C2^3:2D8 - GroupNames

G = C₂³⋊₂D₈ order 128 = 2⁷

1^st semidirect product of C₂³ and D₈ acting via D₈/C₄=C₂²

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

Aliases: C₂³⋊₂D₈, C₂⁴.₇₉D₄, C₄⋊C₄⋊₄D₄, (C₂×C₈)⋊₅D₄, (C₂×D₄)⋊₅D₄, (C₂²×D₈)⋊₁C₂, C₄.₆₀C₂²≀C₂, C₂².₈₀(C₂×D₈), C₄.₂₆(C₄⋊₁D₄), C₄.₁₃(C₄⋊D₄), C₂.₁₂(C₈⋊₇D₄), C₂.₁₂(C₈⋊₂D₄), (C₂²×C₄).₁₃₇D₄, C₂³.₈₉₆(C₂×D₄), C₂.₂₇(D₄⋊D₄), C₂.₁₈(C₂²⋊D₈), C₂².₄Q₁₆⋊₃₆C₂, C₂.₄(C₂³⋊₂D₄), C₂².₁₉₂C₂²≀C₂, C₂².₁₀₀(C₄○D₈), (C₂²×C₈).₁₀₅C₂², (C₂³×C₄).₂₆₇C₂², (C₂²×D₄).₅₆C₂², C₂².₂₁₇(C₄⋊D₄), C₂².₁₂₇(C₈⋊C₂²), (C₂²×C₄).₁₄₃₀C₂³, (C₂×C₄⋊D₄)⋊₁C₂, (C₂×C₂²⋊C₈)⋊₁₈C₂, (C₂×D₄⋊C₄)⋊₁₀C₂, (C₂×C₄).₁₀₂₀(C₂×D₄), (C₂×C₄).₆₁₃(C₄○D₄), (C₂×C₄⋊C₄).₁₀₁C₂², SmallGroup(128,731)

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

Derived series

C₁ — C₂²×C₄ — C₂³⋊₂D₈

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂×C₄⋊D₄ — C₂³⋊₂D₈

Lower central

C₁ — C₂ — C₂²×C₄ — C₂³⋊₂D₈

Upper central

C₁ — C₂³ — C₂³×C₄ — C₂³⋊₂D₈

Jennings

C₁ — C₂ — C₂ — C₂²×C₄ — C₂³⋊₂D₈

Generators and relations for C₂³⋊₂D₈
G = < a,b,c,d,e | a²=b²=c²=d⁸=e²=1, eae=ab=ba, ac=ca, dad^-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d^-1 >

Subgroups: 656 in 248 conjugacy classes, 56 normal (22 characteristic)
C₁, C₂, C₂, C₂, C₄, 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, D₈, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂⁴, C₂²⋊C₈, D₄⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄⋊D₄, C₂²×C₈, C₂×D₈, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂².₄Q₁₆, C₂×C₂²⋊C₈, C₂×D₄⋊C₄, C₂×C₄⋊D₄, C₂²×D₈, C₂³⋊₂D₈
Quotients: C₁, C₂, C₂², D₄, C₂³, D₈, C₂×D₄, C₄○D₄, C₂²≀C₂, C₄⋊D₄, C₄⋊₁D₄, C₂×D₈, C₄○D₈, C₈⋊C₂², C₂³⋊₂D₄, C₂²⋊D₈, D₄⋊D₄, C₈⋊₇D₄, C₈⋊₂D₄, C₂³⋊₂D₈

Smallest permutation representation of C₂³⋊₂D₈
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

D₈

C₄○D₈

C₈⋊C₂²

kernel

C₂³⋊₂D₈

C₂².₄Q₁₆

C₂×C₂²⋊C₈

C₂×D₄⋊C₄

C₂×C₄⋊D₄

C₂²×D₈

C₄⋊C₄

C₂×C₈

C₂²×C₄

C₂×D₄

C₂⁴

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂³⋊₂D₈ ►in GL₆(𝔽₁₇)

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

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	16	0
0	0	0	0	0	16

14	3	0	0	0	0
14	14	0	0	0	0
0	0	16	0	0	0
0	0	0	16	0	0
0	0	0	0	0	16
0	0	0	0	1	0

14	3	0	0	0	0
3	3	0	0	0	0
0	0	16	0	0	0
0	0	0	1	0	0
0	0	0	0	0	16
0	0	0	0	16	0

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

C₂³⋊₂D₈ in GAP, Magma, Sage, TeX

C_2^3\rtimes_2D_8

% in TeX

G:=Group("C2^3:2D8");

// GroupNames label

G:=SmallGroup(128,731);

// by ID

G=gap.SmallGroup(128,731);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,2019,1018,248]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=e^2=1,e*a*e=a*b=b*a,a*c=c*a,d*a*d^-1=a*b*c,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;

// generators/relations

G = C23⋊2D8 order 128 = 27

1st semidirect product of C23 and D8 acting via D8/C4=C22

G = C₂³⋊₂D₈ order 128 = 2⁷

1^st semidirect product of C₂³ and D₈ acting via D₈/C₄=C₂²