C4^2.367C2^3 - GroupNames

G = C₄².₃₆₇C₂³ order 128 = 2⁷

228^th non-split extension by C₄² of C₂³ acting via C₂³/C₂=C₂²

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

Aliases: C₄².₃₆₇C₂³, C₄⋊C₄.₂₃₉D₄, C₄⋊C₄.₉₀C₂³, C₂²⋊C₄.₇₉D₄, C₂.₂₁(Q₈○D₈), (C₄×C₈).₃₅₀C₂², (C₂×C₄).₃₃₅C₂⁴, (C₂×C₈).₄₅₆C₂³, C₄.SD₁₆⋊₄₀C₂, C₂³.₄₅₃(C₂×D₄), C₄⋊Q₈.₁₁₀C₂², (C₂×Q₈).₉₁C₂³, C₈○₂M₄(2)⋊₃₈C₂, C₂.₃₄(D₄○SD₁₆), C₄.₄₆(C₄.₄D₄), (C₂×D₄).₁₀₃C₂³, C₈⋊C₄.₁₆₈C₂², C₂³.₃₈D₄⋊₃₆C₂, (C₂²×C₈).₄₆₂C₂², C₄.₄D₄.₃₂C₂², C₂².₅₉₅(C₂²×D₄), C₄².C₂.₁₆C₂², D₄⋊C₄.₁₃₂C₂², C₂³.₄₁C₂³⋊₇C₂, (C₂²×C₄).₁₀₃₃C₂³, Q₈⋊C₄.₂₀₃C₂², C₂³.₂₄D₄.₁₂C₂, C₂³.₃₆D₄.₁₄C₂, C₂².₁₆(C₄.₄D₄), (C₂²×Q₈).₃₀₁C₂², C₄².₇₈C₂²⋊₂₆C₂, C₄².₂₈C₂²⋊₃₀C₂, C₄².₃₀C₂²⋊₁₇C₂, C₄²⋊C₂.₁₄₀C₂², (C₂×M₄(2)).₃₇₂C₂², C₂³.₃₈C₂³.₁₃C₂, C₄.₄₄(C₂×C₄○D₄), (C₂×C₄).₁₃₇(C₂×D₄), (C₂×Q₈⋊C₄)⋊₅₈C₂, C₂.₄₆(C₂×C₄.₄D₄), (C₂×C₄).₄₉₀(C₄○D₄), (C₂×C₄⋊C₄).₆₂₅C₂², (C₂×C₄○D₄).₁₅₀C₂², SmallGroup(128,1869)

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

Derived series

C₁ — C₂×C₄ — C₄².₃₆₇C₂³

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂²×C₈ — C₈○₂M₄(2) — C₄².₃₆₇C₂³

Lower central

C₁ — C₂ — C₂×C₄ — C₄².₃₆₇C₂³

Upper central

C₁ — C₂² — C₄²⋊C₂ — C₄².₃₆₇C₂³

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₄².₃₆₇C₂³

Generators and relations for C₄².₃₆₇C₂³
G = < a,b,c,d,e | a⁴=b⁴=1, c²=b², d²=a²b^-1, e²=a²b², ab=ba, cac^-1=a^-1b², ad=da, eae^-1=ab², cbc^-1=b^-1, bd=db, be=eb, dcd^-1=bc, ece^-1=a²b²c, de=ed >

Subgroups: 340 in 186 conjugacy classes, 92 normal (38 characteristic)
C₁, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₄², C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂×Q₈, C₄○D₄, C₄×C₈, C₈⋊C₄, D₄⋊C₄, D₄⋊C₄, Q₈⋊C₄, Q₈⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₄².C₂, C₄².C₂, C₄⋊Q₈, C₄⋊Q₈, C₂²×C₈, C₂×M₄(2), C₂²×Q₈, C₂×C₄○D₄, C₈○₂M₄(2), C₂×Q₈⋊C₄, C₂³.₂₄D₄, C₂³.₃₆D₄, C₂³.₃₈D₄, C₄.SD₁₆, C₄².₇₈C₂², C₄².₂₈C₂², C₄².₃₀C₂², C₂³.₃₈C₂³, C₂³.₄₁C₂³, C₄².₃₆₇C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₄.₄D₄, C₂²×D₄, C₂×C₄○D₄, C₂×C₄.₄D₄, D₄○SD₁₆, Q₈○D₈, C₄².₃₆₇C₂³

Smallest permutation representation of C₄².₃₆₇C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

D₄○SD₁₆

Q₈○D₈

kernel

C₄².₃₆₇C₂³

C₈○₂M₄(2)

C₂×Q₈⋊C₄

C₂³.₂₄D₄

C₂³.₃₆D₄

C₂³.₃₈D₄

C₄.SD₁₆

C₄².₇₈C₂²

C₄².₂₈C₂²

C₄².₃₀C₂²

C₂³.₃₈C₂³

C₂³.₄₁C₂³

C₂²⋊C₄

C₄⋊C₄

C₂×C₄

C₂

# reps

Matrix representation of C₄².₃₆₇C₂³ ►in GL₆(𝔽₁₇)

13	8	0	0	0	0
0	4	0	0	0	0
0	0	4	0	0	0
0	0	0	4	0	0
0	0	0	0	13	0
0	0	0	0	0	13

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

13	8	0	0	0	0
13	4	0	0	0	0
0	0	13	0	0	0
0	0	0	4	0	0
0	0	0	0	13	0
0	0	0	0	0	4

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

4	9	0	0	0	0
0	13	0	0	0	0
0	0	0	0	13	0
0	0	0	0	0	13
0	0	4	0	0	0
0	0	0	4	0	0

G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,8,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[13,13,0,0,0,0,8,4,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,3,14,0,0,0,0,3,3,0,0,0,0,0,0,3,14,0,0,0,0,3,3],[4,0,0,0,0,0,9,13,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,13,0,0,0,0,0,0,13,0,0] >;

C₄².₃₆₇C₂³ in GAP, Magma, Sage, TeX

C_4^2._{367}C_2^3

% in TeX

G:=Group("C4^2.367C2^3");

// GroupNames label

G:=SmallGroup(128,1869);

// by ID

G=gap.SmallGroup(128,1869);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,232,758,100,1018,2804,172,4037,124]);

// Polycyclic

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

// generators/relations

G = C42.367C23 order 128 = 27

228th non-split extension by C42 of C23 acting via C23/C2=C22

G = C₄².₃₆₇C₂³ order 128 = 2⁷

228^th non-split extension by C₄² of C₂³ acting via C₂³/C₂=C₂²