C2^3.693C2^4 - GroupNames

G = C₂³.₆₉₃C₂⁴ order 128 = 2⁷

410^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₉₂C₂³, C₂³.₆₉₃C₂⁴, C₂².₃₅₆2_-¹⁺⁴, C₂².₄₆₆2₊¹⁺⁴, C₄²⋊₉C₄⋊₃₇C₂, (C₂²×C₄).₆₀₃C₂³, (C₂×C₄²).₁₀₉C₂², C₂³.Q₈.₄₃C₂, C₂³.₁₁D₄.₅₇C₂, C₂⁴.C₂².₇₉C₂, C₂³.₈₃C₂³⋊₁₂₄C₂, C₂³.₆₃C₂³⋊₁₈₉C₂, C₂³.₈₁C₂³⋊₁₃₀C₂, C₂.C₄².₃₉₇C₂², C₂.₆₄(C₂².₃₄C₂⁴), C₂.₅₂(C₂².₅₇C₂⁴), C₂.₇₃(C₂².₅₀C₂⁴), C₂.₁₂₀(C₂².₄₆C₂⁴), C₂.₁₁₁(C₂².₃₃C₂⁴), C₂.₁₁₇(C₂².₃₆C₂⁴), C₂.₁₁₆(C₂².₄₇C₂⁴), (C₂×C₄).₂₃₄(C₄○D₄), (C₂×C₄⋊C₄).₅₀₃C₂², C₂².₅₅₄(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₇₇C₂², SmallGroup(128,1525)

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

Derived series

C₁ — C₂³ — C₂³.₆₉₃C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₂⁴.C₂² — C₂³.₆₉₃C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₉₃C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₉₃C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₉₃C₂⁴

Generators and relations for C₂³.₆₉₃C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=1, d²=ca=ac, e²=f²=b, g²=ba=ab, 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: 356 in 189 conjugacy classes, 88 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄²⋊₉C₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.Q₈, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂³.₈₃C₂³, C₂³.₆₉₃C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₃C₂⁴, C₂².₃₄C₂⁴, C₂².₃₆C₂⁴, C₂².₄₆C₂⁴, C₂².₄₇C₂⁴, C₂².₅₀C₂⁴, C₂².₅₇C₂⁴, C₂³.₆₉₃C₂⁴

Smallest permutation representation of C₂³.₆₉₃C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	4A	···	4R	4S	···	4W
order	1	2	···	2	2	4	···	4	4	···	4
size	1	1	···	1	8	4	···	4	8	···	8

32 irreducible representations

dim	1	1	1	1	1	1	1	1	2	4	4
type	+	+	+	+	+	+	+	+		+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₄○D₄	2₊¹⁺⁴	2_-¹⁺⁴
kernel	C₂³.₆₉₃C₂⁴	C₄²⋊₉C₄	C₂³.₆₃C₂³	C₂⁴.C₂²	C₂³.Q₈	C₂³.₁₁D₄	C₂³.₈₁C₂³	C₂³.₈₃C₂³	C₂×C₄	C₂²	C₂²
# reps	1	1	5	3	2	2	1	1	12	2	2

Matrix representation of C₂³.₆₉₃C₂⁴ ►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	4	0	0	0
0	0	0	4	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

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

0	1	0	0	0	0
1	0	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

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

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

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[2,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,3,4,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,2,2,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,2,2,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C₂³.₆₉₃C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{693}C_2^4

% in TeX

G:=Group("C2^3.693C2^4");

// GroupNames label

G:=SmallGroup(128,1525);

// by ID

G=gap.SmallGroup(128,1525);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,232,758,723,184,1571,346,192]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=c*a=a*c,e^2=f^2=b,g^2=b*a=a*b,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 = C23.693C24 order 128 = 27

410th central stem extension by C23 of C24

G = C₂³.₆₉₃C₂⁴ order 128 = 2⁷

410^th central stem extension by C₂³ of C₂⁴