C4^2.389C2^3 - GroupNames

G = C₄².₃₈₉C₂³ order 128 = 2⁷

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

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

Aliases: C₄².₃₈₉C₂³, C₄⋊C₄.₂₄₉D₄, (C₄×Q₁₆)⋊₄₁C₂, C₄⋊Q₁₆⋊₂₀C₂, C₈.₁₂(C₄○D₄), C₂²⋊C₄.₈₉D₄, C₂.₂₅(Q₈○D₈), C₈.₅Q₈⋊₁₉C₂, C₈.D₄.₄C₂, C₂³.₈₆(C₂×D₄), Q₁₆⋊C₄⋊₁₄C₂, C₄⋊C₄.₁₁₆C₂³, (C₄×C₈).₂₂₅C₂², (C₂×C₄).₃₇₅C₂⁴, (C₂×C₈).₅₆₈C₂³, C₄⋊Q₈.₁₁₇C₂², (C₄×Q₈).₉₂C₂², C₄.Q₈.₂₇C₂², C₈.₁₈D₄.₁₁C₂, (C₂×Q₁₆).₆₅C₂², (C₂×Q₈).₁₁₇C₂³, C₂.D₈.₁₈₆C₂², C₈⋊C₄.₁₃₂C₂², C₂²⋊Q₈.₃₆C₂², C₈○₂M₄(2).₁₁C₂, (C₂²×C₈).₃₀₄C₂², C₂².₆₃₅(C₂²×D₄), C₄².C₂.₂₁C₂², (C₂²×C₄).₁₀₅₅C₂³, Q₈⋊C₄.₂₀₈C₂², C₄²⋊C₂.₃₃₂C₂², C₄².₃₀C₂²⋊₂₁C₂, (C₂×M₄(2)).₂₈₅C₂², C₂².₃₅C₂⁴.₂C₂, C₂.₇₂(C₂².₂₆C₂⁴), C₄.₆₀(C₂×C₄○D₄), (C₂×C₄).₁₄₇(C₂×D₄), SmallGroup(128,1909)

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

Derived series

C₁ — C₂×C₄ — C₄².₃₈₉C₂³

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₄² — C₄×C₈ — C₈○₂M₄(2) — C₄².₃₈₉C₂³

Lower central

C₁ — C₂ — C₂×C₄ — C₄².₃₈₉C₂³

Upper central

C₁ — C₂² — C₄²⋊C₂ — C₄².₃₈₉C₂³

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₄².₃₈₉C₂³

Generators and relations for C₄².₃₈₉C₂³
G = < a,b,c,d,e | a⁴=b⁴=1, c²=d²=b², e²=cbc^-1=b^-1, ab=ba, ac=ca, dad^-1=ab², ae=ea, bd=db, be=eb, dcd^-1=a²c, ece^-1=b^-1c, de=ed >

Subgroups: 268 in 169 conjugacy classes, 88 normal (26 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, C₂×C₄, Q₈, C₂³, C₄², C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, M₄(2), Q₁₆, C₂²×C₄, C₂×Q₈, C₄×C₈, C₈⋊C₄, Q₈⋊C₄, C₄.Q₈, C₂.D₈, C₄²⋊C₂, C₄×Q₈, C₂²⋊Q₈, C₄².C₂, C₄².C₂, C₄²⋊₂C₂, C₄⋊Q₈, C₂²×C₈, C₂×M₄(2), C₂×Q₁₆, C₈○₂M₄(2), C₄×Q₁₆, Q₁₆⋊C₄, C₈.₁₈D₄, C₈.D₄, C₄².₃₀C₂², C₄⋊Q₁₆, C₈.₅Q₈, C₂².₃₅C₂⁴, C₄².₃₈₉C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, C₂².₂₆C₂⁴, Q₈○D₈, C₄².₃₈₉C₂³

Smallest permutation representation of C₄².₃₈₉C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

(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)

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

Q₈○D₈

kernel

C₄².₃₈₉C₂³

C₈○₂M₄(2)

C₄×Q₁₆

Q₁₆⋊C₄

C₈.₁₈D₄

C₈.D₄

C₄².₃₀C₂²

C₄⋊Q₁₆

C₈.₅Q₈

C₂².₃₅C₂⁴

C₂²⋊C₄

C₄⋊C₄

C₈

C₂

# reps

Matrix representation of C₄².₃₈₉C₂³ ►in GL₆(𝔽₁₇)

4	0	0	0	0	0
0	4	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	1	0	0	0
0	0	0	1	0	0

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	15	0	0	0	0
16	4	0	0	0	0
0	0	0	0	12	12
0	0	0	0	12	5
0	0	12	12	0	0
0	0	12	5	0	0

1	0	0	0	0	0
13	16	0	0	0	0
0	0	0	16	0	0
0	0	1	0	0	0
0	0	0	0	0	1
0	0	0	0	16	0

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

G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[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],[13,16,0,0,0,0,15,4,0,0,0,0,0,0,0,0,12,12,0,0,0,0,12,5,0,0,12,12,0,0,0,0,12,5,0,0],[1,13,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,14,14,0,0,0,0,3,14,0,0,0,0,0,0,14,14,0,0,0,0,3,14] >;

C₄².₃₈₉C₂³ in GAP, Magma, Sage, TeX

C_4^2._{389}C_2^3

% in TeX

G:=Group("C4^2.389C2^3");

// GroupNames label

G:=SmallGroup(128,1909);

// by ID

G=gap.SmallGroup(128,1909);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,520,1018,80,4037,124]);

// Polycyclic

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

// generators/relations

G = C42.389C23 order 128 = 27

250th non-split extension by C42 of C23 acting via C23/C2=C22

G = C₄².₃₈₉C₂³ order 128 = 2⁷

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