C2^4.258C2^3 - GroupNames

G = C₂⁴.₂₅₈C₂³ order 128 = 2⁷

98^th non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²

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

Aliases: C₂⁴.₂₅₈C₂³, C₂³.₃₂₅C₂⁴, C₂².₉₉2_-¹⁺⁴, C₂².₁₃₈2₊¹⁺⁴, C₂²⋊C₄⋊₃₆D₄, (C₂×D₄).₂₀₈D₄, C₂³.₃₈(C₂×D₄), C₂.₂₅(D₄⋊₅D₄), C₂.₁₆(D₄⋊₆D₄), C₂³.Q₈⋊₆C₂, C₂³.₁₁D₄⋊₉C₂, C₂².₅(C₄⋊D₄), C₂³.₈Q₈⋊₃₄C₂, C₂³.₇Q₈⋊₄₂C₂, C₂³.₂₃₀(C₄○D₄), C₂³.₂₃D₄⋊₃₂C₂, C₂³.₁₀D₄⋊₁₆C₂, (C₂²×C₄).₅₀₆C₂³, (C₂×C₄²).₄₇₂C₂², (C₂³×C₄).₃₃₈C₂², C₂².₂₀₅(C₂²×D₄), C₂³.₈₁C₂³⋊₇C₂, C₂⁴.C₂²⋊₃₅C₂, (C₂²×D₄).₅₀₆C₂², C₂³.₆₅C₂³⋊₄₂C₂, C₂.C₄².₅₃₈C₂², C₂.₂₁(C₂³.₃₆C₂³), C₂.₁₁(C₂².₃₃C₂⁴), (C₂×C₄×D₄)⋊₂₇C₂, (C₂×C₄).₄₈(C₂×D₄), C₂.₂₃(C₂×C₄⋊D₄), (C₂×C₄⋊D₄).₂₄C₂, (C₂×C₄).₃₆₀(C₄○D₄), (C₂×C₄⋊C₄).₂₁₂C₂², C₂².₂₀₄(C₂×C₄○D₄), (C₂×C₂².D₄)⋊₈C₂, (C₂×C₂.C₄²)⋊₂₉C₂, (C₂×C₂²⋊C₄).₁₁₅C₂², SmallGroup(128,1157)

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

Derived series

C₁ — C₂³ — C₂⁴.₂₅₈C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂×C₂.C₄² — C₂⁴.₂₅₈C₂³

Lower central

C₁ — C₂³ — C₂⁴.₂₅₈C₂³

Upper central

C₁ — C₂³ — C₂⁴.₂₅₈C₂³

Jennings

C₁ — C₂³ — C₂⁴.₂₅₈C₂³

Generators and relations for C₂⁴.₂₅₈C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=g²=1, d²=e²=b, f²=a, ab=ba, ac=ca, ede^-1=gdg=ad=da, 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, eg=ge, fg=gf >

Subgroups: 660 in 333 conjugacy classes, 112 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄×D₄, C₄⋊D₄, C₂².D₄, C₂³×C₄, C₂²×D₄, C₂×C₂.C₄², C₂³.₇Q₈, C₂³.₈Q₈, C₂³.₂₃D₄, C₂⁴.C₂², C₂³.₆₅C₂³, C₂³.₁₀D₄, C₂³.Q₈, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂×C₄×D₄, C₂×C₄⋊D₄, C₂×C₂².D₄, C₂⁴.₂₅₈C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₄⋊D₄, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂×C₄⋊D₄, C₂³.₃₆C₂³, C₂².₃₃C₂⁴, D₄⋊₅D₄, D₄⋊₆D₄, C₂⁴.₂₅₈C₂³

Smallest permutation representation of C₂⁴.₂₅₈C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	2L	2M	2N	4A	4B	4C	4D	4E	···	4T	4U	4V	4W
order	1	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	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

C₂⁴.₂₅₈C₂³

C₂×C₂.C₄²

C₂³.₇Q₈

C₂³.₈Q₈

C₂³.₂₃D₄

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂³.₁₀D₄

C₂³.Q₈

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂×C₄×D₄

C₂×C₄⋊D₄

C₂×C₂².D₄

C₂²⋊C₄

C₂×D₄

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂⁴.₂₅₈C₂³ ►in GL₆(𝔽₅)

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	4	0	0	0
0	0	0	4	0	0
0	0	0	0	4	0
0	0	0	0	0	4

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

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

0	2	0	0	0	0
2	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

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

0	1	0	0	0	0
1	0	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

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

C₂⁴.₂₅₈C₂³ in GAP, Magma, Sage, TeX

C_2^4._{258}C_2^3

% in TeX

G:=Group("C2^4.258C2^3");

// GroupNames label

G:=SmallGroup(128,1157);

// by ID

G=gap.SmallGroup(128,1157);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,232,758,723,100,675]);

// Polycyclic

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

// generators/relations

G = C24.258C23 order 128 = 27

98th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₂₅₈C₂³ order 128 = 2⁷

98^th non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²