C2^3.700C2^4 - GroupNames

G = C₂³.₇₀₀C₂⁴ order 128 = 2⁷

417^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₉₆C₂³, C₂³.₇₀₀C₂⁴, C₂².₄₇₃2₊¹⁺⁴, C₂².₃₆₂2_-¹⁺⁴, C₂³.Q₈⋊₈₉C₂, (C₂×C₄²).₁₁₄C₂², (C₂²×C₄).₆₀₉C₂³, C₂³.₁₁D₄⋊₁₂₃C₂, C₂³.₁₀D₄.₆₈C₂, (C₂²×D₄).₂₈₆C₂², C₂⁴.C₂²⋊₁₇₄C₂, C₂⁴.₃C₂².₇₇C₂, C₂³.₆₅C₂³⋊₁₅₉C₂, C₂³.₆₃C₂³⋊₁₉₄C₂, C₂.₁₀₇(C₂².₃₂C₂⁴), C₂.C₄².₄₀₄C₂², C₂.₇₅(C₂².₅₀C₂⁴), C₂.₆₆(C₂².₃₄C₂⁴), C₂.₄₄(C₂².₅₆C₂⁴), C₂.₁₂₂(C₂².₃₆C₂⁴), C₂.₁₂₀(C₂².₄₇C₂⁴), (C₂×C₄).₂₄₁(C₄○D₄), (C₂×C₄⋊C₄).₅₁₀C₂², C₂².₅₆₁(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₃₂₈C₂², SmallGroup(128,1532)

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

Derived series

C₁ — C₂³ — C₂³.₇₀₀C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₂⁴.C₂² — C₂³.₇₀₀C₂⁴

Lower central

C₁ — C₂³ — C₂³.₇₀₀C₂⁴

Upper central

C₁ — C₂³ — C₂³.₇₀₀C₂⁴

Jennings

C₁ — C₂³ — C₂³.₇₀₀C₂⁴

Generators and relations for C₂³.₇₀₀C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=1, d²=cb=bc, e²=a, f²=ca=ac, g²=b, ab=ba, ede^-1=ad=da, geg^-1=ae=ea, af=fa, ag=ga, 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: 436 in 210 conjugacy classes, 88 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂²×D₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₅C₂³, C₂⁴.₃C₂², C₂³.₁₀D₄, C₂³.Q₈, C₂³.₁₁D₄, C₂³.₇₀₀C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₂C₂⁴, C₂².₃₄C₂⁴, C₂².₃₆C₂⁴, C₂².₄₇C₂⁴, C₂².₅₀C₂⁴, C₂².₅₆C₂⁴, C₂³.₇₀₀C₂⁴

Smallest permutation representation of C₂³.₇₀₀C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

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	1	1	2	4	4
type	+	+	+	+	+	+	+	+		+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₄○D₄	2₊¹⁺⁴	2_-¹⁺⁴
kernel	C₂³.₇₀₀C₂⁴	C₂³.₆₃C₂³	C₂⁴.C₂²	C₂³.₆₅C₂³	C₂⁴.₃C₂²	C₂³.₁₀D₄	C₂³.Q₈	C₂³.₁₁D₄	C₂×C₄	C₂²	C₂²
# reps	1	2	4	2	1	2	2	2	12	3	1

Matrix representation of C₂³.₇₀₀C₂⁴ ►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

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

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

3	0	0	0	0	0
0	3	0	0	0	0
0	0	2	0	0	0
0	0	3	3	0	0
0	0	0	0	0	2
0	0	0	0	3	0

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

C₂³.₇₀₀C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{700}C_2^4

% in TeX

G:=Group("C2^3.700C2^4");

// GroupNames label

G:=SmallGroup(128,1532);

// by ID

G=gap.SmallGroup(128,1532);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,672,253,120,758,723,436,1571,346,136]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=c*b=b*c,e^2=a,f^2=c*a=a*c,g^2=b,a*b=b*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,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 = C23.700C24 order 128 = 27

417th central stem extension by C23 of C24

G = C₂³.₇₀₀C₂⁴ order 128 = 2⁷

417^th central stem extension by C₂³ of C₂⁴