C2^4.420C2^3 - GroupNames

G = C₂⁴.₄₂₀C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₄₂₀C₂³, C₂³.₆₂₈C₂⁴, C₂².₄₀₁2₊¹⁺⁴, C₂².₃₀₃2_-¹⁺⁴, (C₂×Q₈)⋊₁₂D₄, C₂.₅₄(Q₈⋊₅D₄), C₂³.₈₅(C₄○D₄), C₂.₅₂(C₂³⋊₃D₄), (C₂²×C₄).₁₉₇C₂³, (C₂×C₄²).₆₇₉C₂², (C₂³×C₄).₄₇₅C₂², C₂³.₈Q₈⋊₁₂₁C₂, C₂².₄₃₇(C₂²×D₄), C₂³.₁₁D₄⋊₁₀₀C₂, C₂³.₁₀D₄.₅₀C₂, C₂³.₂₃D₄.₅₈C₂, (C₂²×D₄).₂₅₅C₂², (C₂²×Q₈).₁₉₇C₂², C₂³.₈₃C₂³⋊₈₉C₂, C₂³.₆₇C₂³⋊₈₉C₂, C₂⁴.C₂²⋊₁₄₆C₂, C₂³.₈₁C₂³⋊₁₀₁C₂, C₂.₈₅(C₂².₄₅C₂⁴), C₂.C₄².₃₃₄C₂², C₂.₂₉(C₂².₅₇C₂⁴), C₂.₇₉(C₂².₃₃C₂⁴), (C₂×C₄).₄₂₉(C₂×D₄), (C₂×C₂²⋊Q₈)⋊₄₄C₂, (C₂×C₄⋊C₄).₄₄₁C₂², C₂².₄₉₀(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₂₉₁C₂², SmallGroup(128,1460)

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

Derived series

C₁ — C₂³ — C₂⁴.₄₂₀C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂³.₂₃D₄ — C₂⁴.₄₂₀C₂³

Lower central

C₁ — C₂³ — C₂⁴.₄₂₀C₂³

Upper central

C₁ — C₂³ — C₂⁴.₄₂₀C₂³

Jennings

C₁ — C₂³ — C₂⁴.₄₂₀C₂³

Generators and relations for C₂⁴.₄₂₀C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=e²=f²=1, d²=g²=ba=ab, ac=ca, ede=ad=da, geg^-1=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, gdg^-1=abd, fg=gf >

Subgroups: 516 in 256 conjugacy classes, 96 normal (22 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂⁴, C₂.C₄², C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂²⋊Q₈, C₂³×C₄, C₂²×D₄, C₂²×Q₈, C₂³.₈Q₈, C₂³.₂₃D₄, C₂⁴.C₂², C₂³.₆₇C₂³, C₂³.₁₀D₄, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂³.₈₃C₂³, C₂×C₂²⋊Q₈, C₂⁴.₄₂₀C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂³⋊₃D₄, C₂².₃₃C₂⁴, Q₈⋊₅D₄, C₂².₄₅C₂⁴, C₂².₅₇C₂⁴, C₂⁴.₄₂₀C₂³

Smallest permutation representation of C₂⁴.₄₂₀C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

(2 44)(4 42)(5 50)(6 17)(7 52)(8 19)(9 38)(10 35)(11 40)(12 33)(14 48)(16 46)(18 62)(20 64)(21 34)(22 39)(23 36)(24 37)(26 30)(28 32)(49 63)(51 61)(54 58)(56 60)

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	···	4N	4O	···	4T
order	1	2	···	2	2	2	2	2	4	···	4	4	···	4
size	1	1	···	1	4	4	4	4	4	···	4	8	···	8

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂⁴.₄₂₀C₂³

C₂³.₈Q₈

C₂³.₂₃D₄

C₂⁴.C₂²

C₂³.₆₇C₂³

C₂³.₁₀D₄

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂³.₈₃C₂³

C₂×C₂²⋊Q₈

C₂×Q₈

C₂³

C₂²

# reps

Matrix representation of C₂⁴.₄₂₀C₂³ ►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

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

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

1	0	0	0	0	0
0	4	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	0	1

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

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

C₂⁴.₄₂₀C₂³ in GAP, Magma, Sage, TeX

C_2^4._{420}C_2^3

% in TeX

G:=Group("C2^4.420C2^3");

// GroupNames label

G:=SmallGroup(128,1460);

// by ID

G=gap.SmallGroup(128,1460);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,758,723,1571,346,192]);

// Polycyclic

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

// generators/relations

G = C24.420C23 order 128 = 27

260th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₄₂₀C₂³ order 128 = 2⁷

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