C2^3.698C2^4 - GroupNames

G = C₂³.₆₉₈C₂⁴ order 128 = 2⁷

415^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₉₅C₂³, C₂³.₆₉₈C₂⁴, C₂².₄₇₁2₊¹⁺⁴, C₂².₃₆₀2_-¹⁺⁴, C₂³⋊Q₈.₃₀C₂, (C₂×C₄²).₁₁₃C₂², (C₂²×C₄).₆₀₈C₂³, C₂³.₄Q₈.₂₉C₂, C₂³.₁₁D₄.₅₈C₂, (C₂²×Q₈).₂₂₃C₂², C₂⁴.C₂².₈₀C₂, C₂³.₈₃C₂³⋊₁₂₇C₂, C₂³.₆₃C₂³⋊₁₉₃C₂, C₂³.₆₅C₂³⋊₁₅₈C₂, C₂³.₆₇C₂³⋊₁₀₃C₂, C₂.C₄².₄₀₂C₂², C₂.₁₂₃(C₂².₄₅C₂⁴), C₂.₇₄(C₂².₅₀C₂⁴), C₂.₄₅(C₂².₅₃C₂⁴), C₂.₄₃(C₂².₅₆C₂⁴), C₂.₆₂(C₂².₃₅C₂⁴), C₂.₁₁₅(C₂².₃₃C₂⁴), C₂.₁₂₀(C₂².₃₆C₂⁴), (C₂×C₄).₂₃₉(C₄○D₄), (C₂×C₄⋊C₄).₅₀₈C₂², C₂².₅₅₉(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₃₂₇C₂², SmallGroup(128,1530)

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

Derived series

C₁ — C₂³ — C₂³.₆₉₈C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₂³.₆₇C₂³ — C₂³.₆₉₈C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₉₈C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₉₈C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₉₈C₂⁴

Generators and relations for C₂³.₆₉₈C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=1, d²=c, e²=g²=a, f²=cb=bc, ab=ba, ac=ca, ede^-1=ad=da, geg^-1=ae=ea, af=fa, ag=ga, 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: 372 in 193 conjugacy classes, 88 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂²×Q₈, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₅C₂³, C₂³.₆₇C₂³, C₂³⋊Q₈, C₂³.₁₁D₄, C₂³.₄Q₈, C₂³.₈₃C₂³, C₂³.₆₉₈C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₃C₂⁴, C₂².₃₅C₂⁴, C₂².₃₆C₂⁴, C₂².₄₅C₂⁴, C₂².₅₀C₂⁴, C₂².₅₃C₂⁴, C₂².₅₆C₂⁴, C₂³.₆₉₈C₂⁴

Smallest permutation representation of C₂³.₆₉₈C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	4A	···	4R	4S	···	4W
order	1	2	···	2	2	4	···	4	4	···	4
size	1	1	···	1	8	4	···	4	8	···	8

32 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂³.₆₉₈C₂⁴

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂³.₆₇C₂³

C₂³⋊Q₈

C₂³.₁₁D₄

C₂³.₄Q₈

C₂³.₈₃C₂³

C₂×C₄

C₂²

# reps

Matrix representation of C₂³.₆₉₈C₂⁴ ►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

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

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	3
0	0	0	0	2	0

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

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

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

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],[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],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,2,0,0,0,0,3,0],[1,0,0,0,0,0,3,4,0,0,0,0,0,0,0,2,0,0,0,0,2,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[3,3,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,3,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C₂³.₆₉₈C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{698}C_2^4

% in TeX

G:=Group("C2^3.698C2^4");

// GroupNames label

G:=SmallGroup(128,1530);

// by ID

G=gap.SmallGroup(128,1530);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,723,520,1571,346,80]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=c,e^2=g^2=a,f^2=c*b=b*c,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,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 = C23.698C24 order 128 = 27

415th central stem extension by C23 of C24

G = C₂³.₆₉₈C₂⁴ order 128 = 2⁷

415^th central stem extension by C₂³ of C₂⁴