C2^4.565C2^3 - GroupNames

G = C₂⁴.₅₆₅C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₅₆₅C₂³, C₂³.₃₃₆C₂⁴, C₂².₁₀₆2_-¹⁺⁴, (C₂×Q₈)⋊₃₃D₄, (Q₈×C₂³)⋊₃C₂, C₄.₃₂C₂²≀C₂, C₂³⋊Q₈⋊₆C₂, (C₂²×C₄).₃₇₉D₄, C₂³.₆₁₀(C₂×D₄), C₂.₁₇(Q₈⋊₅D₄), C₂²⋊₄(C₄.₄D₄), (C₂²×C₄).₅₇C₂³, C₂³.₃₂₇(C₄○D₄), C₂³.₂₃D₄⋊₃₇C₂, (C₂³×C₄).₃₄₉C₂², (C₂×C₄²).₄₈₀C₂², C₂².₂₁₆(C₂²×D₄), (C₂²×D₄).₁₂₉C₂², (C₂²×Q₈).₄₂₄C₂², C₂³.₆₇C₂³⋊₃₈C₂, C₂.C₄².₉₅C₂², C₂.₁₄(C₂³.₃₈C₂³), (C₄×C₂²⋊C₄)⋊₅₉C₂, (C₂×C₄.₄D₄)⋊₉C₂, (C₂×C₄).₃₂₀(C₂×D₄), C₂.₂₄(C₂×C₂²≀C₂), (C₂×C₄⋊D₄).₂₇C₂, C₂.₁₂(C₂×C₄.₄D₄), (C₂×C₄⋊C₄).₂₂₀C₂², C₂².₂₁₃(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₁₂₃C₂², SmallGroup(128,1168)

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

Derived series

C₁ — C₂³ — C₂⁴.₅₆₅C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₄×C₂²⋊C₄ — C₂⁴.₅₆₅C₂³

Lower central

C₁ — C₂³ — C₂⁴.₅₆₅C₂³

Upper central

C₁ — C₂³ — C₂⁴.₅₆₅C₂³

Jennings

C₁ — C₂³ — C₂⁴.₅₆₅C₂³

Generators and relations for C₂⁴.₅₆₅C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=g²=b, f²=cb=bc, ab=ba, ac=ca, faf^-1=ad=da, ae=ea, ag=ga, bd=db, geg^-1=be=eb, bf=fb, bg=gb, cd=dc, fef^-1=ce=ec, cf=fc, cg=gc, de=ed, gfg^-1=df=fd, dg=gd >

Subgroups: 804 in 426 conjugacy classes, 124 normal (16 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂⁴, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄⋊D₄, C₄.₄D₄, C₂³×C₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂²×Q₈, C₂²×Q₈, C₄×C₂²⋊C₄, C₂³.₂₃D₄, C₂³.₆₇C₂³, C₂³⋊Q₈, C₂×C₄⋊D₄, C₂×C₄.₄D₄, Q₈×C₂³, C₂⁴.₅₆₅C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²≀C₂, C₄.₄D₄, C₂²×D₄, C₂×C₄○D₄, 2_-¹⁺⁴, C₂×C₂²≀C₂, C₂×C₄.₄D₄, C₂³.₃₈C₂³, Q₈⋊₅D₄, C₂⁴.₅₆₅C₂³

Smallest permutation representation of C₂⁴.₅₆₅C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	2L	2M	4A	4B	4C	4D	4E	···	4V	4W	4X
order	1	2	···	2	2	2	2	2	2	2	4	4	4	4	4	···	4	4	4
size	1	1	···	1	2	2	2	2	8	8	2	2	2	2	4	···	4	8	8

38 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2_-¹⁺⁴

kernel

C₂⁴.₅₆₅C₂³

C₄×C₂²⋊C₄

C₂³.₂₃D₄

C₂³.₆₇C₂³

C₂³⋊Q₈

C₂×C₄⋊D₄

C₂×C₄.₄D₄

Q₈×C₂³

C₂²×C₄

C₂×Q₈

C₂³

C₂²

# reps

Matrix representation of C₂⁴.₅₆₅C₂³ ►in GL₆(𝔽₅)

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	0	4
0	0	0	0	4	0

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	4	0
0	0	0	0	0	4

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

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

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

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

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

C₂⁴.₅₆₅C₂³ in GAP, Magma, Sage, TeX

C_2^4._{565}C_2^3

% in TeX

G:=Group("C2^4.565C2^3");

// GroupNames label

G:=SmallGroup(128,1168);

// by ID

G=gap.SmallGroup(128,1168);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,723,268,675,80]);

// Polycyclic

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

// generators/relations

G = C24.565C23 order 128 = 27

46th non-split extension by C24 of C23 acting via C23/C22=C2

G = C₂⁴.₅₆₅C₂³ order 128 = 2⁷

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