C2^4.599C2^3 - GroupNames

G = C₂⁴.₅₉₉C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₅₉₉C₂³, C₂³.₇₅₅C₂⁴, (C₂²×C₄)⋊₂₁Q₈, C₂³.₆₃₄(C₂×D₄), (C₂²×C₄).₇₇₁D₄, C₂³.₁₀₅(C₂×Q₈), (C₂²×C₄²).₂₇C₂, C₂³.₂₅₅(C₄○D₄), (C₂³×C₄).₆₅₅C₂², (C₂²×C₄).₂₆₃C₂³, C₂².₄₆₅(C₂²×D₄), C₂².₄₄(C₂²⋊Q₈), C₂³.₇Q₈.₇₉C₂, C₂³.₈Q₈.₇₁C₂, C₂².₁₈₀(C₂²×Q₈), (C₂×C₄²).₁₀₁₆C₂², C₂³.₃₄D₄.₃₇C₂, C₄.₉₈(C₂².D₄), (C₂²×Q₈).₂₄₈C₂², C₂.₉₈(C₂².₁₉C₂⁴), C₂³.₆₃C₂³⋊₂₀₅C₂, C₂³.₆₅C₂³⋊₁₆₉C₂, C₂³.₆₇C₂³⋊₁₀₉C₂, C₂.C₄².₄₅₂C₂², C₂.₄₆(C₂³.₃₇C₂³), C₂.₁₁₃(C₂³.₃₆C₂³), (C₂×C₄).₁₇₂(C₂×Q₈), C₂.₄₇(C₂×C₂²⋊Q₈), (C₂×C₄).₁₂₀₈(C₂×D₄), (C₂×C₂²⋊Q₈).₄₉C₂, (C₂×C₄).₆₇₁(C₄○D₄), (C₂×C₄⋊C₄).₅₅₈C₂², C₂².₅₉₆(C₂×C₄○D₄), C₂.₄₇(C₂×C₂².D₄), (C₂×C₂²⋊C₄).₃₆₅C₂², SmallGroup(128,1587)

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

Derived series

C₁ — C₂³ — C₂⁴.₅₉₉C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂²×C₄² — C₂⁴.₅₉₉C₂³

Lower central

C₁ — C₂³ — C₂⁴.₅₉₉C₂³

Upper central

C₁ — C₂³ — C₂⁴.₅₉₉C₂³

Jennings

C₁ — C₂³ — C₂⁴.₅₉₉C₂³

Generators and relations for C₂⁴.₅₉₉C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=g²=b, f²=cb=bc, ab=ba, eae^-1=ac=ca, ad=da, af=fa, ag=ga, bd=db, fef^-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, geg^-1=de=ed, df=fd, dg=gd, fg=gf >

Subgroups: 468 in 282 conjugacy classes, 120 normal (20 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂²⋊Q₈, C₂³×C₄, C₂²×Q₈, C₂³.₇Q₈, C₂³.₃₄D₄, C₂³.₈Q₈, C₂³.₆₃C₂³, C₂³.₆₅C₂³, C₂³.₆₇C₂³, C₂²×C₄², C₂×C₂²⋊Q₈, C₂⁴.₅₉₉C₂³
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, C₂²⋊Q₈, C₂².D₄, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, C₂×C₂²⋊Q₈, C₂×C₂².D₄, C₂².₁₉C₂⁴, C₂³.₃₆C₂³, C₂³.₃₇C₂³, C₂⁴.₅₉₉C₂³

Smallest permutation representation of C₂⁴.₅₉₉C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

dim

type

image

C₁

C₂

D₄

Q₈

C₄○D₄

kernel

C₂⁴.₅₉₉C₂³

C₂³.₇Q₈

C₂³.₃₄D₄

C₂³.₈Q₈

C₂³.₆₃C₂³

C₂³.₆₅C₂³

C₂³.₆₇C₂³

C₂²×C₄²

C₂×C₂²⋊Q₈

C₂²×C₄

C₂×C₄

C₂³

# reps

Matrix representation of C₂⁴.₅₉₉C₂³ ►in GL₆(𝔽₅)

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

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

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

3	2	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	0	2

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

C₂⁴.₅₉₉C₂³ in GAP, Magma, Sage, TeX

C_2^4._{599}C_2^3

% in TeX

G:=Group("C2^4.599C2^3");

// GroupNames label

G:=SmallGroup(128,1587);

// by ID

G=gap.SmallGroup(128,1587);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,344,758,2019,80]);

// 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,e*a*e^-1=a*c=c*a,a*d=d*a,a*f=f*a,a*g=g*a,b*d=d*b,f*e*f^-1=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,g*e*g^-1=d*e=e*d,d*f=f*d,d*g=g*d,f*g=g*f>;

// generators/relations

G = C24.599C23 order 128 = 27

80th non-split extension by C24 of C23 acting via C23/C22=C2

G = C₂⁴.₅₉₉C₂³ order 128 = 2⁷

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