C₄○D₄), (C₂³×C₄).₈₇C₂², C₂³.Q₈⋊₁₆C₂, C₂³.₈Q₈⋊₄₉C₂, C₂³.₂₃D₄⋊₄₄C₂, C₂³.₁₀D₄⋊₂₉C₂, (C₂²×C₄).₈₁₄C₂³, (C₂×C₄²).₅₀₄C₂², C₂².₂₄₁(C₂²×D₄), C₂⁴.₃C₂²⋊₄₁C₂, C₂⁴.C₂²⋊₄₆C₂, (C₂²×D₄).₁₃₅C₂², (C₂²×Q₈).₁₀₉C₂², C₂³.₇₈C₂³⋊₁₀C₂, C₂.₃₃(C₂².₁₉C₂⁴), C₂.₁₇(C₂².₄₅C₂⁴), C₂.C₄².₁₁₈C₂², C₂.₁₉(C₂².₂₆C₂⁴), C₂.₂₁(C₂².₃₆C₂⁴), (C₂×C₄×D₄)⋊₃₇C₂, (C₄×C₄⋊C₄)⋊₆₀C₂, (C₂×C₄).₅₅(C₂×D₄), (C₂×C₂²⋊Q₈)⋊₁₄C₂, (C₂×C₄.₄D₄)⋊₁₁C₂, (C₂×C₄).₃₆₅(C₄○D₄), (C₂×C₄⋊C₄).₂₄₂C₂², C₂².₂₃₈(C₂×C₄○D₄), (C₂×C₂².D₄)⋊₁₅C₂, (C₂×C₂²⋊C₄).₁₃₇C₂², SmallGroup(128,1193)

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

Derived series

C₁ — C₂³ — C₂⁴.₂₈₂C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₄×C₄⋊C₄ — C₂⁴.₂₈₂C₂³

Lower central

C₁ — C₂³ — C₂⁴.₂₈₂C₂³

Upper central

C₁ — C₂³ — C₂⁴.₂₈₂C₂³

Jennings

C₁ — C₂³ — C₂⁴.₂₈₂C₂³

Generators and relations for C₂⁴.₂₈₂C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=f²=a, g²=b, ab=ba, ac=ca, ede^-1=gdg^-1=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: 644 in 323 conjugacy classes, 108 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄×D₄, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₂³×C₄, C₂²×D₄, C₂²×Q₈, C₄×C₄⋊C₄, C₂³.₈Q₈, C₂³.₂₃D₄, C₂⁴.C₂², C₂⁴.₃C₂², C₂³⋊₂D₄, C₂³.₁₀D₄, C₂³.₇₈C₂³, C₂³.Q₈, C₂×C₄×D₄, C₂×C₂²⋊Q₈, C₂×C₂².D₄, C₂×C₄.₄D₄, C₂⁴.₂₈₂C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₁₉C₂⁴, C₂².₂₆C₂⁴, C₂².₃₆C₂⁴, D₄², D₄⋊₆D₄, Q₈⋊₅D₄, C₂².₄₅C₂⁴, C₂⁴.₂₈₂C₂³

Smallest permutation representation of C₂⁴.₂₈₂C₂³
►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 41)(2 42)(3 43)(4 44)(5 20)(6 17)(7 18)(8 19)(9 57)(10 58)(11 59)(12 60)(13 25)(14 26)(15 27)(16 28)(21 34)(22 35)(23 36)(24 33)(29 45)(30 46)(31 47)(32 48)(37 56)(38 53)(39 54)(40 55)(49 61)(50 62)(51 63)(52 64)

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

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

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	2L	4A	···	4H	4I	···	4V	4W	4X	4Y
order	1	2	···	2	2	2	2	2	2	4	···	4	4	···	4	4	4	4
size	1	1	···	1	4	4	4	4	8	2	···	2	4	···	4	8	8	8

38 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂⁴.₂₈₂C₂³

C₄×C₄⋊C₄

C₂³.₈Q₈

C₂³.₂₃D₄

C₂⁴.C₂²

C₂⁴.₃C₂²

C₂³⋊₂D₄

C₂³.₁₀D₄

C₂³.₇₈C₂³

C₂³.Q₈

C₂×C₄×D₄

C₂×C₂²⋊Q₈

C₂×C₂².D₄

C₂×C₄.₄D₄

C₂²⋊C₄

C₄⋊C₄

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂⁴.₂₈₂C₂³ ►in 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	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	4	0
0	0	0	0	0	4

0	3	0	0	0	0
2	0	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	4

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

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

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

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

C₂⁴.₂₈₂C₂³ in GAP, Magma, Sage, TeX

C_2^4._{282}C_2^3

% in TeX

G:=Group("C2^4.282C2^3");

// GroupNames label

G:=SmallGroup(128,1193);

// by ID

G=gap.SmallGroup(128,1193);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,120,758,723,100,675,136]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=f^2=a,g^2=b,a*b=b*a,a*c=c*a,e*d*e^-1=g*d*g^-1=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 = C24.282C23 order 128 = 27

122nd non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₂₈₂C₂³ order 128 = 2⁷

122^nd non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²