C2^3.633C2^4 - GroupNames

G = C₂³.₆₃₃C₂⁴ order 128 = 2⁷

350^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₇₅C₂³, C₂³.₆₃₃C₂⁴, C₂².₄₀₆2₊¹⁺⁴, (C₂×Q₈)⋊₁₅D₄, C₂³⋊₂D₄⋊₄₄C₂, C₂.₃₆(Q₈⋊₆D₄), C₂³.₁₀D₄⋊₉₈C₂, C₂.₅₇(C₂³⋊₃D₄), (C₂²×C₄).₈₉₁C₂³, (C₂×C₄²).₆₈₄C₂², C₂².₄₄₂(C₂²×D₄), C₂⁴.₃C₂²⋊₉₀C₂, C₂.₇(C₂⁴⋊C₂²), (C₂²×D₄).₂₅₉C₂², (C₂²×Q₈).₂₀₂C₂², C₂⁴.C₂²⋊₁₄₉C₂, C₂³.₆₇C₂³⋊₉₄C₂, C₂.₇₆(C₂².₃₂C₂⁴), C₂.C₄².₃₃₉C₂², C₂.₃₇(C₂².₄₉C₂⁴), (C₂×C₄).₁₂₇(C₂×D₄), (C₂×C₄.₄D₄)⋊₃₂C₂, (C₂×C₄).₂₁₃(C₄○D₄), (C₂×C₄⋊C₄).₄₄₆C₂², C₂².₄₉₅(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₂₉₆C₂², SmallGroup(128,1465)

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

Derived series

C₁ — C₂³ — C₂³.₆₃₃C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₂⁴.C₂² — C₂³.₆₃₃C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₃₃C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₃₃C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₃₃C₂⁴

Generators and relations for C₂³.₆₃₃C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=1, d²=g²=ba=ab, e²=b, f²=a, 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: 692 in 296 conjugacy classes, 96 normal (14 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, C₂×Q₈, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄.₄D₄, C₂²×D₄, C₂²×Q₈, C₂⁴.C₂², C₂⁴.₃C₂², C₂³.₆₇C₂³, C₂³⋊₂D₄, C₂³.₁₀D₄, C₂×C₄.₄D₄, C₂³.₆₃₃C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, C₂³⋊₃D₄, C₂².₃₂C₂⁴, Q₈⋊₆D₄, C₂².₄₉C₂⁴, C₂⁴⋊C₂², C₂³.₆₃₃C₂⁴

Smallest permutation representation of C₂³.₆₃₃C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	···	4R	4S	4T
order	1	2	···	2	2	2	2	2	4	···	4	4	4
size	1	1	···	1	8	8	8	8	4	···	4	8	8

32 irreducible representations

dim	1	1	1	1	1	1	1	2	2	4
type	+	+	+	+	+	+	+	+		+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	D₄	C₄○D₄	2₊¹⁺⁴
kernel	C₂³.₆₃₃C₂⁴	C₂⁴.C₂²	C₂⁴.₃C₂²	C₂³.₆₇C₂³	C₂³⋊₂D₄	C₂³.₁₀D₄	C₂×C₄.₄D₄	C₂×Q₈	C₂×C₄	C₂²
# reps	1	4	2	1	4	2	2	4	8	4

Matrix representation of C₂³.₆₃₃C₂⁴ ►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	3	3	0	0
0	0	0	2	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	2	0	0	0
0	0	0	2	0	0
0	0	0	0	4	0
0	0	0	0	0	1

3	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	0	1
0	0	0	0	1	0

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

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

C₂³.₆₃₃C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{633}C_2^4

% in TeX

G:=Group("C2^3.633C2^4");

// GroupNames label

G:=SmallGroup(128,1465);

// by ID

G=gap.SmallGroup(128,1465);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,232,758,723,100,1571,346,192]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=g^2=b*a=a*b,e^2=b,f^2=a,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 = C23.633C24 order 128 = 27

350th central stem extension by C23 of C24

G = C₂³.₆₃₃C₂⁴ order 128 = 2⁷

350^th central stem extension by C₂³ of C₂⁴