C2xC2^3.32C2^3 - GroupNames

G = C₂xC₂³.₃₂C₂³ order 128 = 2⁷

Direct product of C₂ and C₂³.₃₂C₂³

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

Aliases: C₂xC₂³.₃₂C₂³, C₂².₁₂C₂⁵, C₄².₅₃₄C₂³, C₂³.₁₀₆C₂⁴, C₂⁴.₆₀₄C₂³, C₂².₇₁2_-¹⁺⁴, C₂.₈(C₂⁴xC₄), C₄.₃₈(C₂³xC₄), (C₄xQ₈):₈₁C₂², (C₂²xQ₈):₂₄C₄, C₄:C₄.₅₁₃C₂³, (C₂xC₄).₁₅₈C₂⁴, Q₈.₂₄(C₂²xC₄), (Q₈xC₂³).₁₃C₂, C₂².₁₉(C₂³xC₄), (C₂xQ₈).₄₇₉C₂³, C₂.₁(C₂x2_-¹⁺⁴), C₂²:C₄.₁₂₅C₂³, C₂³.₂₃₄(C₂²xC₄), (C₂³xC₄).₅₇₆C₂², (C₂xC₄²).₉₁₂C₂², (C₂²xC₄).₁₂₉₃C₂³, (C₂²xQ₈).₄₈₁C₂², C₄²:C₂.₃₃₅C₂², (C₂xC₄xQ₈):₄₁C₂, Q₈ o(C₂xC₂²:C₄), C₂²:C₄ o₂(C₂xQ₈), (C₂xQ₈):₄₃(C₂xC₄), (C₂xC₄:C₄).₉₈₂C₂², (C₂xC₄).₂₈₂(C₂²xC₄), (C₂²xC₄).₃₇₂(C₂xC₄), (C₂xC₄²:C₂).₆₁C₂, (C₂xC₂²:C₄).₅₆₁C₂², SmallGroup(128,2158)

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

Derived series

C₁ — C₂ — C₂xC₂³.₃₂C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³xC₄ — Q₈xC₂³ — C₂xC₂³.₃₂C₂³

Lower central

C₁ — C₂ — C₂xC₂³.₃₂C₂³

Upper central

C₁ — C₂³ — C₂xC₂³.₃₂C₂³

Jennings

C₁ — C₂² — C₂xC₂³.₃₂C₂³

Generators and relations for C₂xC₂³.₃₂C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=d, f²=g²=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, ebe^-1=bc=cb, bd=db, bf=fb, bg=gb, cd=dc, fef^-1=ce=ec, gfg^-1=cf=fc, cg=gc, de=ed, df=fd, dg=gd, eg=ge >

Subgroups: 796 in 744 conjugacy classes, 692 normal (7 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂xC₄, C₂xC₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²:C₄, C₄:C₄, C₂²xC₄, C₂xQ₈, C₂⁴, C₂xC₄², C₂xC₂²:C₄, C₂xC₄:C₄, C₄²:C₂, C₄xQ₈, C₂³xC₄, C₂²xQ₈, C₂xC₄²:C₂, C₂xC₄xQ₈, C₂³.₃₂C₂³, Q₈xC₂³, C₂xC₂³.₃₂C₂³
Quotients: C₁, C₂, C₄, C₂², C₂xC₄, C₂³, C₂²xC₄, C₂⁴, C₂³xC₄, 2_-¹⁺⁴, C₂⁵, C₂³.₃₂C₂³, C₂⁴xC₄, C₂x2_-¹⁺⁴, C₂xC₂³.₃₂C₂³

Smallest permutation representation of C₂xC₂³.₃₂C₂³
►On 64 points

Generators in S₆₄

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

(2 20)(4 18)(6 62)(8 64)(10 56)(12 54)(14 52)(16 50)(22 60)(24 58)(26 48)(28 46)(30 44)(32 42)(34 40)(36 38)

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

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

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

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

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

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

68 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	···	4BD
order	1	2	···	2	2	2	2	2	4	···	4
size	1	1	···	1	2	2	2	2	2	···	2

68 irreducible representations

dim	1	1	1	1	1	1	4
type	+	+	+	+	+		-
image	C₁	C₂	C₂	C₂	C₂	C₄	2_-¹⁺⁴
kernel	C₂xC₂³.₃₂C₂³	C₂xC₄²:C₂	C₂xC₄xQ₈	C₂³.₃₂C₂³	Q₈xC₂³	C₂²xQ₈	C₂²
# reps	1	6	8	16	1	32	4

Matrix representation of C₂xC₂³.₃₂C₂³ ►in GL₆(F₅)

1	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

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	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	4	0
0	0	0	0	0	4

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

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

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

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

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

C₂xC₂³.₃₂C₂³ in GAP, Magma, Sage, TeX

C_2\times C_2^3._{32}C_2^3

% in TeX

G:=Group("C2xC2^3.32C2^3");

// GroupNames label

G:=SmallGroup(128,2158);

// by ID

G=gap.SmallGroup(128,2158);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,232,387,184,1123]);

// Polycyclic

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

// generators/relations

G = C2xC23.32C23 order 128 = 27

Direct product of C2 and C23.32C23

G = C₂xC₂³.₃₂C₂³ order 128 = 2⁷

Direct product of C₂ and C₂³.₃₂C₂³