C2^3.595C2^4 - GroupNames

G = C₂³.₅₉₅C₂⁴ order 128 = 2⁷

312^nd central stem extension by C₂³ of C₂⁴

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

Aliases: C₂³.₅₉₅C₂⁴, C₂⁴.₄₀₂C₂³, C₂².₃₆₉2₊¹⁺⁴, C₂².₂₇₅2_-¹⁺⁴, C₂²⋊C₄.₁₅D₄, C₂³.₆₈(C₂×D₄), C₂.₆₁(D₄⋊₆D₄), C₂.₁₀₀(D₄⋊₅D₄), C₂³.Q₈⋊₆₀C₂, C₂³.₇Q₈⋊₈₉C₂, C₂³.₁₇₃(C₄○D₄), C₂³.₁₁D₄⋊₈₄C₂, (C₂³×C₄).₄₅₈C₂², (C₂×C₄²).₆₄₈C₂², (C₂²×C₄).₅₅₉C₂³, C₂³.₈Q₈⋊₁₀₇C₂, C₂².₄₀₄(C₂²×D₄), C₂³.₂₃D₄.₅₂C₂, C₂³.₁₀D₄.₄₂C₂, (C₂²×D₄).₂₃₂C₂², C₂³.₈₁C₂³⋊₈₅C₂, C₂³.₈₃C₂³⋊₇₉C₂, C₂⁴.C₂²⋊₁₂₇C₂, C₂.₆₅(C₂².₃₂C₂⁴), C₂³.₆₅C₂³⋊₁₂₀C₂, C₂.C₄².₃₀₂C₂², C₂.₄₅(C₂².₃₁C₂⁴), C₂.₆₅(C₂².₃₃C₂⁴), C₂.₁₃(C₂².₅₇C₂⁴), C₂.₈₃(C₂².₄₆C₂⁴), (C₂×C₄).₄₂₁(C₂×D₄), (C₂×C₄²⋊₂C₂)⋊₁₉C₂, (C₂×C₄).₄₂₄(C₄○D₄), (C₂×C₄⋊C₄).₄₀₉C₂², C₂².₄₅₇(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₂₆₂C₂², (C₂×C₂².D₄).₂₆C₂, SmallGroup(128,1427)

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

Derived series

C₁ — C₂³ — C₂³.₅₉₅C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂³.₇Q₈ — C₂³.₅₉₅C₂⁴

Lower central

C₁ — C₂³ — C₂³.₅₉₅C₂⁴

Upper central

C₁ — C₂³ — C₂³.₅₉₅C₂⁴

Jennings

C₁ — C₂³ — C₂³.₅₉₅C₂⁴

Generators and relations for C₂³.₅₉₅C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=ba=ab, f²=g²=a, 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: 500 in 252 conjugacy classes, 96 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂².D₄, C₄²⋊₂C₂, C₂³×C₄, C₂²×D₄, C₂³.₇Q₈, C₂³.₈Q₈, C₂³.₂₃D₄, C₂⁴.C₂², C₂³.₆₅C₂³, C₂³.₁₀D₄, C₂³.Q₈, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂³.₈₃C₂³, C₂×C₂².D₄, C₂×C₄²⋊₂C₂, C₂³.₅₉₅C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₁C₂⁴, C₂².₃₂C₂⁴, C₂².₃₃C₂⁴, D₄⋊₅D₄, D₄⋊₆D₄, C₂².₄₆C₂⁴, C₂².₅₇C₂⁴, C₂³.₅₉₅C₂⁴

Smallest permutation representation of C₂³.₅₉₅C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	···	4N	4O	···	4T
order	1	2	···	2	2	2	2	2	4	···	4	4	···	4
size	1	1	···	1	4	4	4	4	4	···	4	8	···	8

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂³.₅₉₅C₂⁴

C₂³.₇Q₈

C₂³.₈Q₈

C₂³.₂₃D₄

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂³.₁₀D₄

C₂³.Q₈

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂³.₈₃C₂³

C₂×C₂².D₄

C₂×C₄²⋊₂C₂

C₂²⋊C₄

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₅₉₅C₂⁴ ►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

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

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

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

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

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

C₂³.₅₉₅C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{595}C_2^4

% in TeX

G:=Group("C2^3.595C2^4");

// GroupNames label

G:=SmallGroup(128,1427);

// by ID

G=gap.SmallGroup(128,1427);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,344,758,723,100,1571,346,80]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=b*a=a*b,f^2=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,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 = C23.595C24 order 128 = 27

312nd central stem extension by C23 of C24

G = C₂³.₅₉₅C₂⁴ order 128 = 2⁷

312^nd central stem extension by C₂³ of C₂⁴