C2^3.753C2^4 - GroupNames

G = C₂³.₇₅₃C₂⁴ order 128 = 2⁷

470^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₁₂₁C₂³, C₂³.₇₅₃C₂⁴, (C₂²×C₄²)⋊₈C₂, (C₂²×C₄).₇₀₆D₄, C₂³.₆₃₂(C₂×D₄), C₂³.₂₅₃(C₄○D₄), C₂³.₃₄D₄⋊₆₄C₂, (C₂³×C₄).₆₅₃C₂², C₂³.₈Q₈⋊₁₄₈C₂, C₂².₄₆₃(C₂²×D₄), (C₂²×C₄).₁₂₆₄C₂³, (C₂×C₄²).₁₀₁₄C₂², C₂³.₂₃D₄.₈₀C₂, (C₂²×D₄).₃₁₁C₂², C₂⁴.C₂²⋊₁₈₄C₂, C₂.₉₆(C₂².₁₉C₂⁴), C₂³.₆₃C₂³⋊₂₀₄C₂, C₂.C₄².₄₅₀C₂², C₂².₃₉(C₂².D₄), C₂.₁₁₁(C₂³.₃₆C₂³), (C₂×C₄).₁₂₀₇(C₂×D₄), (C₂×C₄).₅₂₉(C₄○D₄), (C₂×C₄⋊C₄).₅₅₆C₂², C₂².₅₉₄(C₂×C₄○D₄), C₂.₄₅(C₂×C₂².D₄), (C₂×C₂²⋊C₄).₃₆₃C₂², (C₂×C₂².D₄).₃₂C₂, SmallGroup(128,1585)

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

Derived series

C₁ — C₂³ — C₂³.₇₅₃C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂²×C₄² — C₂³.₇₅₃C₂⁴

Lower central

C₁ — C₂³ — C₂³.₇₅₃C₂⁴

Upper central

C₁ — C₂³ — C₂³.₇₅₃C₂⁴

Jennings

C₁ — C₂³ — C₂³.₇₅₃C₂⁴

Generators and relations for C₂³.₇₅₃C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=f²=1, e²=b, g²=a, ab=ba, ac=ca, ede^-1=ad=da, ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, gdg^-1=cd=dc, ce=ec, cf=fc, cg=gc, ef=fe, eg=ge, fg=gf >

Subgroups: 516 in 291 conjugacy classes, 108 normal (14 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂².D₄, C₂³×C₄, C₂³×C₄, C₂²×D₄, C₂³.₃₄D₄, C₂³.₈Q₈, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂²×C₄², C₂×C₂².D₄, C₂³.₇₅₃C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂².D₄, C₂²×D₄, C₂×C₄○D₄, C₂×C₂².D₄, C₂².₁₉C₂⁴, C₂³.₃₆C₂³, C₂³.₇₅₃C₂⁴

Smallest permutation representation of C₂³.₇₅₃C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2
type	+	+	+	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	D₄	C₄○D₄	C₄○D₄
kernel	C₂³.₇₅₃C₂⁴	C₂³.₃₄D₄	C₂³.₈Q₈	C₂³.₂₃D₄	C₂³.₆₃C₂³	C₂⁴.C₂²	C₂²×C₄²	C₂×C₂².D₄	C₂²×C₄	C₂×C₄	C₂³
# reps	1	1	2	2	4	4	1	1	4	16	8

Matrix representation of C₂³.₇₅₃C₂⁴ ►in GL₆(𝔽₅)

4	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	1	0
0	0	0	0	0	1

4	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	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

1	0	0	0	0	0
1	4	0	0	0	0
0	0	1	0	0	0
0	0	1	4	0	0
0	0	0	0	1	0
0	0	0	0	1	4

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

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

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

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

C₂³.₇₅₃C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{753}C_2^4

% in TeX

G:=Group("C2^3.753C2^4");

// GroupNames label

G:=SmallGroup(128,1585);

// by ID

G=gap.SmallGroup(128,1585);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,232,758,2019,80]);

// Polycyclic

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

// generators/relations

G = C23.753C24 order 128 = 27

470th central stem extension by C23 of C24

G = C₂³.₇₅₃C₂⁴ order 128 = 2⁷

470^th central stem extension by C₂³ of C₂⁴