C2^4.427C2^3 - GroupNames

G = C₂⁴.₄₂₇C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₄₂₇C₂³, C₂³.₆₃₉C₂⁴, C₂².₃₁₁2_-¹⁺⁴, C₂².₄₁₂2₊¹⁺⁴, (C₂×C₄²).₈₉C₂², C₂³.₁₈₃(C₄○D₄), (C₂²×C₄).₅₆₅C₂³, (C₂³×C₄).₄₈₁C₂², C₂³.₇Q₈.₆₈C₂, C₂³.Q₈.₃₃C₂, C₂³.₈Q₈.₅₃C₂, C₂³.₁₁D₄.₄₁C₂, C₂³.₃₄D₄.₂₉C₂, C₂³.₈₃C₂³⋊₉₂C₂, C₂⁴.C₂².₅₈C₂, C₂³.₆₅C₂³⋊₁₄₀C₂, C₂³.₈₁C₂³⋊₁₀₅C₂, C₂³.₆₃C₂³⋊₁₅₅C₂, C₂.₉₁(C₂².₄₅C₂⁴), C₂.C₄².₃₄₃C₂², C₂.₉₀(C₂².₃₆C₂⁴), C₂.₂₅(C₂².₅₆C₂⁴), C₂.₈₃(C₂².₃₃C₂⁴), C₂.₉₁(C₂².₄₇C₂⁴), C₂.₄₈(C₂².₃₅C₂⁴), C₂.₉₀(C₂².₄₆C₂⁴), (C₂×C₄).₄₄₁(C₄○D₄), (C₂×C₄⋊C₄).₄₅₀C₂², C₂².₅₀₀(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₆₀C₂², SmallGroup(128,1471)

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

Derived series

C₁ — C₂³ — C₂⁴.₄₂₇C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂³.₃₄D₄ — C₂⁴.₄₂₇C₂³

Lower central

C₁ — C₂³ — C₂⁴.₄₂₇C₂³

Upper central

C₁ — C₂³ — C₂⁴.₄₂₇C₂³

Jennings

C₁ — C₂³ — C₂⁴.₄₂₇C₂³

Generators and relations for C₂⁴.₄₂₇C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=g²=1, d²=c, e²=ba=ab, f²=a, ac=ca, ede^-1=ad=da, geg=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, gdg=abd, fg=gf >

Subgroups: 372 in 196 conjugacy classes, 88 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂³×C₄, C₂³.₇Q₈, C₂³.₃₄D₄, C₂³.₈Q₈, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₅C₂³, C₂³.Q₈, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂³.₈₃C₂³, C₂⁴.₄₂₇C₂³
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₃C₂⁴, C₂².₃₅C₂⁴, C₂².₃₆C₂⁴, C₂².₄₅C₂⁴, C₂².₄₆C₂⁴, C₂².₄₇C₂⁴, C₂².₅₆C₂⁴, C₂⁴.₄₂₇C₂³

Smallest permutation representation of C₂⁴.₄₂₇C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

(2 24)(4 22)(5 36)(6 39)(7 34)(8 37)(10 50)(12 52)(14 54)(16 56)(17 45)(18 30)(19 47)(20 32)(26 42)(28 44)(29 57)(31 59)(33 63)(35 61)(38 62)(40 64)(46 58)(48 60)

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	4A	···	4P	4Q	···	4V
order	1	2	···	2	2	2	4	···	4	4	···	4
size	1	1	···	1	4	4	4	···	4	8	···	8

32 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂⁴.₄₂₇C₂³

C₂³.₇Q₈

C₂³.₃₄D₄

C₂³.₈Q₈

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂³.Q₈

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂³.₈₃C₂³

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂⁴.₄₂₇C₂³ ►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

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

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

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

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

C₂⁴.₄₂₇C₂³ in GAP, Magma, Sage, TeX

C_2^4._{427}C_2^3

% in TeX

G:=Group("C2^4.427C2^3");

// GroupNames label

G:=SmallGroup(128,1471);

// by ID

G=gap.SmallGroup(128,1471);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,344,758,723,100,1571,346]);

// Polycyclic

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

// generators/relations

G = C24.427C23 order 128 = 27

267th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₄₂₇C₂³ order 128 = 2⁷

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