C2^3.696C2^4 - GroupNames

G = C₂³.₆₉₆C₂⁴ order 128 = 2⁷

413^rd central stem extension by C₂³ of C₂⁴

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

Aliases: C₂³.₆₉₆C₂⁴, C₂⁴.₄₅₅C₂³, C₂².₄₆₉2₊¹⁺⁴, C₂².₃₅₉2_-¹⁺⁴, C₂³.₄Q₈⋊₆₂C₂, C₂³.₁₀₂(C₄○D₄), (C₂³×C₄).₁₇₆C₂², (C₂²×C₄).₆₀₆C₂³, (C₂×C₄²).₁₁₂C₂², C₂³.₈Q₈⋊₁₃₉C₂, C₂³.₁₁D₄⋊₁₂₁C₂, C₂³.₁₀D₄.₆₇C₂, C₂³.₂₃D₄.₇₆C₂, (C₂²×D₄).₂₈₄C₂², C₂⁴.C₂²⋊₁₇₂C₂, C₂⁴.₃C₂².₇₆C₂, C₂³.₆₅C₂³⋊₁₅₇C₂, C₂³.₈₃C₂³⋊₁₂₆C₂, C₂³.₆₃C₂³⋊₁₉₁C₂, C₂.₃₇(C₂².₅₄C₂⁴), C₂.C₄².₄₀₀C₂², C₂.₁₂₁(C₂².₄₅C₂⁴), C₂.₄₃(C₂².₅₃C₂⁴), C₂.₁₁₉(C₂².₃₆C₂⁴), C₂.₁₁₈(C₂².₄₇C₂⁴), C₂.₁₁₄(C₂².₃₃C₂⁴), (C₂×C₄).₂₃₇(C₄○D₄), (C₂×C₄⋊C₄).₅₀₆C₂², C₂².₅₅₇(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₇₈C₂², SmallGroup(128,1528)

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

Derived series

C₁ — C₂³ — C₂³.₆₉₆C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂³.₂₃D₄ — C₂³.₆₉₆C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₉₆C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₉₆C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₉₆C₂⁴

Generators and relations for C₂³.₆₉₆C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=d, f²=b, g²=bcd, faf^-1=ab=ba, ac=ca, ad=da, eae^-1=abc, ag=ga, bc=cb, bd=db, fef^-1=be=eb, bf=fb, bg=gb, cd=dc, geg^-1=ce=ec, cf=fc, cg=gc, de=ed, gfg^-1=df=fd, dg=gd >

Subgroups: 452 in 217 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, C₂³.₆₅C₂³, C₂⁴.₃C₂², C₂³.₁₀D₄, C₂³.₁₁D₄, C₂³.₄Q₈, C₂³.₈₃C₂³, C₂³.₆₉₆C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₃C₂⁴, C₂².₃₆C₂⁴, C₂².₄₅C₂⁴, C₂².₄₇C₂⁴, C₂².₅₃C₂⁴, C₂².₅₄C₂⁴, C₂³.₆₉₆C₂⁴

Smallest permutation representation of C₂³.₆₉₆C₂⁴
►On 64 points

Generators in S₆₄

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

(1 9)(2 10)(3 11)(4 12)(5 38)(6 39)(7 40)(8 37)(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 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 63)(34 64)(35 61)(36 62)

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	4A	···	4P	4Q	···	4U
order	1	2	···	2	2	2	2	4	···	4	4	···	4
size	1	1	···	1	4	4	8	4	···	4	8	···	8

32 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂³.₆₉₆C₂⁴

C₂³.₈Q₈

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂⁴.₃C₂²

C₂³.₁₀D₄

C₂³.₁₁D₄

C₂³.₄Q₈

C₂³.₈₃C₂³

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₆₉₆C₂⁴ ►in GL₆(𝔽₅)

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

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	0	1	0	0
0	0	0	0	3	0
0	0	0	0	0	3

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	1	0
0	0	0	0	0	4

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

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

C₂³.₆₉₆C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{696}C_2^4

% in TeX

G:=Group("C2^3.696C2^4");

// GroupNames label

G:=SmallGroup(128,1528);

// by ID

G=gap.SmallGroup(128,1528);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,723,604,1571,346]);

// Polycyclic

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

// generators/relations

G = C23.696C24 order 128 = 27

413rd central stem extension by C23 of C24

G = C₂³.₆₉₆C₂⁴ order 128 = 2⁷

413^rd central stem extension by C₂³ of C₂⁴