C2xC2^3.46D4 - GroupNames

G = C₂×C₂³.₄₆D₄ order 128 = 2⁷

Direct product of C₂ and C₂³.₄₆D₄

direct product, p-group, metabelian, nilpotent (class 3), monomial

Aliases: C₂×C₂³.₄₆D₄, C₂⁴.₁₈₁D₄, C₂³.₄₄SD₁₆, C₄⋊C₄.₅₀C₂³, C₄.Q₈⋊₆₃C₂², C₂²⋊C₈⋊₆₇C₂², (C₂×C₈).₃₁₁C₂³, (C₂×C₄).₂₈₇C₂⁴, (C₂×D₄).₇₇C₂³, C₂³.₆₆₃(C₂×D₄), (C₂²×C₄).₄₃₈D₄, D₄⋊C₄⋊₇₇C₂², C₂².₃₅(C₂×SD₁₆), C₂.₁₃(C₂²×SD₁₆), C₄⋊D₄.₁₅₃C₂², (C₂³×C₄).₅₅₇C₂², (C₂²×C₈).₃₄₈C₂², C₂².₅₄₇(C₂²×D₄), C₂².₁₂₁(C₈⋊C₂²), (C₂²×C₄).₁₀₀₄C₂³, C₄.₅₉(C₂².D₄), (C₂²×D₄).₃₅₈C₂², C₂².₁₁₀(C₂².D₄), (C₂×C₄.Q₈)⋊₃₄C₂, C₄.₉₇(C₂×C₄○D₄), (C₂²×C₄⋊C₄)⋊₃₄C₂, (C₂×C₂²⋊C₈)⋊₃₅C₂, (C₂×C₄).₈₄₇(C₂×D₄), C₂.₂₆(C₂×C₈⋊C₂²), (C₂×D₄⋊C₄)⋊₃₉C₂, (C₂×C₄⋊C₄)⋊₁₁₇C₂², (C₂×C₄⋊D₄).₅₇C₂, (C₂×C₄).₈₄₅(C₄○D₄), C₂.₅₂(C₂×C₂².D₄), SmallGroup(128,1821)

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

Derived series

C₁ — C₂×C₄ — C₂×C₂³.₄₆D₄

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂×C₄⋊C₄ — C₂²×C₄⋊C₄ — C₂×C₂³.₄₆D₄

Lower central

C₁ — C₂ — C₂×C₄ — C₂×C₂³.₄₆D₄

Upper central

C₁ — C₂³ — C₂³×C₄ — C₂×C₂³.₄₆D₄

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₂×C₂³.₄₆D₄

Generators and relations for C₂×C₂³.₄₆D₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=f²=1, e⁴=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe^-1=fbf=bc=cb, bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=ce³ >

Subgroups: 540 in 256 conjugacy classes, 108 normal (18 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂⁴, C₂²⋊C₈, D₄⋊C₄, C₄.Q₈, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄⋊D₄, C₄⋊D₄, C₂²×C₈, C₂³×C₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂×C₂²⋊C₈, C₂×D₄⋊C₄, C₂×C₄.Q₈, C₂³.₄₆D₄, C₂²×C₄⋊C₄, C₂×C₄⋊D₄, C₂×C₂³.₄₆D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, SD₁₆, C₂×D₄, C₄○D₄, C₂⁴, C₂².D₄, C₂×SD₁₆, C₈⋊C₂², C₂²×D₄, C₂×C₄○D₄, C₂³.₄₆D₄, C₂×C₂².D₄, C₂²×SD₁₆, C₂×C₈⋊C₂², C₂×C₂³.₄₆D₄

Smallest permutation representation of C₂×C₂³.₄₆D₄
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	2L	2M	4A	4B	4C	4D	4E	···	4N	4O	4P	8A	···	8H
order	1	2	···	2	2	2	2	2	2	2	4	4	4	4	4	···	4	4	4	8	···	8
size	1	1	···	1	2	2	2	2	8	8	2	2	2	2	4	···	4	8	8	4	···	4

38 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

SD₁₆

C₈⋊C₂²

kernel

C₂×C₂³.₄₆D₄

C₂×C₂²⋊C₈

C₂×D₄⋊C₄

C₂×C₄.Q₈

C₂³.₄₆D₄

C₂²×C₄⋊C₄

C₂×C₄⋊D₄

C₂²×C₄

C₂⁴

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂×C₂³.₄₆D₄ ►in GL₆(𝔽₁₇)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	16	0	0	0
0	0	0	16	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	8	0	0	0	0
0	16	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	16	0
0	0	0	0	0	16

16	0	0	0	0	0
0	16	0	0	0	0
0	0	16	0	0	0
0	0	0	16	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	16	0
0	0	0	0	0	16

13	2	0	0	0	0
1	4	0	0	0	0
0	0	4	0	0	0
0	0	0	13	0	0
0	0	0	0	12	5
0	0	0	0	12	12

16	0	0	0	0	0
13	1	0	0	0	0
0	0	1	0	0	0
0	0	0	16	0	0
0	0	0	0	16	0
0	0	0	0	0	1

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,8,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[13,1,0,0,0,0,2,4,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,12,12,0,0,0,0,5,12],[16,13,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1] >;

C₂×C₂³.₄₆D₄ in GAP, Magma, Sage, TeX

C_2\times C_2^3._{46}D_4

% in TeX

G:=Group("C2xC2^3.46D4");

// GroupNames label

G:=SmallGroup(128,1821);

// by ID

G=gap.SmallGroup(128,1821);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,436,4037,1027,124]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=f^2=1,e^4=d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,e*b*e^-1=f*b*f=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=c*e^3>;

// generators/relations

G = C2×C23.46D4 order 128 = 27

Direct product of C2 and C23.46D4

G = C₂×C₂³.₄₆D₄ order 128 = 2⁷

Direct product of C₂ and C₂³.₄₆D₄