C2^4.389C2^3 - GroupNames

G = C₂⁴.₃₈₉C₂³ order 128 = 2⁷

229^th non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²

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

Aliases: C₂⁴.₃₈₉C₂³, C₂³.₅₈₂C₂⁴, C₂².₂₆₅2_-¹⁺⁴, C₂².₃₅₆2₊¹⁺⁴, C₂.₄₆D₄², C₄⋊C₄⋊₁₁D₄, (C₂×D₄)⋊₁₇D₄, C₂³⋊₂D₄⋊₃₉C₂, C₂³.₂₀₈(C₂×D₄), C₂.₅₈(D₄⋊₆D₄), C₂.₂₈(Q₈⋊₆D₄), C₂³.Q₈⋊₅₃C₂, C₂³.₈Q₈⋊₉₉C₂, C₂³.₁₀D₄⋊₇₈C₂, C₂.₄₁(C₂³⋊₃D₄), (C₂×C₄²).₆₃₉C₂², (C₂³×C₄).₄₅₀C₂², (C₂²×C₄).₁₇₈C₂³, C₂².₃₉₁(C₂²×D₄), C₂⁴.₃C₂²⋊₇₆C₂, (C₂²×D₄).₂₂₁C₂², C₂.₆₀(C₂².₃₂C₂⁴), C₂³.₆₅C₂³⋊₁₁₆C₂, C₂.C₄².₂₉₁C₂², C₂.₁₀(C₂².₅₆C₂⁴), C₂.₄₁(C₂².₃₁C₂⁴), (C₂×C₄).₉₀(C₂×D₄), (C₂×C₄⋊D₄)⋊₃₄C₂, (C₂×C₄).₁₉₀(C₄○D₄), (C₂×C₄⋊C₄).₃₉₈C₂², C₂².₄₄₅(C₂×C₄○D₄), (C₂×C₂².D₄)⋊₃₃C₂, (C₂×C₂²⋊C₄).₂₅₁C₂², SmallGroup(128,1414)

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

Derived series

C₁ — C₂³ — C₂⁴.₃₈₉C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂×C₂².D₄ — C₂⁴.₃₈₉C₂³

Lower central

C₁ — C₂³ — C₂⁴.₃₈₉C₂³

Upper central

C₁ — C₂³ — C₂⁴.₃₈₉C₂³

Jennings

C₁ — C₂³ — C₂⁴.₃₈₉C₂³

Generators and relations for C₂⁴.₃₈₉C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=f²=b, g²=cb=bc, ab=ba, faf^-1=ac=ca, ad=da, ae=ea, gag^-1=abc, bd=db, geg^-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, gfg^-1=cf=fc, cg=gc, fef^-1=de=ed, df=fd, dg=gd >

Subgroups: 740 in 334 conjugacy classes, 104 normal (30 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄⋊D₄, C₂².D₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂³.₈Q₈, C₂³.₆₅C₂³, C₂⁴.₃C₂², C₂³⋊₂D₄, C₂³.₁₀D₄, C₂³.Q₈, C₂×C₄⋊D₄, C₂×C₂².D₄, C₂⁴.₃₈₉C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂³⋊₃D₄, C₂².₃₁C₂⁴, C₂².₃₂C₂⁴, D₄², D₄⋊₆D₄, Q₈⋊₆D₄, C₂².₅₆C₂⁴, C₂⁴.₃₈₉C₂³

Smallest permutation representation of C₂⁴.₃₈₉C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	2L	2M	4A	···	4N	4O	4P	4Q	4R
order	1	2	···	2	2	2	2	2	2	2	4	···	4	4	4	4	4
size	1	1	···	1	4	4	4	4	8	8	4	···	4	8	8	8	8

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂⁴.₃₈₉C₂³

C₂³.₈Q₈

C₂³.₆₅C₂³

C₂⁴.₃C₂²

C₂³⋊₂D₄

C₂³.₁₀D₄

C₂³.Q₈

C₂×C₄⋊D₄

C₂×C₂².D₄

C₄⋊C₄

C₂×D₄

C₂×C₄

C₂²

# reps

Matrix representation of C₂⁴.₃₈₉C₂³ ►in GL₆(𝔽₅)

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

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

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

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

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

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

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

C₂⁴.₃₈₉C₂³ in GAP, Magma, Sage, TeX

C_2^4._{389}C_2^3

% in TeX

G:=Group("C2^4.389C2^3");

// GroupNames label

G:=SmallGroup(128,1414);

// by ID

G=gap.SmallGroup(128,1414);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,120,758,723,100,1571,346]);

// Polycyclic

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

// generators/relations

G = C24.389C23 order 128 = 27

229th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₃₈₉C₂³ order 128 = 2⁷

229^th non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²