C2^3.354C2^4 - GroupNames

G = C₂³.₃₅₄C₂⁴ order 128 = 2⁷

71^st central stem extension by C₂³ of C₂⁴

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

Aliases: C₂³.₃₅₄C₂⁴, C₂⁴.₂₇₅C₂³, C₂².₁₆₁2₊¹⁺⁴, C₄⋊C₄.₃₂₇D₄, (C₂×D₄).₂₉Q₈, C₂³.₁₄(C₂×Q₈), C₄.₅₅(C₂²⋊Q₈), C₂.₄₁(D₄⋊₅D₄), C₂³.₄Q₈⋊₈C₂, C₂.₁₅(Q₈⋊₆D₄), C₂.₁₅(D₄⋊₃Q₈), C₂³.₈Q₈⋊₄₄C₂, C₂³.Q₈⋊₁₂C₂, C₂³.₇Q₈⋊₄₆C₂, C₂².₈₀(C₂²×Q₈), (C₂²×C₄).₈₀₉C₂³, (C₂×C₄²).₄₉₇C₂², (C₂³×C₄).₃₅₇C₂², C₂².₂₃₄(C₂²×D₄), (C₂²×D₄).₅₁₃C₂², C₂³.₆₅C₂³⋊₅₇C₂, C₂⁴.₃C₂².₃₅C₂, C₂.C₄².₁₁₁C₂², C₂.₁₁(C₂².₃₄C₂⁴), C₂.₂₉(C₂³.₃₆C₂³), (C₄×C₄⋊C₄)⋊₅₉C₂, (C₂×C₄×D₄).₅₁C₂, (C₂×C₄).₃₃₃(C₂×D₄), (C₂×C₄².C₂)⋊₄C₂, (C₂×C₄).₃₀₇(C₂×Q₈), C₂.₂₇(C₂×C₂²⋊Q₈), (C₂×C₄).₁₀₉(C₄○D₄), (C₂×C₄⋊C₄).₂₃₆C₂², C₂².₂₃₁(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₁₃₁C₂², SmallGroup(128,1186)

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

Derived series

C₁ — C₂³ — C₂³.₃₅₄C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₄×C₄⋊C₄ — C₂³.₃₅₄C₂⁴

Lower central

C₁ — C₂³ — C₂³.₃₅₄C₂⁴

Upper central

C₁ — C₂³ — C₂³.₃₅₄C₂⁴

Jennings

C₁ — C₂³ — C₂³.₃₅₄C₂⁴

Generators and relations for C₂³.₃₅₄C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=g²=1, e²=c, f²=cb=bc, eae^-1=gag=ab=ba, ac=ca, ad=da, faf^-1=acd, bd=db, geg=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef^-1=de=ed, df=fd, dg=gd, gfg=cdf >

Subgroups: 516 in 274 conjugacy classes, 112 normal (42 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊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, C₄².C₂, C₂³×C₄, C₂²×D₄, C₄×C₄⋊C₄, C₂³.₇Q₈, C₂³.₈Q₈, C₂³.₆₅C₂³, C₂⁴.₃C₂², C₂³.Q₈, C₂³.₄Q₈, C₂×C₄×D₄, C₂×C₄².C₂, C₂³.₃₅₄C₂⁴
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, C₂²⋊Q₈, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, 2₊¹⁺⁴, C₂×C₂²⋊Q₈, C₂³.₃₆C₂³, C₂².₃₄C₂⁴, D₄⋊₅D₄, Q₈⋊₆D₄, D₄⋊₃Q₈, C₂³.₃₅₄C₂⁴

Smallest permutation representation of C₂³.₃₅₄C₂⁴
►On 64 points

Generators in S₆₄

(2 40)(4 38)(5 61)(6 35)(7 63)(8 33)(9 43)(11 41)(13 59)(14 30)(15 57)(16 32)(17 36)(18 64)(19 34)(20 62)(22 52)(24 50)(25 55)(27 53)(29 47)(31 45)(46 58)(48 60)

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	···	4H	4I	···	4V	4W	4X	4Y	4Z
order	1	2	···	2	2	2	2	2	4	···	4	4	···	4	4	4	4	4
size	1	1	···	1	4	4	4	4	2	···	2	4	···	4	8	8	8	8

38 irreducible representations

dim

type

image

C₁

C₂

D₄

Q₈

C₄○D₄

2₊¹⁺⁴

kernel

C₂³.₃₅₄C₂⁴

C₄×C₄⋊C₄

C₂³.₇Q₈

C₂³.₈Q₈

C₂³.₆₅C₂³

C₂⁴.₃C₂²

C₂³.Q₈

C₂³.₄Q₈

C₂×C₄×D₄

C₂×C₄².C₂

C₄⋊C₄

C₂×D₄

C₂×C₄

C₂²

# reps

Matrix representation of C₂³.₃₅₄C₂⁴ ►in GL₆(𝔽₅)

4	0	0	0	0	0
4	1	0	0	0	0
0	0	1	0	0	0
0	0	1	4	0	0
0	0	0	0	1	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

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

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

2	1	0	0	0	0
2	3	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

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

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

C₂³.₃₅₄C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{354}C_2^4

% in TeX

G:=Group("C2^3.354C2^4");

// GroupNames label

G:=SmallGroup(128,1186);

// by ID

G=gap.SmallGroup(128,1186);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,232,758,723,268,675,80]);

// Polycyclic

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

// generators/relations

G = C23.354C24 order 128 = 27

71st central stem extension by C23 of C24

G = C₂³.₃₅₄C₂⁴ order 128 = 2⁷

71^st central stem extension by C₂³ of C₂⁴