C4^2.185D4 - GroupNames

G = C₄².₁₈₅D₄ order 128 = 2⁷

167^th non-split extension by C₄² of D₄ acting via D₄/C₂=C₂²

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

Aliases: C₄².₁₈₅D₄, C₂⁴.₃₅₈C₂³, C₂³.₅₁₁C₂⁴, C₂².₂₁₁2_-¹⁺⁴, C₄²⋊₄C₄⋊₂₉C₂, C₂³.₁₆₄(C₄○D₄), (C₂×C₄²).₅₉₈C₂², (C₂³×C₄).₄₁₅C₂², (C₂²×C₄).₁₂₆C₂³, C₂².₃₃₇(C₂²×D₄), C₂³.₄Q₈.₁₄C₂, C₂³.₈Q₈.₄₀C₂, C₂³.₁₁D₄.₂₅C₂, C₂³.₈₃C₂³⋊₅₆C₂, C₂³.₈₁C₂³⋊₅₅C₂, C₂.₈₀(C₂².₁₉C₂⁴), C₂⁴.C₂².₄₂C₂, C₂³.₆₅C₂³⋊₁₀₁C₂, C₂³.₆₃C₂³⋊₁₁₁C₂, C₂.C₄².₂₄₀C₂², C₂.₄₉(C₂².₂₆C₂⁴), C₂.₃₀(C₂³.₃₈C₂³), C₂.₇₈(C₂².₄₆C₂⁴), (C₂×C₄).₃₇₂(C₂×D₄), (C₂×C₄².C₂)⋊₁₅C₂, (C₂×C₄).₄₁₃(C₄○D₄), (C₂×C₄⋊C₄).₃₅₀C₂², C₂².₃₈₇(C₂×C₄○D₄), (C₂×C₄²⋊C₂).₄₄C₂, (C₂×C₂²⋊C₄).₅₁₈C₂², SmallGroup(128,1343)

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

Derived series

C₁ — C₂³ — C₄².₁₈₅D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂×C₄²⋊C₂ — C₄².₁₈₅D₄

Lower central

C₁ — C₂³ — C₄².₁₈₅D₄

Upper central

C₁ — C₂³ — C₄².₁₈₅D₄

Jennings

C₁ — C₂³ — C₄².₁₈₅D₄

Generators and relations for C₄².₁₈₅D₄
G = < a,b,c,d | a⁴=b⁴=c⁴=1, d²=a²b², ab=ba, cac^-1=a^-1b², dad^-1=a^-1, cbc^-1=dbd^-1=a²b^-1, dcd^-1=a²b²c^-1 >

Subgroups: 404 in 238 conjugacy classes, 100 normal (34 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₄², C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂.C₄², C₂.C₄², C₂×C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₄².C₂, C₂³×C₄, C₄²⋊₄C₄, C₂³.₈Q₈, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₅C₂³, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂³.₄Q₈, C₂³.₈₃C₂³, C₂×C₄²⋊C₂, C₂×C₄².C₂, C₄².₁₈₅D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2_-¹⁺⁴, C₂².₁₉C₂⁴, C₂².₂₆C₂⁴, C₂³.₃₈C₂³, C₂².₄₆C₂⁴, C₄².₁₈₅D₄

Smallest permutation representation of C₄².₁₈₅D₄
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	4A	···	4H	4I	···	4X	4Y	4Z	4AA	4AB
order	1	2	···	2	2	2	4	···	4	4	···	4	4	4	4	4
size	1	1	···	1	4	4	2	···	2	4	···	4	8	8	8	8

38 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2_-¹⁺⁴

kernel

C₄².₁₈₅D₄

C₄²⋊₄C₄

C₂³.₈Q₈

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂³.₄Q₈

C₂³.₈₃C₂³

C₂×C₄²⋊C₂

C₂×C₄².C₂

C₄²

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₄².₁₈₅D₄ ►in GL₆(𝔽₅)

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

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

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

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

C₄².₁₈₅D₄ in GAP, Magma, Sage, TeX

C_4^2._{185}D_4

% in TeX

G:=Group("C4^2.185D4");

// GroupNames label

G:=SmallGroup(128,1343);

// by ID

G=gap.SmallGroup(128,1343);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,568,758,723,675,248]);

// Polycyclic

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

// generators/relations

G = C42.185D4 order 128 = 27

167th non-split extension by C42 of D4 acting via D4/C2=C22

G = C₄².₁₈₅D₄ order 128 = 2⁷

167^th non-split extension by C₄² of D₄ acting via D₄/C₂=C₂²