C2^4.450C2^3 - GroupNames

G = C₂⁴.₄₅₀C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₄₅₀C₂³, C₂³.₆₈₄C₂⁴, C₂².₄₅₇2₊¹⁺⁴, (C₂×D₄)⋊₇Q₈, C₂³.₄₃(C₂×Q₈), C₂³⋊Q₈⋊₅₆C₂, C₂.₆₁(D₄⋊₃Q₈), C₂³.Q₈⋊₈₆C₂, (C₂³×C₄).₄₉₄C₂², (C₂×C₄²).₇₁₁C₂², (C₂²×C₄).₂₁₄C₂³, C₂³.₈Q₈⋊₁₃₇C₂, C₂³.₇Q₈⋊₁₁₁C₂, C₂.₁₈(C₂³⋊₂Q₈), C₂².₁₅₉(C₂²×Q₈), C₂³.₂₃D₄.₇₄C₂, (C₂²×D₄).₂₈₀C₂², (C₂²×Q₈).₂₁₈C₂², C₂³.₇₈C₂³⋊₅₉C₂, C₂⁴.₃C₂².₇₄C₂, C₂³.₈₃C₂³⋊₁₂₀C₂, C₂³.₆₃C₂³⋊₁₈₄C₂, C₂.₁₀₂(C₂².₃₂C₂⁴), C₂.₃₃(C₂².₅₄C₂⁴), C₂.C₄².₃₈₈C₂², C₂.₄₅(C₂².₄₉C₂⁴), (C₂×C₄).₈₃(C₂×Q₈), (C₂×C₄).₄₇₀(C₄○D₄), (C₂×C₄⋊C₄).₄₉₄C₂², C₂².₅₄₅(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₃₂₀C₂², SmallGroup(128,1516)

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

Derived series

C₁ — C₂³ — C₂⁴.₄₅₀C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂³.₈Q₈ — C₂⁴.₄₅₀C₂³

Lower central

C₁ — C₂³ — C₂⁴.₄₅₀C₂³

Upper central

C₁ — C₂³ — C₂⁴.₄₅₀C₂³

Jennings

C₁ — C₂³ — C₂⁴.₄₅₀C₂³

Generators and relations for C₂⁴.₄₅₀C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=f²=1, d²=ca=ac, e²=g²=ba=ab, ede^-1=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: 500 in 236 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×D₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂³×C₄, C₂²×D₄, C₂²×Q₈, C₂³.₇Q₈, C₂³.₈Q₈, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂⁴.₃C₂², C₂³⋊Q₈, C₂³.₇₈C₂³, C₂³.Q₈, C₂³.₈₃C₂³, C₂⁴.₄₅₀C₂³
Quotients: C₁, C₂, C₂², Q₈, C₂³, C₂×Q₈, C₄○D₄, C₂⁴, C₂²×Q₈, C₂×C₄○D₄, 2₊¹⁺⁴, C₂².₃₂C₂⁴, C₂³⋊₂Q₈, D₄⋊₃Q₈, C₂².₄₉C₂⁴, C₂².₅₄C₂⁴, C₂⁴.₄₅₀C₂³

Smallest permutation representation of C₂⁴.₄₅₀C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

Q₈

C₄○D₄

2₊¹⁺⁴

kernel

C₂⁴.₄₅₀C₂³

C₂³.₇Q₈

C₂³.₈Q₈

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.₃C₂²

C₂³⋊Q₈

C₂³.₇₈C₂³

C₂³.Q₈

C₂³.₈₃C₂³

C₂×D₄

C₂×C₄

C₂²

# reps

Matrix representation of C₂⁴.₄₅₀C₂³ ►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	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
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

0	3	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	2	0
0	0	0	0	0	2

3	0	0	0	0	0
0	2	0	0	0	0
0	0	3	0	0	0
0	0	0	3	0	0
0	0	0	0	0	2
0	0	0	0	3	0

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

0	1	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	0
0	0	0	0	0	1

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

C₂⁴.₄₅₀C₂³ in GAP, Magma, Sage, TeX

C_2^4._{450}C_2^3

% in TeX

G:=Group("C2^4.450C2^3");

// GroupNames label

G:=SmallGroup(128,1516);

// by ID

G=gap.SmallGroup(128,1516);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,344,758,723,1571,346,192]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=c*a=a*c,e^2=g^2=b*a=a*b,e*d*e^-1=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 = C24.450C23 order 128 = 27

290th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₄₅₀C₂³ order 128 = 2⁷

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