C2^4.278C2^3 - GroupNames

G = C₂⁴.₂₇₈C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₂₇₈C₂³, C₂³.₃₅₇C₂⁴, C₂².₁₆₄2₊¹⁺⁴, C₂²⋊C₄⋊₃₉D₄, C₂³⋊₂D₄.₄C₂, C₂³.₁₇₀(C₂×D₄), C₂.₄₃(D₄⋊₅D₄), C₂³.₂₈(C₄○D₄), (C₂³×C₄).₈₃C₂², (C₂²×C₄).₆₂C₂³, C₂³.Q₈⋊₁₄C₂, C₂³.₇Q₈⋊₄₇C₂, C₂³.₃₄D₄⋊₂₅C₂, C₂³.₁₀D₄⋊₂₇C₂, C₂³.₂₃D₄⋊₄₂C₂, (C₂×C₄²).₅₀₀C₂², C₂².₂₃₇(C₂²×D₄), C₂⁴.C₂²⋊₄₂C₂, C₂³.₈₃C₂³⋊₉C₂, (C₂²×D₄).₅₁₆C₂², C₂³.₆₃C₂³⋊₃₉C₂, C₂.₂₉(C₂².₁₉C₂⁴), C₂.₁₆(C₂².₄₅C₂⁴), C₂.C₄².₁₁₄C₂², C₂.₃₀(C₂³.₃₆C₂³), C₂.₂₁(C₂².₄₇C₂⁴), C₂.₁₂(C₂².₃₄C₂⁴), (C₂×C₄×D₄)⋊₃₅C₂, (C₄×C₂²⋊C₄)⋊₆₀C₂, (C₂×C₄).₈₉₅(C₂×D₄), (C₂×C₄).₇₃₁(C₄○D₄), (C₂×C₄⋊C₄).₈₄₉C₂², C₂².₂₃₄(C₂×C₄○D₄), (C₂×C₂².D₄)⋊₁₄C₂, (C₂×C₂²⋊C₄).₁₃₄C₂², SmallGroup(128,1189)

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

Derived series

C₁ — C₂³ — C₂⁴.₂₇₈C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₄×C₂²⋊C₄ — C₂⁴.₂₇₈C₂³

Lower central

C₁ — C₂³ — C₂⁴.₂₇₈C₂³

Upper central

C₁ — C₂³ — C₂⁴.₂₇₈C₂³

Jennings

C₁ — C₂³ — C₂⁴.₂₇₈C₂³

Generators and relations for C₂⁴.₂₇₈C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=f²=1, e²=a, g²=b, ab=ba, ac=ca, ede^-1=gdg^-1=ad=da, 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, eg=ge, fg=gf >

Subgroups: 612 in 298 conjugacy classes, 100 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄×D₄, C₂².D₄, C₂³×C₄, C₂²×D₄, C₄×C₂²⋊C₄, C₂³.₇Q₈, C₂³.₃₄D₄, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³⋊₂D₄, C₂³.₁₀D₄, C₂³.Q₈, C₂³.₈₃C₂³, C₂×C₄×D₄, C₂×C₂².D₄, C₂⁴.₂₇₈C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, C₂².₁₉C₂⁴, C₂³.₃₆C₂³, C₂².₃₄C₂⁴, D₄⋊₅D₄, C₂².₄₅C₂⁴, C₂².₄₇C₂⁴, C₂⁴.₂₇₈C₂³

Smallest permutation representation of C₂⁴.₂₇₈C₂³
►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 25)(2 26)(3 27)(4 28)(5 58)(6 59)(7 60)(8 57)(9 49)(10 50)(11 51)(12 52)(13 37)(14 38)(15 39)(16 40)(17 48)(18 45)(19 46)(20 47)(21 42)(22 43)(23 44)(24 41)(29 33)(30 34)(31 35)(32 36)(53 61)(54 62)(55 63)(56 64)

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

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

(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)

(2 12)(4 10)(5 62)(6 59)(7 64)(8 57)(14 36)(16 34)(17 23)(18 45)(19 21)(20 47)(22 43)(24 41)(26 52)(28 50)(30 40)(32 38)(42 46)(44 48)(53 61)(54 58)(55 63)(56 60)

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	···	2M	4A	···	4H	4I	···	4T	4U	4V	4W	4X
order	1	2	···	2	2	···	2	4	···	4	4	···	4	4	4	4	4
size	1	1	···	1	4	···	4	2	···	2	4	···	4	8	8	8	8

38 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

kernel

C₂⁴.₂₇₈C₂³

C₄×C₂²⋊C₄

C₂³.₇Q₈

C₂³.₃₄D₄

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³⋊₂D₄

C₂³.₁₀D₄

C₂³.Q₈

C₂³.₈₃C₂³

C₂×C₄×D₄

C₂×C₂².D₄

C₂²⋊C₄

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂⁴.₂₇₈C₂³ ►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

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	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

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

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

4	2	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	2	0
0	0	0	0	0	2

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

C₂⁴.₂₇₈C₂³ in GAP, Magma, Sage, TeX

C_2^4._{278}C_2^3

% in TeX

G:=Group("C2^4.278C2^3");

// GroupNames label

G:=SmallGroup(128,1189);

// by ID

G=gap.SmallGroup(128,1189);

# by ID

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

// Polycyclic

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

// generators/relations

G = C24.278C23 order 128 = 27

118th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₂₇₈C₂³ order 128 = 2⁷

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