C2^4.327C2^3 - GroupNames

G = C₂⁴.₃₂₇C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₃₂₇C₂³, C₂³.₄₅₄C₂⁴, C₂².₂₃₉2₊¹⁺⁴, C₄⋊C₄⋊₂₈D₄, C₂.₇₄(D₄⋊₅D₄), C₂.₂₆(Q₈⋊₆D₄), C₂³⋊₂D₄.₁₀C₂, C₂³.Q₈⋊₂₉C₂, C₂³.₁₅₃(C₄○D₄), C₂³.₁₀D₄⋊₄₃C₂, C₂³.₂₃D₄⋊₅₆C₂, (C₂×C₄²).₅₅₉C₂², (C₂³×C₄).₁₁₄C₂², (C₂²×C₄).₅₄₀C₂³, C₂².₃₀₅(C₂²×D₄), C₂⁴.C₂²⋊₈₁C₂, C₂⁴.₃C₂²⋊₅₇C₂, (C₂²×D₄).₁₆₉C₂², C₂³.₈₁C₂³⋊₃₉C₂, C₂.₂₂(C₂².₃₂C₂⁴), C₂.₄₇(C₂².₄₅C₂⁴), C₂.C₄².₁₉₁C₂², C₂.₃₄(C₂².₂₆C₂⁴), C₂.₇₈(C₂³.₃₆C₂³), C₂.₅₄(C₂².₄₇C₂⁴), (C₄×C₄⋊C₄)⋊₉₂C₂, (C₂×C₄).₇₉(C₂×D₄), (C₄×C₂²⋊C₄)⋊₈₅C₂, (C₂×C₄²⋊₂C₂)⋊₁₀C₂, (C₂×C₄).₃₈₇(C₄○D₄), (C₂×C₄⋊C₄).₈₇₂C₂², C₂².₃₃₀(C₂×C₄○D₄), (C₂×C₂².D₄)⋊₂₁C₂, (C₂×C₂²⋊C₄).₁₈₁C₂², SmallGroup(128,1286)

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

Derived series

C₁ — C₂³ — C₂⁴.₃₂₇C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₄×C₂²⋊C₄ — C₂⁴.₃₂₇C₂³

Lower central

C₁ — C₂³ — C₂⁴.₃₂₇C₂³

Upper central

C₁ — C₂³ — C₂⁴.₃₂₇C₂³

Jennings

C₁ — C₂³ — C₂⁴.₃₂₇C₂³

Generators and relations for C₂⁴.₃₂₇C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=f²=1, d²=a, e²=ca=ac, g²=b, ab=ba, ede^-1=gdg^-1=ad=da, ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 580 in 280 conjugacy classes, 100 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂².D₄, C₄²⋊₂C₂, C₂³×C₄, C₂²×D₄, C₄×C₂²⋊C₄, C₄×C₄⋊C₄, C₂³.₂₃D₄, C₂⁴.C₂², C₂⁴.₃C₂², C₂³⋊₂D₄, C₂³.₁₀D₄, C₂³.Q₈, C₂³.₈₁C₂³, C₂×C₂².D₄, C₂×C₄²⋊₂C₂, C₂⁴.₃₂₇C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, C₂³.₃₆C₂³, C₂².₂₆C₂⁴, C₂².₃₂C₂⁴, D₄⋊₅D₄, Q₈⋊₆D₄, C₂².₄₅C₂⁴, C₂².₄₇C₂⁴, C₂⁴.₃₂₇C₂³

Smallest permutation representation of C₂⁴.₃₂₇C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	···	4H	4I	···	4X	4Y	4Z
order	1	2	···	2	2	2	2	2	4	···	4	4	···	4	4	4
size	1	1	···	1	4	4	8	8	2	···	2	4	···	4	8	8

38 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

kernel

C₂⁴.₃₂₇C₂³

C₄×C₂²⋊C₄

C₄×C₄⋊C₄

C₂³.₂₃D₄

C₂⁴.C₂²

C₂⁴.₃C₂²

C₂³⋊₂D₄

C₂³.₁₀D₄

C₂³.Q₈

C₂³.₈₁C₂³

C₂×C₂².D₄

C₂×C₄²⋊₂C₂

C₄⋊C₄

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂⁴.₃₂₇C₂³ ►in GL₆(𝔽₅)

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

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

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

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

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

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

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

C₂⁴.₃₂₇C₂³ in GAP, Magma, Sage, TeX

C_2^4._{327}C_2^3

% in TeX

G:=Group("C2^4.327C2^3");

// GroupNames label

G:=SmallGroup(128,1286);

// by ID

G=gap.SmallGroup(128,1286);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,568,758,723,675,136]);

// Polycyclic

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

// generators/relations

G = C24.327C23 order 128 = 27

167th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₃₂₇C₂³ order 128 = 2⁷

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