C2^4.345C2^3 - GroupNames

G = C₂⁴.₃₄₅C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₃₄₅C₂³, C₂³.₄₈₇C₂⁴, C₂².₂₆₉2₊¹⁺⁴, C₂²⋊C₄.₁₀Q₈, C₂³.₂₉(C₂×Q₈), C₂.₄₅(D₄⋊₃Q₈), (C₂×C₄²).₇₃C₂², (C₂³×C₄).₁₂₆C₂², (C₂²×C₄).₁₁₃C₂³, C₂³.Q₈.₁₆C₂, C₂³.₈Q₈.₃₇C₂, C₂².₁₂₂(C₂²×Q₈), C₂³.₈₃C₂³⋊₄₇C₂, C₂³.₆₃C₂³⋊₉₇C₂, C₂³.₈₁C₂³⋊₄₇C₂, C₂³.₆₅C₂³⋊₉₃C₂, C₂.₃₁(C₂².₃₂C₂⁴), C₂⁴.C₂².₃₆C₂, C₂.C₄².₂₂₁C₂², C₂.₆₇(C₂².₄₇C₂⁴), C₂.₉₁(C₂³.₃₆C₂³), C₂.₃₃(C₂³.₃₇C₂³), (C₄×C₄⋊C₄)⋊₁₀₃C₂, (C₂×C₄).₂₅₇(C₂×Q₈), (C₄×C₂²⋊C₄).₆₉C₂, (C₂×C₄).₅₂₆(C₄○D₄), (C₂×C₄⋊C₄).₈₇₈C₂², C₂².₃₆₃(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₅₁₂C₂², SmallGroup(128,1319)

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

Derived series

C₁ — C₂³ — C₂⁴.₃₄₅C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₄×C₄⋊C₄ — C₂⁴.₃₄₅C₂³

Lower central

C₁ — C₂³ — C₂⁴.₃₄₅C₂³

Upper central

C₁ — C₂³ — C₂⁴.₃₄₅C₂³

Jennings

C₁ — C₂³ — C₂⁴.₃₄₅C₂³

Generators and relations for C₂⁴.₃₄₅C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=bcd, f²=d, g²=c, eae^-1=gag^-1=ab=ba, faf^-1=ac=ca, ad=da, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef^-1=de=ed, df=fd, dg=gd, eg=ge, fg=gf >

Subgroups: 388 in 214 conjugacy classes, 100 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂³×C₄, C₄×C₂²⋊C₄, C₄×C₄⋊C₄, C₂³.₈Q₈, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₅C₂³, C₂³.Q₈, C₂³.₈₁C₂³, C₂³.₈₃C₂³, C₂⁴.₃₄₅C₂³
Quotients: C₁, C₂, C₂², Q₈, C₂³, C₂×Q₈, C₄○D₄, C₂⁴, C₂²×Q₈, C₂×C₄○D₄, 2₊¹⁺⁴, C₂³.₃₆C₂³, C₂³.₃₇C₂³, C₂².₃₂C₂⁴, C₂².₄₇C₂⁴, D₄⋊₃Q₈, C₂⁴.₃₄₅C₂³

Smallest permutation representation of C₂⁴.₃₄₅C₂³
►On 64 points

Generators in S₆₄

(2 56)(4 54)(5 52)(6 17)(7 50)(8 19)(9 43)(11 41)(13 45)(15 47)(18 63)(20 61)(21 36)(22 40)(23 34)(24 38)(26 29)(28 31)(33 59)(35 57)(37 60)(39 58)(49 62)(51 64)

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

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

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

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	4A	···	4H	4I	···	4X	4Y	4Z	4AA	4AB
order	1	2	···	2	2	2	4	···	4	4	···	4	4	4	4	4
size	1	1	···	1	4	4	2	···	2	4	···	4	8	8	8	8

38 irreducible representations

dim

type

image

C₁

C₂

Q₈

C₄○D₄

2₊¹⁺⁴

kernel

C₂⁴.₃₄₅C₂³

C₄×C₂²⋊C₄

C₄×C₄⋊C₄

C₂³.₈Q₈

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂³.Q₈

C₂³.₈₁C₂³

C₂³.₈₃C₂³

C₂²⋊C₄

C₂×C₄

C₂²

# reps

Matrix representation of C₂⁴.₃₄₅C₂³ ►in GL₆(𝔽₅)

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	1	0
0	0	0	0	4	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

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

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

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

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

C₂⁴.₃₄₅C₂³ in GAP, Magma, Sage, TeX

C_2^4._{345}C_2^3

% in TeX

G:=Group("C2^4.345C2^3");

// GroupNames label

G:=SmallGroup(128,1319);

// by ID

G=gap.SmallGroup(128,1319);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,792,758,723,352,675,136]);

// Polycyclic

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

// generators/relations

G = C24.345C23 order 128 = 27

185th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₃₄₅C₂³ order 128 = 2⁷

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