C2^4.412C2^3 - GroupNames

G = C₂⁴.₄₁₂C₂³ order 128 = 2⁷

252^nd non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²

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

Aliases: C₂⁴.₄₁₂C₂³, C₂³.₆₁₀C₂⁴, C₂².₂₈₆2_-¹⁺⁴, C₂².₃₈₄2₊¹⁺⁴, C₂²⋊C₄⋊₁₈D₄, C₂³⋊Q₈⋊₄₈C₂, C₂³.₂₁₉(C₂×D₄), C₂.₆₆(D₄⋊₆D₄), C₂³⋊₂D₄.₂₅C₂, C₂.₁₁₅(D₄⋊₅D₄), C₂³.₄Q₈⋊₄₇C₂, C₂³.₁₁D₄⋊₉₃C₂, C₂³.₂₃D₄⋊₉₇C₂, C₂³.₁₀D₄⋊₉₃C₂, (C₂²×C₄).₁₈₈C₂³, (C₂×C₄²).₆₆₁C₂², (C₂³×C₄).₁₅₄C₂², C₂³.₈Q₈⋊₁₁₂C₂, C₂².₄₁₉(C₂²×D₄), C₂⁴.₃C₂²⋊₈₆C₂, (C₂²×D₄).₂₄₅C₂², (C₂²×Q₈).₁₈₉C₂², C₂⁴.C₂²⋊₁₃₈C₂, C₂³.₆₇C₂³⋊₈₅C₂, C₂.₆₁(C₂².₂₉C₂⁴), C₂.₇₁(C₂².₃₂C₂⁴), C₂³.₆₅C₂³⋊₁₂₇C₂, C₂.C₄².₃₁₆C₂², C₂.₂₁(C₂².₅₆C₂⁴), C₂.₇₈(C₂².₃₆C₂⁴), C₂.₃₄(C₂².₅₃C₂⁴), (C₂×C₄).₁₁₀(C₂×D₄), (C₂×C₄.₄D₄)⋊₂₉C₂, (C₂×C₄).₁₉₇(C₄○D₄), (C₂×C₄⋊C₄).₄₂₃C₂², C₂².₄₇₂(C₂×C₄○D₄), (C₂×C₂².D₄)⋊₄₃C₂, (C₂×C₂²⋊C₄).₂₇₆C₂², SmallGroup(128,1442)

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

Derived series

C₁ — C₂³ — C₂⁴.₄₁₂C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂³.₂₃D₄ — C₂⁴.₄₁₂C₂³

Lower central

C₁ — C₂³ — C₂⁴.₄₁₂C₂³

Upper central

C₁ — C₂³ — C₂⁴.₄₁₂C₂³

Jennings

C₁ — C₂³ — C₂⁴.₄₁₂C₂³

Generators and relations for C₂⁴.₄₁₂C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=f²=b, g²=cb=bc, faf^-1=ab=ba, ac=ca, ad=da, eae^-1=abc, ag=ga, bd=db, fef^-1=be=eb, bf=fb, bg=gb, cd=dc, geg^-1=ce=ec, cf=fc, cg=gc, de=ed, gfg^-1=df=fd, dg=gd >

Subgroups: 596 in 274 conjugacy classes, 96 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂².D₄, C₄.₄D₄, C₂³×C₄, C₂²×D₄, C₂²×Q₈, C₂³.₈Q₈, C₂³.₂₃D₄, C₂⁴.C₂², C₂³.₆₅C₂³, C₂⁴.₃C₂², C₂³.₆₇C₂³, C₂³⋊₂D₄, C₂³⋊Q₈, C₂³.₁₀D₄, C₂³.₁₁D₄, C₂³.₄Q₈, C₂×C₂².D₄, C₂×C₄.₄D₄, C₂⁴.₄₁₂C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₂₉C₂⁴, C₂².₃₂C₂⁴, C₂².₃₆C₂⁴, D₄⋊₅D₄, D₄⋊₆D₄, C₂².₅₃C₂⁴, C₂².₅₆C₂⁴, C₂⁴.₄₁₂C₂³

Smallest permutation representation of C₂⁴.₄₁₂C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	···	4P	4Q	4R	4S	4T
order	1	2	···	2	2	2	2	2	4	···	4	4	4	4	4
size	1	1	···	1	4	4	8	8	4	···	4	8	8	8	8

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂⁴.₄₁₂C₂³

C₂³.₈Q₈

C₂³.₂₃D₄

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂⁴.₃C₂²

C₂³.₆₇C₂³

C₂³⋊₂D₄

C₂³⋊Q₈

C₂³.₁₀D₄

C₂³.₁₁D₄

C₂³.₄Q₈

C₂×C₂².D₄

C₂×C₄.₄D₄

C₂²⋊C₄

C₂×C₄

C₂²

# reps

Matrix representation of C₂⁴.₄₁₂C₂³ ►in GL₆(𝔽₅)

0	1	0	0	0	0
1	0	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	0	1
0	0	0	0	1	0

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	2	0	0	0	0
3	0	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

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

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

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

C₂⁴.₄₁₂C₂³ in GAP, Magma, Sage, TeX

C_2^4._{412}C_2^3

% in TeX

G:=Group("C2^4.412C2^3");

// GroupNames label

G:=SmallGroup(128,1442);

// by ID

G=gap.SmallGroup(128,1442);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,723,268,1571,346]);

// Polycyclic

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

// generators/relations

G = C24.412C23 order 128 = 27

252nd non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₄₁₂C₂³ order 128 = 2⁷

252^nd non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²