C2^3.612C2^4 - GroupNames

G = C₂³.₆₁₂C₂⁴ order 128 = 2⁷

329^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₆₈C₂³, C₂³.₆₁₂C₂⁴, C₂².₃₈₆2₊¹⁺⁴, C₂².₂₈₈2_-¹⁺⁴, C₂²⋊C₄⋊₁₉D₄, C₂³.₇₁(C₂×D₄), C₂³⋊Q₈⋊₄₉C₂, C₂³⋊₂D₄.₂₆C₂, C₂.₁₁₇(D₄⋊₅D₄), C₂³.₂₃D₄⋊₉₈C₂, C₂³.₁₀D₄⋊₉₅C₂, C₂³.₁₁D₄⋊₉₄C₂, (C₂²×C₄).₁₉₀C₂³, (C₂×C₄²).₆₆₃C₂², (C₂³×C₄).₁₅₅C₂², C₂³.₈Q₈⋊₁₁₄C₂, C₂².₄₂₁(C₂²×D₄), C₂.₆(C₂⁴⋊C₂²), (C₂²×D₄).₂₄₇C₂², (C₂²×Q₈).₁₉₀C₂², C₂⁴.C₂²⋊₁₄₀C₂, C₂³.₆₇C₂³⋊₈₆C₂, C₂.C₄².₃₁₈C₂², C₂.₃₅(C₂².₅₃C₂⁴), C₂.₇₉(C₂².₃₆C₂⁴), C₂.₅₀(C₂².₃₁C₂⁴), (C₂×C₄).₄₂₅(C₂×D₄), (C₂×C₄.₄D₄)⋊₃₀C₂, (C₂×C₄).₁₉₉(C₄○D₄), (C₂×C₄⋊C₄).₄₂₅C₂², C₂².₄₇₄(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₂₇₈C₂², SmallGroup(128,1444)

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

Derived series

C₁ — C₂³ — C₂³.₆₁₂C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₂³.₆₇C₂³ — C₂³.₆₁₂C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₁₂C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₁₂C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₁₂C₂⁴

Generators and relations for C₂³.₆₁₂C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=g²=a, f²=b, ab=ba, ac=ca, ede^-1=ad=da, geg^-1=ae=ea, af=fa, ag=ga, bc=cb, fdf^-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef^-1=ce=ec, cf=fc, cg=gc, gdg^-1=abd, fg=gf >

Subgroups: 612 in 278 conjugacy classes, 96 normal (22 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂⁴, C₂⁴, C₂.C₄², C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄.₄D₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂²×Q₈, C₂³.₈Q₈, C₂³.₂₃D₄, C₂⁴.C₂², C₂³.₆₇C₂³, C₂³⋊₂D₄, C₂³⋊Q₈, C₂³.₁₀D₄, C₂³.₁₁D₄, C₂×C₄.₄D₄, C₂³.₆₁₂C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₁C₂⁴, C₂².₃₆C₂⁴, D₄⋊₅D₄, C₂².₅₃C₂⁴, C₂⁴⋊C₂², C₂³.₆₁₂C₂⁴

Smallest permutation representation of C₂³.₆₁₂C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	···	4P	4Q	4R	4S	4T
order	1	2	···	2	2	2	2	2	4	···	4	4	4	4	4
size	1	1	···	1	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

C₂³.₆₁₂C₂⁴

C₂³.₈Q₈

C₂³.₂₃D₄

C₂⁴.C₂²

C₂³.₆₇C₂³

C₂³⋊₂D₄

C₂³⋊Q₈

C₂³.₁₀D₄

C₂³.₁₁D₄

C₂×C₄.₄D₄

C₂²⋊C₄

C₂×C₄

C₂²

# reps

Matrix representation of C₂³.₆₁₂C₂⁴ ►in GL₆(𝔽₅)

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

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

3	2	0	0	0	0
0	2	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	1	0

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

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

C₂³.₆₁₂C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{612}C_2^4

% in TeX

G:=Group("C2^3.612C2^4");

// GroupNames label

G:=SmallGroup(128,1444);

// by ID

G=gap.SmallGroup(128,1444);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,120,758,723,184,1571,346,80]);

// Polycyclic

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

// generators/relations

G = C23.612C24 order 128 = 27

329th central stem extension by C23 of C24

G = C₂³.₆₁₂C₂⁴ order 128 = 2⁷

329^th central stem extension by C₂³ of C₂⁴