C2^3.41D8 - GroupNames

G = C₂³.₄₁D₈ order 128 = 2⁷

12^nd non-split extension by C₂³ of D₈ acting via D₈/D₄=C₂

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C₂³.₄₁D₈, C₈.₉₅(C₂×D₄), (C₂×Q₁₆)⋊₁₄C₄, (C₂×C₄).₁₃₉D₈, (C₂×C₈).₁₁₉D₄, Q₁₆.₈(C₂×C₄), C₈.₆(C₂²⋊C₄), C₈.₃₂(C₂²×C₄), C₄.₁₃(C₂×SD₁₆), (C₂×C₄).₄₉SD₁₆, C₂².₅₆(C₂×D₈), C₂.Q₃₂⋊₁₅C₂, (C₂×C₈).₄₉₇C₂³, (C₂×C₁₆).₅₁C₂², C₂.₂(Q₃₂⋊C₂), (C₂²×C₄).₃₃₂D₄, C₄.₁₂(D₄⋊C₄), (C₂×M₅(2)).₂₁C₂, (C₂²×Q₁₆).₁₃C₂, C₂.D₈.₁₄₅C₂², (C₂²×C₈).₂₃₁C₂², (C₂×Q₁₆).₁₀₁C₂², C₂².₃₀(D₄⋊C₄), C₂³.₂₅D₄.₁₄C₂, (C₂×C₈).₈₁(C₂×C₄), (C₂×C₄).₇₅₉(C₂×D₄), C₄.₅₃(C₂×C₂²⋊C₄), C₂.₃₁(C₂×D₄⋊C₄), (C₂×C₄).₁₅₀(C₂²⋊C₄), SmallGroup(128,873)

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

Derived series

C₁ — C₈ — C₂³.₄₁D₈

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂×C₈ — C₂²×C₈ — C₂²×Q₁₆ — C₂³.₄₁D₈

Lower central

C₁ — C₂ — C₄ — C₈ — C₂³.₄₁D₈

Upper central

C₁ — C₂² — C₂²×C₄ — C₂²×C₈ — C₂³.₄₁D₈

Jennings

C₁ — C₂ — C₂ — C₂ — C₂ — C₄ — C₄ — C₂×C₈ — C₂³.₄₁D₈

Generators and relations for C₂³.₄₁D₈
G = < a,b,c,d,e | a²=b²=c²=1, d⁸=c, e²=b, ab=ba, dad^-1=eae^-1=ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede^-1=bd⁷ >

Subgroups: 228 in 110 conjugacy classes, 52 normal (18 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, C₂×C₄, Q₈, C₂³, C₁₆, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, Q₁₆, Q₁₆, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₄.Q₈, C₂.D₈, C₂×C₁₆, M₅(2), C₄²⋊C₂, C₂²×C₈, C₂×Q₁₆, C₂×Q₁₆, C₂²×Q₈, C₂.Q₃₂, C₂³.₂₅D₄, C₂×M₅(2), C₂²×Q₁₆, C₂³.₄₁D₈
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, D₈, SD₁₆, C₂²×C₄, C₂×D₄, D₄⋊C₄, C₂×C₂²⋊C₄, C₂×D₈, C₂×SD₁₆, C₂×D₄⋊C₄, Q₃₂⋊C₂, C₂³.₄₁D₈

Smallest permutation representation of C₂³.₄₁D₈
►On 64 points

Generators in S₆₄

(2 10)(4 12)(6 14)(8 16)(17 25)(19 27)(21 29)(23 31)(34 42)(36 44)(38 46)(40 48)(49 57)(51 59)(53 61)(55 63)

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

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

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

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

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

32 conjugacy classes

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	···	4L	8A	8B	8C	8D	8E	8F	16A	···	16H
order	1	2	2	2	2	2	4	4	4	4	4	···	4	8	8	8	8	8	8	16	···	16
size	1	1	1	1	2	2	2	2	2	2	8	···	8	2	2	2	2	4	4	4	···	4

32 irreducible representations

dim

type

image

C₁

C₂

C₄

D₄

D₈

SD₁₆

D₈

Q₃₂⋊C₂

kernel

C₂³.₄₁D₈

C₂.Q₃₂

C₂³.₂₅D₄

C₂×M₅(2)

C₂²×Q₁₆

C₂×Q₁₆

C₂×C₈

C₂²×C₄

C₂×C₄

C₂³

C₂

# reps

Matrix representation of C₂³.₄₁D₈ ►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	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	16	0
0	0	0	0	0	16

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

2	10	0	0	0	0
13	5	0	0	0	0
0	0	0	0	7	10
0	0	0	0	12	10
0	0	14	3	0	0
0	0	8	3	0	0

5	2	0	0	0	0
4	12	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	16	0	0	0
0	0	0	16	0	0

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[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],[2,13,0,0,0,0,10,5,0,0,0,0,0,0,0,0,14,8,0,0,0,0,3,3,0,0,7,12,0,0,0,0,10,10,0,0],[5,4,0,0,0,0,2,12,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0] >;

C₂³.₄₁D₈ in GAP, Magma, Sage, TeX

C_2^3._{41}D_8

% in TeX

G:=Group("C2^3.41D8");

// GroupNames label

G:=SmallGroup(128,873);

// by ID

G=gap.SmallGroup(128,873);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,456,723,352,1123,570,360,4037,2028,124]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^8=c,e^2=b,a*b=b*a,d*a*d^-1=e*a*e^-1=a*c=c*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^7>;

// generators/relations

G = C23.41D8 order 128 = 27

12nd non-split extension by C23 of D8 acting via D8/D4=C2

G = C₂³.₄₁D₈ order 128 = 2⁷

12^nd non-split extension by C₂³ of D₈ acting via D₈/D₄=C₂