C2^3.685C2^4 - GroupNames

G = C₂³.₆₈₅C₂⁴ order 128 = 2⁷

402^nd central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₉₁C₂³, C₂³.₆₈₅C₂⁴, C₂².₄₅₈2₊¹⁺⁴, C₂².₃₄₈2_-¹⁺⁴, C₄²⋊₈C₄⋊₇₁C₂, C₂³⋊Q₈⋊₅₇C₂, (C₂×C₄²).₇₁₂C₂², (C₂²×C₄).₅₉₈C₂³, C₂³.₁₀D₄.₆₄C₂, (C₂²×D₄).₂₈₁C₂², (C₂²×Q₈).₂₁₉C₂², C₂⁴.C₂²⋊₁₆₉C₂, C₂³.₇₈C₂³⋊₆₀C₂, C₂.₁₂(C₂⁴⋊C₂²), C₂.C₄².₃₈₉C₂², C₂.₄₆(C₂².₄₉C₂⁴), C₂.₁₀₄(C₂².₃₃C₂⁴), (C₂×C₄).₄₇₁(C₄○D₄), (C₂×C₄⋊C₄).₄₉₅C₂², C₂².₅₄₆(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₃₂₁C₂², SmallGroup(128,1517)

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

Derived series

C₁ — C₂³ — C₂³.₆₈₅C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₂⁴.C₂² — C₂³.₆₈₅C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₈₅C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₈₅C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₈₅C₂⁴

Generators and relations for C₂³.₆₈₅C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=1, d²=c, e²=b, f²=a, g²=ba=ab, ac=ca, ede^-1=ad=da, geg^-1=ae=ea, af=fa, ag=ga, bc=cb, fdf^-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef^-1=ce=ec, cf=fc, cg=gc, gdg^-1=abd, fg=gf >

Subgroups: 452 in 214 conjugacy classes, 88 normal (9 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂²×D₄, C₂²×Q₈, C₄²⋊₈C₄, C₂⁴.C₂², C₂³⋊Q₈, C₂³.₁₀D₄, C₂³.₇₈C₂³, C₂³.₆₈₅C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₃C₂⁴, C₂².₄₉C₂⁴, C₂⁴⋊C₂², C₂³.₆₈₅C₂⁴

Smallest permutation representation of C₂³.₆₈₅C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	4A	···	4R	4S	4T	4U	4V
order	1	2	···	2	2	2	4	···	4	4	4	4	4
size	1	1	···	1	8	8	4	···	4	8	8	8	8

32 irreducible representations

dim	1	1	1	1	1	1	2	4	4
type	+	+	+	+	+	+		+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄○D₄	2₊¹⁺⁴	2_-¹⁺⁴
kernel	C₂³.₆₈₅C₂⁴	C₄²⋊₈C₄	C₂⁴.C₂²	C₂³⋊Q₈	C₂³.₁₀D₄	C₂³.₇₈C₂³	C₂×C₄	C₂²	C₂²
# reps	1	3	6	2	3	1	12	3	1

Matrix representation of C₂³.₆₈₅C₂⁴ ►in GL₆(𝔽₅)

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

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

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

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

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

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

C₂³.₆₈₅C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{685}C_2^4

% in TeX

G:=Group("C2^3.685C2^4");

// GroupNames label

G:=SmallGroup(128,1517);

// by ID

G=gap.SmallGroup(128,1517);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,232,758,723,100,1571,346,192]);

// Polycyclic

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

// generators/relations

G = C23.685C24 order 128 = 27

402nd central stem extension by C23 of C24

G = C₂³.₆₈₅C₂⁴ order 128 = 2⁷

402^nd central stem extension by C₂³ of C₂⁴