C2^2.96C2^5 - GroupNames

G = C₂².₉₆C₂⁵ order 128 = 2⁷

77^th central stem extension by C₂² of C₂⁵

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

Aliases: C₂².₉₆C₂⁵, C₄².₈₈C₂³, C₂³.₁₃₉C₂⁴, C₄.₄₁2_-¹⁺⁴, Q₈⋊₃Q₈⋊₁₈C₂, D₄⋊₃Q₈⋊₂₃C₂, (C₂×C₄).₈₆C₂⁴, C₄⋊C₄.₅₂₀C₂³, C₄⋊Q₈.₃₄₄C₂², (C₂×D₄).₄₇₃C₂³, (C₄×D₄).₂₃₅C₂², (C₄×Q₈).₂₂₂C₂², (C₂×Q₈).₄₅₁C₂³, C₄⋊D₄.₂₄₂C₂², C₄⋊₁D₄.₁₈₆C₂², C₂²⋊C₄.₁₀₆C₂³, C₄²⋊₂C₂.₁C₂², (C₂×C₄²).₉₄₉C₂², (C₂²×C₄).₃₆₇C₂³, C₂²⋊Q₈.₁₁₇C₂², C₂.₂₆(C₂×2_-¹⁺⁴), C₂.₃₀(C₂.C₂⁵), C₂².₃₃C₂⁴⋊₅C₂, C₄.₄D₄.₁₇₆C₂², C₄².C₂.₁₅₃C₂², (C₂²×Q₈).₅₀₀C₂², C₂².D₄.₈C₂², C₂³.₃₈C₂³⋊₂₄C₂, C₄²⋊C₂.₂₂₉C₂², C₂³.₃₆C₂³⋊₃₀C₂, C₂².₅₃C₂⁴⋊₁₂C₂, C₂³.₃₇C₂³⋊₃₉C₂, C₂³.₃₃C₂³⋊₂₂C₂, C₂².₅₀C₂⁴⋊₂₁C₂, C₂².₄₆C₂⁴⋊₁₇C₂, C₂².₂₆C₂⁴.₅₀C₂, (C₂×C₄×Q₈)⋊₆₀C₂, C₄.₁₇₉(C₂×C₄○D₄), C₂².₁₇(C₂×C₄○D₄), C₂.₅₂(C₂²×C₄○D₄), (C₂×C₄).₃₀₆(C₄○D₄), (C₂×C₄⋊C₄).₇₀₆C₂², (C₂×C₄○D₄).₂₂₉C₂², SmallGroup(128,2239)

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

Derived series

C₁ — C₂² — C₂².₉₆C₂⁵

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₂×C₄×Q₈ — C₂².₉₆C₂⁵

Lower central

C₁ — C₂² — C₂².₉₆C₂⁵

Upper central

C₁ — C₂² — C₂².₉₆C₂⁵

Jennings

C₁ — C₂² — C₂².₉₆C₂⁵

Generators and relations for C₂².₉₆C₂⁵
G = < a,b,c,d,e,f,g | a²=b²=c²=g²=1, d²=ba=ab, e²=b, f²=a, dcd^-1=gcg=ac=ca, fdf^-1=ad=da, ae=ea, af=fa, ag=ga, ece^-1=bc=cb, bd=db, be=eb, bf=fb, bg=gb, cf=fc, de=ed, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 660 in 506 conjugacy classes, 390 normal (30 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₄², C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄○D₄, C₂×C₄², C₂×C₄², C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₄×D₄, C₄×Q₈, C₄⋊D₄, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₄².C₂, C₄²⋊₂C₂, C₄⋊₁D₄, C₄⋊Q₈, C₄⋊Q₈, C₂²×Q₈, C₂×C₄○D₄, C₂×C₄×Q₈, C₂³.₃₃C₂³, C₂³.₃₆C₂³, C₂².₂₆C₂⁴, C₂³.₃₇C₂³, C₂³.₃₈C₂³, C₂².₃₃C₂⁴, C₂².₄₆C₂⁴, D₄⋊₃Q₈, C₂².₅₀C₂⁴, Q₈⋊₃Q₈, C₂².₅₃C₂⁴, C₂².₉₆C₂⁵
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2_-¹⁺⁴, C₂⁵, C₂²×C₄○D₄, C₂×2_-¹⁺⁴, C₂.C₂⁵, C₂².₉₆C₂⁵

Smallest permutation representation of C₂².₉₆C₂⁵
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	4A	···	4P	4Q	···	4AH
order	1	2	2	2	2	2	2	2	2	2	4	···	4	4	···	4
size	1	1	1	1	2	2	4	4	4	4	2	···	2	4	···	4

44 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2_-¹⁺⁴

C₂.C₂⁵

kernel

C₂².₉₆C₂⁵

C₂×C₄×Q₈

C₂³.₃₃C₂³

C₂³.₃₆C₂³

C₂².₂₆C₂⁴

C₂³.₃₇C₂³

C₂³.₃₈C₂³

C₂².₃₃C₂⁴

C₂².₄₆C₂⁴

D₄⋊₃Q₈

C₂².₅₀C₂⁴

Q₈⋊₃Q₈

C₂².₅₃C₂⁴

C₂×C₄

C₄

C₂

# reps

Matrix representation of C₂².₉₆C₂⁵ ►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	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	1	0	0	0	0
0	0	0	3	0	0
0	0	2	0	0	0
0	0	1	1	3	4
0	0	0	1	3	2

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

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

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

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

C₂².₉₆C₂⁵ in GAP, Magma, Sage, TeX

C_2^2._{96}C_2^5

% in TeX

G:=Group("C2^2.96C2^5");

// GroupNames label

G:=SmallGroup(128,2239);

// by ID

G=gap.SmallGroup(128,2239);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,680,1430,352,570,136,1684]);

// Polycyclic

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

// generators/relations

G = C22.96C25 order 128 = 27

77th central stem extension by C22 of C25

G = C₂².₉₆C₂⁵ order 128 = 2⁷

77^th central stem extension by C₂² of C₂⁵