C2^4.407C2^3 - GroupNames

G = C₂⁴.₄₀₇C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₄₀₇C₂³, C₂³.₆₀₁C₂⁴, C₂².₃₇₅2₊¹⁺⁴, C₂².₂₇₉2_-¹⁺⁴, (C₂×D₄).₁₄₁D₄, C₂³.₂₁₆(C₂×D₄), C₂.₆₃(D₄⋊₆D₄), C₂.₁₀₆(D₄⋊₅D₄), C₂³.Q₈⋊₆₃C₂, C₂³.₂₃D₄⋊₉₁C₂, C₂³.₃₄D₄⋊₅₀C₂, C₂³.₁₀D₄⋊₈₇C₂, C₂³.₁₁D₄⋊₈₈C₂, C₂.₄₇(C₂³⋊₃D₄), (C₂×C₄²).₆₅₃C₂², (C₂²×C₄).₈₇₈C₂³, (C₂³×C₄).₁₅₀C₂², C₂³.₈Q₈⋊₁₀₈C₂, C₂².₄₁₀(C₂²×D₄), C₂⁴.₃C₂²⋊₈₂C₂, (C₂²×D₄).₂₃₇C₂², C₂³.₈₁C₂³⋊₈₈C₂, C₂⁴.C₂²⋊₁₃₂C₂, C₂³.₆₅C₂³⋊₁₂₂C₂, C₂.C₄².₃₀₇C₂², C₂.₈₄(C₂².₄₇C₂⁴), C₂.₇₅(C₂².₃₆C₂⁴), C₂.₁₇(C₂².₅₆C₂⁴), C₂.₄₂(C₂².₃₄C₂⁴), (C₂×C₄).₁₀₃(C₂×D₄), (C₂×C₄⋊D₄).₄₆C₂, (C₂×C₄).₁₉₃(C₄○D₄), (C₂×C₄⋊C₄).₄₁₄C₂², C₂².₄₆₃(C₂×C₄○D₄), (C₂×C₂².D₄)⋊₃₇C₂, (C₂×C₂²⋊C₄).₂₆₇C₂², SmallGroup(128,1433)

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

Derived series

C₁ — C₂³ — C₂⁴.₄₀₇C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂³.₃₄D₄ — C₂⁴.₄₀₇C₂³

Lower central

C₁ — C₂³ — C₂⁴.₄₀₇C₂³

Upper central

C₁ — C₂³ — C₂⁴.₄₀₇C₂³

Jennings

C₁ — C₂³ — C₂⁴.₄₀₇C₂³

Generators and relations for C₂⁴.₄₀₇C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=g²=cb=bc, f²=b, ab=ba, faf^-1=ac=ca, ad=da, ae=ea, gag^-1=abc, bd=db, geg^-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, gfg^-1=cf=fc, cg=gc, fef^-1=de=ed, df=fd, dg=gd >

Subgroups: 596 in 277 conjugacy classes, 96 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄⋊D₄, C₂².D₄, C₂³×C₄, C₂²×D₄, C₂³.₃₄D₄, C₂³.₈Q₈, C₂³.₂₃D₄, C₂⁴.C₂², C₂³.₆₅C₂³, C₂⁴.₃C₂², C₂³.₁₀D₄, C₂³.Q₈, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂×C₄⋊D₄, C₂×C₂².D₄, C₂⁴.₄₀₇C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂³⋊₃D₄, C₂².₃₄C₂⁴, C₂².₃₆C₂⁴, D₄⋊₅D₄, D₄⋊₆D₄, C₂².₄₇C₂⁴, C₂².₅₆C₂⁴, C₂⁴.₄₀₇C₂³

Smallest permutation representation of C₂⁴.₄₀₇C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂⁴.₄₀₇C₂³

C₂³.₃₄D₄

C₂³.₈Q₈

C₂³.₂₃D₄

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂⁴.₃C₂²

C₂³.₁₀D₄

C₂³.Q₈

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂×C₄⋊D₄

C₂×C₂².D₄

C₂×D₄

C₂×C₄

C₂²

# reps

Matrix representation of C₂⁴.₄₀₇C₂³ ►in GL₆(𝔽₅)

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

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

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

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

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,1,0,0,0,0,0,0,3,3,0,0,0,0,4,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[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],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,1,1,0,0,0,0,3,4,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,2,2,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C₂⁴.₄₀₇C₂³ in GAP, Magma, Sage, TeX

C_2^4._{407}C_2^3

% in TeX

G:=Group("C2^4.407C2^3");

// GroupNames label

G:=SmallGroup(128,1433);

// by ID

G=gap.SmallGroup(128,1433);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,344,758,723,100,1571,346]);

// Polycyclic

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

// generators/relations

G = C24.407C23 order 128 = 27

247th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₄₀₇C₂³ order 128 = 2⁷

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