C2xC2^3.23D4 - GroupNames

G = C₂xC₂³.₂₃D₄ order 128 = 2⁷

Direct product of C₂ and C₂³.₂₃D₄

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

Aliases: C₂xC₂³.₂₃D₄, C₂⁴.₁₃₈D₄, C₂⁵.₁₄C₂², C₂⁴.₅₃₂C₂³, C₂³.₁₆₉C₂⁴, (C₂⁴xC₄):₁C₂, C₂⁴:₈(C₂xC₄), (C₂²xC₄):₅₁D₄, (C₂²xD₄):₁₉C₄, C₂³:₄(C₂²xC₄), (D₄xC₂³).₅C₂, C₂³:₆(C₂²:C₄), (C₂³xC₄):₅₅C₂², C₂³.₅₉₈(C₂xD₄), C₂².₁₀₇(C₄xD₄), C₂².₆₀(C₂³xC₄), C₂².₆₇(C₂²xD₄), C₂².₁₀₇C₂²wrC₂, C₂³.₃₅₈(C₄oD₄), (C₂²xC₄).₄₄₇C₂³, C₂².₁₅₆(C₄:D₄), C₂.C₄²:₅₉C₂², (C₂²xD₄).₄₆₂C₂², C₂².₉₈(C₂².D₄), C₂.₆(C₂xC₄xD₄), (C₂xC₄):₂₀(C₂xD₄), (C₂xD₄):₃₆(C₂xC₄), (C₂xC₄):₅(C₂²xC₄), C₂.₂(C₂xC₄:D₄), (C₂²xC₄):₁₆(C₂xC₄), C₂.₃(C₂xC₂²wrC₂), C₂²:₂(C₂xC₂²:C₄), (C₂²xC₂²:C₄):₄C₂, C₂².₆₁(C₂xC₄oD₄), C₂.₈(C₂²xC₂²:C₄), (C₂xC₂²:C₄):₆₉C₂², C₂.₃(C₂xC₂².D₄), (C₂xC₂.C₄²):₁₆C₂, SmallGroup(128,1019)

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

Derived series

C₁ — C₂² — C₂xC₂³.₂₃D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂⁵ — C₂⁴xC₄ — C₂xC₂³.₂₃D₄

Lower central

C₁ — C₂² — C₂xC₂³.₂₃D₄

Upper central

C₁ — C₂⁴ — C₂xC₂³.₂₃D₄

Jennings

C₁ — C₂³ — C₂xC₂³.₂₃D₄

Generators and relations for C₂xC₂³.₂₃D₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=e⁴=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=de^-1 >

Subgroups: 1452 in 788 conjugacy classes, 236 normal (16 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂xC₄, C₂xC₄, D₄, C₂³, C₂³, C₂³, C₂²:C₄, C₂²xC₄, C₂²xC₄, C₂xD₄, C₂xD₄, C₂⁴, C₂⁴, C₂⁴, C₂.C₄², C₂xC₂²:C₄, C₂xC₂²:C₄, C₂³xC₄, C₂³xC₄, C₂³xC₄, C₂²xD₄, C₂²xD₄, C₂⁵, C₂xC₂.C₄², C₂³.₂₃D₄, C₂²xC₂²:C₄, C₂²xC₂²:C₄, C₂⁴xC₄, D₄xC₂³, C₂xC₂³.₂₃D₄
Quotients: C₁, C₂, C₄, C₂², C₂xC₄, D₄, C₂³, C₂²:C₄, C₂²xC₄, C₂xD₄, C₄oD₄, C₂⁴, C₂xC₂²:C₄, C₄xD₄, C₂²wrC₂, C₄:D₄, C₂².D₄, C₂³xC₄, C₂²xD₄, C₂xC₄oD₄, C₂³.₂₃D₄, C₂²xC₂²:C₄, C₂xC₄xD₄, C₂xC₂²wrC₂, C₂xC₄:D₄, C₂xC₂².D₄, C₂xC₂³.₂₃D₄

Smallest permutation representation of C₂xC₂³.₂₃D₄
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

56 conjugacy classes

class	1	2A	···	2O	2P	···	2W	2X	2Y	2Z	2AA	4A	···	4P	4Q	···	4AB
order	1	2	···	2	2	···	2	2	2	2	2	4	···	4	4	···	4
size	1	1	···	1	2	···	2	4	4	4	4	2	···	2	4	···	4

56 irreducible representations

dim	1	1	1	1	1	1	1	2	2	2
type	+	+	+	+	+	+		+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	D₄	D₄	C₄oD₄
kernel	C₂xC₂³.₂₃D₄	C₂xC₂.C₄²	C₂³.₂₃D₄	C₂²xC₂²:C₄	C₂⁴xC₄	D₄xC₂³	C₂²xD₄	C₂²xC₄	C₂⁴	C₂³
# reps	1	2	8	3	1	1	16	8	8	8

Matrix representation of C₂xC₂³.₂₃D₄ ►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	1	0	0	0	0
0	0	0	4	0	0
0	0	4	0	0	0
0	0	0	0	4	2
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	4	0
0	0	0	0	0	4

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

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

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	4	0
0	0	0	0	4	1

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

C₂xC₂³.₂₃D₄ in GAP, Magma, Sage, TeX

C_2\times C_2^3._{23}D_4

% in TeX

G:=Group("C2xC2^3.23D4");

// GroupNames label

G:=SmallGroup(128,1019);

// by ID

G=gap.SmallGroup(128,1019);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,568,758]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^4=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,f*b*f=b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=d*e^-1>;

// generators/relations

G = C2xC23.23D4 order 128 = 27

Direct product of C2 and C23.23D4

G = C₂xC₂³.₂₃D₄ order 128 = 2⁷

Direct product of C₂ and C₂³.₂₃D₄