C2^4.262C2^3 - GroupNames

G = C₂⁴.₂₆₂C₂³ order 128 = 2⁷

102^nd non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²

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

Aliases: C₂⁴.₂₆₂C₂³, C₂³.₃₃₀C₂⁴, C₂².₁₀₃2_-¹⁺⁴, C₂².₁₄₂2₊¹⁺⁴, (C₂×D₄)⋊₄₄D₄, (C₂²×C₄)⋊₂₅D₄, C₂³⋊Q₈⋊₅C₂, C₂³⋊₂D₄⋊₁₁C₂, C₂².₁C₂²≀C₂, C₂³.₁₅₉(C₂×D₄), C₂.₂₉(D₄⋊₅D₄), C₂.₁₉(D₄⋊₆D₄), C₂³.₈Q₈⋊₃₅C₂, C₂³.₁₀D₄⋊₂₀C₂, C₂³.₂₃D₄⋊₃₅C₂, (C₂²×C₄).₇₉₇C₂³, (C₂³×C₄).₃₄₃C₂², C₂².₂₁₀(C₂²×D₄), C₂³.₇₈C₂³⋊₅C₂, (C₂²×D₄).₁₂₇C₂², (C₂²×Q₈).₄₂₀C₂², C₂.C₄².₉₁C₂², C₂.₉(C₂².₃₁C₂⁴), C₂.₁₄(C₂².₂₆C₂⁴), (C₂×C₄⋊D₄)⋊₉C₂, (C₂×C₄)⋊₁₀(C₄○D₄), (C₂×C₄).₃₁₅(C₂×D₄), (C₂²×C₄○D₄)⋊₄C₂, C₂.₁₈(C₂×C₂²≀C₂), (C₂×C₄⋊C₄).₂₁₆C₂², C₂².₂₀₉(C₂×C₄○D₄), (C₂×C₂².D₄)⋊₉C₂, (C₂×C₂.C₄²)⋊₃₀C₂, (C₂×C₂²⋊C₄).₁₂₀C₂², SmallGroup(128,1162)

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

Derived series

C₁ — C₂³ — C₂⁴.₂₆₂C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂×C₂.C₄² — C₂⁴.₂₆₂C₂³

Lower central

C₁ — C₂³ — C₂⁴.₂₆₂C₂³

Upper central

C₁ — C₂³ — C₂⁴.₂₆₂C₂³

Jennings

C₁ — C₂³ — C₂⁴.₂₆₂C₂³

Generators and relations for C₂⁴.₂₆₂C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=e²=g²=1, f²=c, eae=gag=ab=ba, ac=ca, ad=da, faf^-1=acd, bc=cb, bd=db, geg=be=eb, bf=fb, bg=gb, cd=dc, fef^-1=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >

Subgroups: 900 in 457 conjugacy classes, 120 normal (34 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, C₂⁴, C₂.C₄², C₂.C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄⋊D₄, C₂².D₄, C₂³×C₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, C₂×C₂.C₄², C₂³.₈Q₈, C₂³.₂₃D₄, C₂³⋊₂D₄, C₂³⋊Q₈, C₂³.₁₀D₄, C₂³.₇₈C₂³, C₂×C₄⋊D₄, C₂×C₂².D₄, C₂²×C₄○D₄, C₂⁴.₂₆₂C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²≀C₂, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂×C₂²≀C₂, C₂².₂₆C₂⁴, C₂².₃₁C₂⁴, D₄⋊₅D₄, D₄⋊₆D₄, C₂⁴.₂₆₂C₂³

Smallest permutation representation of C₂⁴.₂₆₂C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	2L	2M	2N	2O	2P	4A	4B	4C	4D	4E	···	4R	4S	4T	4U
order	1	2	···	2	2	2	2	2	2	2	2	2	2	4	4	4	4	4	···	4	4	4	4
size	1	1	···	1	2	2	2	2	4	4	4	4	8	2	2	2	2	4	···	4	8	8	8

38 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂⁴.₂₆₂C₂³

C₂×C₂.C₄²

C₂³.₈Q₈

C₂³.₂₃D₄

C₂³⋊₂D₄

C₂³⋊Q₈

C₂³.₁₀D₄

C₂³.₇₈C₂³

C₂×C₄⋊D₄

C₂×C₂².D₄

C₂²×C₄○D₄

C₂²×C₄

C₂×D₄

C₂×C₄

C₂²

# reps

Matrix representation of C₂⁴.₂₆₂C₂³ ►in GL₆(𝔽₅)

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

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	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

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

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

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

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

C₂⁴.₂₆₂C₂³ in GAP, Magma, Sage, TeX

C_2^4._{262}C_2^3

% in TeX

G:=Group("C2^4.262C2^3");

// GroupNames label

G:=SmallGroup(128,1162);

// by ID

G=gap.SmallGroup(128,1162);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,120,758,723,184,675]);

// Polycyclic

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

// generators/relations

G = C24.262C23 order 128 = 27

102nd non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₂₆₂C₂³ order 128 = 2⁷

102^nd non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²