C2^3.697C2^4 - GroupNames

G = C₂³.₆₉₇C₂⁴ order 128 = 2⁷

414^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₉₄C₂³, C₂³.₆₉₇C₂⁴, C₂².₄₇₀2₊¹⁺⁴, C₂³⋊₂D₄.₃₄C₂, C₂³.₁₀₃(C₄○D₄), (C₂²×C₄).₆₀₇C₂³, (C₂×C₄²).₇₂₀C₂², (C₂³×C₄).₁₇₇C₂², C₂³.₈Q₈⋊₁₄₀C₂, C₂³.₁₁D₄⋊₁₂₂C₂, C₂³.₂₃D₄⋊₁₀₇C₂, C₂³.₁₀D₄⋊₁₀₄C₂, C₂⁴.₃C₂²⋊₉₅C₂, (C₂²×D₄).₂₈₅C₂², C₂⁴.C₂²⋊₁₇₃C₂, C₂³.₆₃C₂³⋊₁₉₂C₂, C₂.₁₀₆(C₂².₃₂C₂⁴), C₂.₃₈(C₂².₅₄C₂⁴), C₂.C₄².₄₀₁C₂², C₂.₁₂₂(C₂².₄₅C₂⁴), C₂.₆₅(C₂².₃₄C₂⁴), C₂.₄₄(C₂².₅₃C₂⁴), C₂.₁₁₉(C₂².₄₇C₂⁴), (C₂×C₄).₂₃₈(C₄○D₄), (C₂×C₄⋊C₄).₅₀₇C₂², C₂².₅₅₈(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₇₉C₂², SmallGroup(128,1529)

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

Derived series

C₁ — C₂³ — C₂³.₆₉₇C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂³.₂₃D₄ — C₂³.₆₉₇C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₉₇C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₉₇C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₉₇C₂⁴

Generators and relations for C₂³.₆₉₇C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=g²=1, d²=ca=ac, e²=a, f²=abc, ab=ba, ede^-1=ad=da, geg=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=abd, fg=gf >

Subgroups: 548 in 242 conjugacy classes, 88 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂³×C₄, C₂²×D₄, C₂³.₈Q₈, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂⁴.₃C₂², C₂³⋊₂D₄, C₂³.₁₀D₄, C₂³.₁₁D₄, C₂³.₆₉₇C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, C₂².₃₂C₂⁴, C₂².₃₄C₂⁴, C₂².₄₅C₂⁴, C₂².₄₇C₂⁴, C₂².₅₃C₂⁴, C₂².₅₄C₂⁴, C₂³.₆₉₇C₂⁴

Smallest permutation representation of C₂³.₆₉₇C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

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

C₄○D₄

2₊¹⁺⁴

kernel

C₂³.₆₉₇C₂⁴

C₂³.₈Q₈

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂⁴.₃C₂²

C₂³⋊₂D₄

C₂³.₁₀D₄

C₂³.₁₁D₄

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₆₉₇C₂⁴ ►in GL₆(𝔽₅)

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

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

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

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

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

4	3	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	2
0	0	0	0	0	4

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

C₂³.₆₉₇C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{697}C_2^4

% in TeX

G:=Group("C2^3.697C2^4");

// GroupNames label

G:=SmallGroup(128,1529);

// by ID

G=gap.SmallGroup(128,1529);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,120,758,723,604,1571,346]);

// Polycyclic

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

// generators/relations

G = C23.697C24 order 128 = 27

414th central stem extension by C23 of C24

G = C₂³.₆₉₇C₂⁴ order 128 = 2⁷

414^th central stem extension by C₂³ of C₂⁴