C2^3.494C2^4 - GroupNames

G = C₂³.₄₉₄C₂⁴ order 128 = 2⁷

211^st central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₃₅C₂³, C₂³.₄₉₄C₂⁴, C₂².₂₇₆2₊¹⁺⁴, C₂².₂₀₃2_-¹⁺⁴, (C₂²×C₄).₁₁₆C₂³, (C₂×C₄²).₅₈₃C₂², C₂³.Q₈.₁₇C₂, C₂³.₈₄C₂³⋊₆C₂, C₂³.₁₁D₄.₂₂C₂, C₂³.₆₅C₂³⋊₉₅C₂, C₂³.₈₁C₂³⋊₅₀C₂, C₂³.₆₃C₂³⋊₉₉C₂, C₂⁴.C₂².₃₈C₂, C₂.C₄².₄₉₃C₂², C₂.₃₄(C₂².₃₃C₂⁴), C₂.₉₆(C₂³.₃₆C₂³), C₂.₇₁(C₂².₄₇C₂⁴), C₂.₄₈(C₂².₅₀C₂⁴), (C₄×C₄⋊C₄)⋊₁₀₇C₂, (C₂×C₄).₁₅₉(C₄○D₄), (C₂×C₄⋊C₄).₃₃₅C₂², C₂².₃₇₀(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₁₉₇C₂², SmallGroup(128,1326)

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

Derived series

C₁ — C₂³ — C₂³.₄₉₄C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₄×C₄⋊C₄ — C₂³.₄₉₄C₂⁴

Lower central

C₁ — C₂³ — C₂³.₄₉₄C₂⁴

Upper central

C₁ — C₂³ — C₂³.₄₉₄C₂⁴

Jennings

C₁ — C₂³ — C₂³.₄₉₄C₂⁴

Generators and relations for C₂³.₄₉₄C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=1, d²=f²=c, e²=abc, g²=b, ab=ba, ac=ca, ede^-1=gdg^-1=ad=da, 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, eg=ge, fg=gf >

Subgroups: 356 in 199 conjugacy classes, 92 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄×C₄⋊C₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₅C₂³, C₂³.Q₈, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂³.₈₄C₂³, C₂³.₄₉₄C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂³.₃₆C₂³, C₂².₃₃C₂⁴, C₂².₄₇C₂⁴, C₂².₅₀C₂⁴, C₂³.₄₉₄C₂⁴

Smallest permutation representation of C₂³.₄₉₄C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	4A	···	4H	4I	···	4Z	4AA	4AB	4AC
order	1	2	···	2	2	4	···	4	4	···	4	4	4	4
size	1	1	···	1	8	2	···	2	4	···	4	8	8	8

38 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂³.₄₉₄C₂⁴

C₄×C₄⋊C₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂³.Q₈

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂³.₈₄C₂³

C₂×C₄

C₂²

# reps

Matrix representation of C₂³.₄₉₄C₂⁴ ►in GL₆(𝔽₅)

4	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	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	4	0
0	0	0	0	0	4

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

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

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

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

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

C₂³.₄₉₄C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{494}C_2^4

% in TeX

G:=Group("C2^3.494C2^4");

// GroupNames label

G:=SmallGroup(128,1326);

// by ID

G=gap.SmallGroup(128,1326);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,792,758,723,352,675,136]);

// Polycyclic

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

// generators/relations

G = C23.494C24 order 128 = 27

211st central stem extension by C23 of C24

G = C₂³.₄₉₄C₂⁴ order 128 = 2⁷

211^st central stem extension by C₂³ of C₂⁴