C2^4.406C2^3 - GroupNames

G = C₂⁴.₄₀₆C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₄₀₆C₂³, C₂³.₅₉₉C₂⁴, C₂².₃₇₃2₊¹⁺⁴, (C₂×D₄)⋊₁₉D₄, C₂³⋊₂D₄⋊₄₁C₂, C₂³.₂₁₄(C₂×D₄), C₂.₁₀₄(D₄⋊₅D₄), C₂³.₂₃D₄⋊₉₀C₂, C₂³.₃₄D₄⋊₄₈C₂, C₂³.₁₀D₄⋊₈₆C₂, C₂.₄₆(C₂³⋊₃D₄), (C₂×C₄²).₆₅₁C₂², (C₂³×C₄).₄₆₁C₂², (C₂²×C₄).₈₇₆C₂³, C₂².₄₀₈(C₂²×D₄), C₂⁴.₃C₂²⋊₈₁C₂, (C₂²×D₄).₂₃₅C₂², C₂⁴.C₂²⋊₁₃₀C₂, C₂³.₈₁C₂³⋊₈₇C₂, C₂.₁₆(C₂².₅₄C₂⁴), C₂.C₄².₃₀₅C₂², C₂.₃₅(C₂².₄₉C₂⁴), C₂.₄₁(C₂².₃₄C₂⁴), (C₂×C₄⋊D₄)⋊₃₆C₂, (C₂×C₄).₁₀₁(C₂×D₄), (C₂×C₄).₄₂₆(C₄○D₄), (C₂×C₄⋊C₄).₄₁₂C₂², C₂².₄₆₁(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₂₆₅C₂², SmallGroup(128,1431)

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

Derived series

C₁ — C₂³ — C₂⁴.₄₀₆C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂³.₃₄D₄ — C₂⁴.₄₀₆C₂³

Lower central

C₁ — C₂³ — C₂⁴.₄₀₆C₂³

Upper central

C₁ — C₂³ — C₂⁴.₄₀₆C₂³

Jennings

C₁ — C₂³ — C₂⁴.₄₀₆C₂³

Generators and relations for C₂⁴.₄₀₆C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=c, f²=b, g²=cb=bc, 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: 708 in 306 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×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄⋊D₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂³.₃₄D₄, C₂³.₂₃D₄, C₂⁴.C₂², C₂⁴.₃C₂², C₂³⋊₂D₄, C₂³.₁₀D₄, C₂³.₁₀D₄, C₂³.₈₁C₂³, C₂×C₄⋊D₄, C₂⁴.₄₀₆C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, C₂³⋊₃D₄, C₂².₃₄C₂⁴, D₄⋊₅D₄, C₂².₄₉C₂⁴, C₂².₅₄C₂⁴, C₂⁴.₄₀₆C₂³

Smallest permutation representation of C₂⁴.₄₀₆C₂³
►On 64 points

Generators in S₆₄

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

(1 39)(2 40)(3 37)(4 38)(5 17)(6 18)(7 19)(8 20)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 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 62)(34 63)(35 64)(36 61)

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	2L	2M	4A	···	4N	4O	4P	4Q	4R
order	1	2	···	2	2	2	2	2	2	2	4	···	4	4	4	4	4
size	1	1	···	1	4	4	4	4	8	8	4	···	4	8	8	8	8

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

kernel

C₂⁴.₄₀₆C₂³

C₂³.₃₄D₄

C₂³.₂₃D₄

C₂⁴.C₂²

C₂⁴.₃C₂²

C₂³⋊₂D₄

C₂³.₁₀D₄

C₂³.₈₁C₂³

C₂×C₄⋊D₄

C₂×D₄

C₂×C₄

C₂²

# reps

Matrix representation of C₂⁴.₄₀₆C₂³ ►in GL₆(𝔽₅)

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

0	1	0	0	0	0
1	0	0	0	0	0
0	0	2	0	0	0
0	0	0	2	0	0
0	0	0	0	0	1
0	0	0	0	1	0

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

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

C₂⁴.₄₀₆C₂³ in GAP, Magma, Sage, TeX

C_2^4._{406}C_2^3

% in TeX

G:=Group("C2^4.406C2^3");

// GroupNames label

G:=SmallGroup(128,1431);

// by ID

G=gap.SmallGroup(128,1431);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,232,758,723,100,1571,346]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=c,f^2=b,g^2=c*b=b*c,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 = C24.406C23 order 128 = 27

246th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₄₀₆C₂³ order 128 = 2⁷

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