C2^4.445C2^3 - GroupNames

G = C₂⁴.₄₄₅C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₄₄₅C₂³, C₂³.₆₇₀C₂⁴, C₂².₄₄₃2₊¹⁺⁴, C₂².₃₃₆2_-¹⁺⁴, C₄²⋊₅C₄⋊₃₄C₂, C₂³.₉₅(C₄○D₄), (C₂³×C₄).₄₉₁C₂², (C₂×C₄²).₁₀₄C₂², (C₂²×C₄).₂₁₀C₂³, C₂³.₇Q₈.₇₄C₂, C₂³.₈Q₈.₆₂C₂, C₂³.₁₁D₄.₄₉C₂, C₂.₉₄(C₂².₃₂C₂⁴), C₂⁴.C₂².₆₉C₂, C₂³.₆₅C₂³⋊₁₄₅C₂, C₂³.₈₁C₂³⋊₁₁₉C₂, C₂³.₆₃C₂³⋊₁₇₄C₂, C₂³.₈₃C₂³⋊₁₀₈C₂, C₂.C₄².₃₇₄C₂², C₂.₄₂(C₂².₅₇C₂⁴), C₂.₉₅(C₂².₃₃C₂⁴), C₂.₁₀₃(C₂².₄₇C₂⁴), C₂.₁₀₈(C₂².₄₆C₂⁴), C₂.₁₀₉(C₂².₃₆C₂⁴), (C₂×C₄).₄₆₄(C₄○D₄), (C₂×C₄⋊C₄).₄₈₀C₂², C₂².₅₃₁(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₃₁₃C₂², SmallGroup(128,1502)

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

Derived series

C₁ — C₂³ — C₂⁴.₄₄₅C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₂³.₆₅C₂³ — C₂⁴.₄₄₅C₂³

Lower central

C₁ — C₂³ — C₂⁴.₄₄₅C₂³

Upper central

C₁ — C₂³ — C₂⁴.₄₄₅C₂³

Jennings

C₁ — C₂³ — C₂⁴.₄₄₅C₂³

Generators and relations for C₂⁴.₄₄₅C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=c, f²=g²=cb=bc, ab=ba, gag^-1=ac=ca, eae^-1=ad=da, faf^-1=acd, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef^-1=de=ed, df=fd, dg=gd, geg^-1=bde, gfg^-1=cdf >

Subgroups: 372 in 196 conjugacy classes, 88 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂³×C₄, C₂³.₇Q₈, C₄²⋊₅C₄, C₂³.₈Q₈, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₅C₂³, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂³.₈₃C₂³, C₂⁴.₄₄₅C₂³
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₂C₂⁴, C₂².₃₃C₂⁴, C₂².₃₆C₂⁴, C₂².₄₆C₂⁴, C₂².₄₇C₂⁴, C₂².₅₇C₂⁴, C₂⁴.₄₄₅C₂³

Smallest permutation representation of C₂⁴.₄₄₅C₂³
►On 64 points

Generators in S₆₄

(2 7)(4 5)(9 56)(10 12)(11 54)(13 15)(14 57)(16 59)(18 63)(20 61)(21 38)(23 40)(25 44)(26 28)(27 42)(29 31)(30 45)(32 47)(33 49)(35 51)(41 43)(46 48)(53 55)(58 60)

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	4A	···	4P	4Q	···	4V
order	1	2	···	2	2	2	4	···	4	4	···	4
size	1	1	···	1	4	4	4	···	4	8	···	8

32 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂⁴.₄₄₅C₂³

C₂³.₇Q₈

C₄²⋊₅C₄

C₂³.₈Q₈

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂³.₈₃C₂³

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂⁴.₄₄₅C₂³ ►in GL₆(𝔽₅)

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

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

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

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

C₂⁴.₄₄₅C₂³ in GAP, Magma, Sage, TeX

C_2^4._{445}C_2^3

% in TeX

G:=Group("C2^4.445C2^3");

// GroupNames label

G:=SmallGroup(128,1502);

// by ID

G=gap.SmallGroup(128,1502);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,232,758,723,268,1571,346,80]);

// Polycyclic

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

// generators/relations

G = C24.445C23 order 128 = 27

285th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₄₄₅C₂³ order 128 = 2⁷

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