C2^3.675C2^4 - GroupNames

G = C₂³.₆₇₅C₂⁴ order 128 = 2⁷

392^nd central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₈₆C₂³, C₂³.₆₇₅C₂⁴, C₂².₄₄₈2₊¹⁺⁴, C₂².₃₄₁2_-¹⁺⁴, C₄²⋊₅C₄⋊₃₅C₂, C₄²⋊₈C₄⋊₆₇C₂, C₂³⋊Q₈.₂₈C₂, (C₂²×C₄).₅₉₁C₂³, (C₂×C₄²).₇₀₆C₂², C₂³.Q₈.₃₈C₂, C₂³.₁₁D₄.₅₃C₂, (C₂²×Q₈).₂₁₇C₂², C₂³.₇₈C₂³⋊₅₈C₂, C₂.₉₇(C₂².₃₂C₂⁴), C₂⁴.C₂².₇₃C₂, C₂³.₆₃C₂³⋊₁₇₈C₂, C₂³.₆₅C₂³⋊₁₄₉C₂, C₂³.₆₇C₂³⋊₁₀₁C₂, C₂³.₈₃C₂³⋊₁₁₂C₂, C₂.C₄².₃₇₉C₂², C₂.₉₈(C₂².₃₃C₂⁴), C₂.₆₅(C₂².₅₀C₂⁴), C₂.₄₂(C₂².₄₉C₂⁴), C₂.₄₆(C₂².₅₇C₂⁴), C₂.₁₁₃(C₂².₃₆C₂⁴), C₂.₁₁₃(C₂².₄₆C₂⁴), (C₂×C₄).₂₂₄(C₄○D₄), (C₂×C₄⋊C₄).₄₈₅C₂², C₂².₅₃₆(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₃₁₄C₂², SmallGroup(128,1507)

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

Derived series

C₁ — C₂³ — C₂³.₆₇₅C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₂³.₆₅C₂³ — C₂³.₆₇₅C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₇₅C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₇₅C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₇₅C₂⁴

Generators and relations for C₂³.₆₇₅C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=1, d²=cb=bc, e²=b, f²=ba=ab, g²=a, 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, C₄²⋊₈C₄, C₄²⋊₅C₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₅C₂³, C₂³.₆₇C₂³, C₂³⋊Q₈, C₂³.₇₈C₂³, C₂³.Q₈, C₂³.₁₁D₄, C₂³.₈₃C₂³, C₂³.₆₇₅C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₂C₂⁴, C₂².₃₃C₂⁴, C₂².₃₆C₂⁴, C₂².₄₆C₂⁴, C₂².₄₉C₂⁴, C₂².₅₀C₂⁴, C₂².₅₇C₂⁴, C₂³.₆₇₅C₂⁴

Smallest permutation representation of C₂³.₆₇₅C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

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

C₂³.₆₇₅C₂⁴

C₄²⋊₈C₄

C₄²⋊₅C₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂³.₆₇C₂³

C₂³⋊Q₈

C₂³.₇₈C₂³

C₂³.Q₈

C₂³.₁₁D₄

C₂³.₈₃C₂³

C₂×C₄

C₂²

# reps

Matrix representation of C₂³.₆₇₅C₂⁴ ►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	3	0	0	0	0
1	4	0	0	0	0
0	0	0	3	0	0
0	0	2	0	0	0
0	0	0	0	3	0
0	0	0	0	0	3

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

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

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

C₂³.₆₇₅C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{675}C_2^4

% in TeX

G:=Group("C2^3.675C2^4");

// GroupNames label

G:=SmallGroup(128,1507);

// by ID

G=gap.SmallGroup(128,1507);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,672,253,232,758,723,268,1571,346,80]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=c*b=b*c,e^2=b,f^2=b*a=a*b,g^2=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 = C23.675C24 order 128 = 27

392nd central stem extension by C23 of C24

G = C₂³.₆₇₅C₂⁴ order 128 = 2⁷

392^nd central stem extension by C₂³ of C₂⁴