C2^4.434C2^3 - GroupNames

G = C₂⁴.₄₃₄C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₄₃₄C₂³, C₂³.₆₄₈C₂⁴, C₂².₄₂₁2₊¹⁺⁴, (C₂×D₄).₁₈Q₈, C₂³.₃₉(C₂×Q₈), C₂.₅₅(D₄⋊₃Q₈), C₂³.Q₈⋊₇₈C₂, C₂³.₄Q₈⋊₅₉C₂, (C₂×C₄²).₆₈₉C₂², (C₂³×C₄).₁₆₀C₂², (C₂²×C₄).₂₀₆C₂³, C₂³.₇Q₈⋊₁₀₄C₂, C₂³.₈Q₈⋊₁₂₅C₂, C₂.₁₆(C₂³⋊₂Q₈), C₂².₁₅₃(C₂²×Q₈), C₂³.₂₃D₄.₆₅C₂, (C₂²×D₄).₂₆₅C₂², C₂⁴.₃C₂².₆₉C₂, C₂³.₆₃C₂³⋊₁₆₄C₂, C₂³.₈₁C₂³⋊₁₁₁C₂, C₂.₂₅(C₂².₅₄C₂⁴), C₂.C₄².₃₅₂C₂², C₂.₁₀₀(C₂².₄₅C₂⁴), C₂.₅₂(C₂².₃₄C₂⁴), (C₂×C₄).₇₇(C₂×Q₈), (C₂×C₄).₄₄₉(C₄○D₄), (C₂×C₄⋊C₄).₄₅₉C₂², C₂².₅₀₉(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₃₀₄C₂², SmallGroup(128,1480)

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

Derived series

C₁ — C₂³ — C₂⁴.₄₃₄C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂³.₇Q₈ — C₂⁴.₄₃₄C₂³

Lower central

C₁ — C₂³ — C₂⁴.₄₃₄C₂³

Upper central

C₁ — C₂³ — C₂⁴.₄₃₄C₂³

Jennings

C₁ — C₂³ — C₂⁴.₄₃₄C₂³

Generators and relations for C₂⁴.₄₃₄C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=f²=1, d²=abc, e²=ba=ab, g²=a, ac=ca, 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: 484 in 232 conjugacy classes, 96 normal (22 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂³×C₄, C₂²×D₄, C₂³.₇Q₈, C₂³.₈Q₈, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂⁴.₃C₂², C₂³.Q₈, C₂³.₈₁C₂³, C₂³.₄Q₈, C₂⁴.₄₃₄C₂³
Quotients: C₁, C₂, C₂², Q₈, C₂³, C₂×Q₈, C₄○D₄, C₂⁴, C₂²×Q₈, C₂×C₄○D₄, 2₊¹⁺⁴, C₂².₃₄C₂⁴, C₂³⋊₂Q₈, C₂².₄₅C₂⁴, D₄⋊₃Q₈, C₂².₅₄C₂⁴, C₂⁴.₄₃₄C₂³

Smallest permutation representation of C₂⁴.₄₃₄C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

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

C₂⁴.₄₃₄C₂³

C₂³.₇Q₈

C₂³.₈Q₈

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.₃C₂²

C₂³.Q₈

C₂³.₈₁C₂³

C₂³.₄Q₈

C₂×D₄

C₂×C₄

C₂²

# reps

Matrix representation of C₂⁴.₄₃₄C₂³ ►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

2	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	0	2	0
0	0	0	0	2	3

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

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

C₂⁴.₄₃₄C₂³ in GAP, Magma, Sage, TeX

C_2^4._{434}C_2^3

% in TeX

G:=Group("C2^4.434C2^3");

// GroupNames label

G:=SmallGroup(128,1480);

// by ID

G=gap.SmallGroup(128,1480);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,784,253,344,758,723,1571,346,80]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=a*b*c,e^2=b*a=a*b,g^2=a,a*c=c*a,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 = C24.434C23 order 128 = 27

274th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₄₃₄C₂³ order 128 = 2⁷

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