C2xC2^3.48D4 - GroupNames

G = C₂×C₂³.₄₈D₄ order 128 = 2⁷

Direct product of C₂ and C₂³.₄₈D₄

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

Aliases: C₂×C₂³.₄₈D₄, C₂⁴.₁₈₂D₄, C₂³.₂₁Q₁₆, C₄⋊C₄.₅₁C₂³, C₂.D₈⋊₅₄C₂², C₂.₈(C₂²×Q₁₆), (C₂×C₄).₂₈₈C₂⁴, (C₂×C₈).₁₄₂C₂³, C₂³.₆₆₄(C₂×D₄), (C₂²×C₄).₄₃₉D₄, (C₂×Q₈).₆₅C₂³, C₂².₁₅(C₂×Q₁₆), Q₈⋊C₄⋊₆₉C₂², C₂²⋊C₈.₁₇₁C₂², (C₂²×C₈).₁₄₇C₂², (C₂³×C₄).₅₅₈C₂², C₂².₅₄₈(C₂²×D₄), C₂²⋊Q₈.₁₅₈C₂², C₂².₁₂₂(C₈⋊C₂²), (C₂²×C₄).₁₀₀₅C₂³, C₄.₆₀(C₂².D₄), (C₂²×Q₈).₂₉₁C₂², C₂².₁₁₁(C₂².D₄), (C₂×C₂.D₈)⋊₂₆C₂, C₄.₉₈(C₂×C₄○D₄), (C₂×C₄).₈₄₈(C₂×D₄), C₂.₂₇(C₂×C₈⋊C₂²), (C₂×Q₈⋊C₄)⋊₂₄C₂, (C₂²×C₄⋊C₄).₄₆C₂, (C₂×C₂²⋊C₈).₃₁C₂, (C₂×C₂²⋊Q₈).₅₆C₂, (C₂×C₄).₈₄₆(C₄○D₄), (C₂×C₄⋊C₄).₉₂₅C₂², C₂.₅₃(C₂×C₂².D₄), SmallGroup(128,1822)

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

Derived series

C₁ — C₂×C₄ — C₂×C₂³.₄₈D₄

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂×C₄⋊C₄ — C₂²×C₄⋊C₄ — C₂×C₂³.₄₈D₄

Lower central

C₁ — C₂ — C₂×C₄ — C₂×C₂³.₄₈D₄

Upper central

C₁ — C₂³ — C₂³×C₄ — C₂×C₂³.₄₈D₄

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₂×C₂³.₄₈D₄

Generators and relations for C₂×C₂³.₄₈D₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=1, e⁴=f²=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe^-1=fbf^-1=bc=cb, bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef^-1=cde³ >

Subgroups: 412 in 230 conjugacy classes, 108 normal (18 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂×Q₈, C₂⁴, C₂²⋊C₈, Q₈⋊C₄, C₂.D₈, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂²⋊Q₈, C₂²⋊Q₈, C₂²×C₈, C₂³×C₄, C₂³×C₄, C₂²×Q₈, C₂×C₂²⋊C₈, C₂×Q₈⋊C₄, C₂×C₂.D₈, C₂³.₄₈D₄, C₂²×C₄⋊C₄, C₂×C₂²⋊Q₈, C₂×C₂³.₄₈D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, Q₁₆, C₂×D₄, C₄○D₄, C₂⁴, C₂².D₄, C₂×Q₁₆, C₈⋊C₂², C₂²×D₄, C₂×C₄○D₄, C₂³.₄₈D₄, C₂×C₂².D₄, C₂²×Q₁₆, C₂×C₈⋊C₂², C₂×C₂³.₄₈D₄

Smallest permutation representation of C₂×C₂³.₄₈D₄
►On 64 points

Generators in S₆₄

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

(2 58)(4 60)(6 62)(8 64)(10 32)(12 26)(14 28)(16 30)(18 51)(20 53)(22 55)(24 49)(33 46)(35 48)(37 42)(39 44)

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

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

Q₁₆

C₈⋊C₂²

kernel

C₂×C₂³.₄₈D₄

C₂×C₂²⋊C₈

C₂×Q₈⋊C₄

C₂×C₂.D₈

C₂³.₄₈D₄

C₂²×C₄⋊C₄

C₂×C₂²⋊Q₈

C₂²×C₄

C₂⁴

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂×C₂³.₄₈D₄ ►in GL₆(𝔽₁₇)

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

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

0	4	0	0	0	0
4	0	0	0	0	0
0	0	0	13	0	0
0	0	4	0	0	0
0	0	0	0	3	14
0	0	0	0	3	3

0	1	0	0	0	0
1	0	0	0	0	0
0	0	0	16	0	0
0	0	16	0	0	0
0	0	0	0	16	10
0	0	0	0	10	1

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,13,0,0,0,0,0,0,0,3,3,0,0,0,0,14,3],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,16,10,0,0,0,0,10,1] >;

C₂×C₂³.₄₈D₄ in GAP, Magma, Sage, TeX

C_2\times C_2^3._{48}D_4

% in TeX

G:=Group("C2xC2^3.48D4");

// GroupNames label

G:=SmallGroup(128,1822);

// by ID

G=gap.SmallGroup(128,1822);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,436,4037,1027,124]);

// Polycyclic

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

// generators/relations

G = C2×C23.48D4 order 128 = 27

Direct product of C2 and C23.48D4

G = C₂×C₂³.₄₈D₄ order 128 = 2⁷

Direct product of C₂ and C₂³.₄₈D₄