A4xC2xC12 - GroupNames

G = A₄xC₂xC₁₂ order 288 = 2⁵·3²

Direct product of C₂xC₁₂ and A₄

direct product, metabelian, soluble, monomial, A-group

Aliases: A₄xC₂xC₁₂, C₂³.₆C₆², (C₂³xC₁₂):C₃, C₂²:(C₆xC₁₂), C₂⁴.(C₃xC₆), (C₂³xC₄):C₃², (C₂²xC₁₂):₄C₆, (C₂²xC₆):₄C₁₂, C₂³:₂(C₃xC₁₂), C₂².₇(C₆xA₄), (C₂³xC₆).₇C₆, (C₂²xA₄).₂C₆, C₆.₂₀(C₂²xA₄), (C₆xA₄).₂₃C₂², C₂.₁(A₄xC₂xC₆), (A₄xC₂xC₆).₄C₂, (C₂xC₆):₆(C₂xC₁₂), (C₂xA₄).₆(C₂xC₆), (C₂xC₆).₂₈(C₂xA₄), (C₂²xC₄):₂(C₃xC₆), (C₂²xC₆).₃₉(C₂xC₆), SmallGroup(288,979)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — A₄xC₂xC₁₂

Chief series

C₁ — C₂² — C₂³ — C₂²xC₆ — C₆xA₄ — A₄xC₂xC₆ — A₄xC₂xC₁₂

Lower central

C₂² — A₄xC₂xC₁₂

Upper central

C₁ — C₂xC₁₂

Generators and relations for A₄xC₂xC₁₂
G = < a,b,c,d,e | a²=b¹²=c²=d²=e³=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece^-1=cd=dc, ede^-1=c >

Subgroups: 396 in 164 conjugacy classes, 64 normal (20 characteristic)
C₁, C₂, C₂, C₂, C₃, C₃, C₄, C₄, C₂², C₂², C₆, C₆, C₆, C₂xC₄, C₂xC₄, C₂³, C₂³, C₂³, C₃², C₁₂, C₁₂, A₄, C₂xC₆, C₂xC₆, C₂²xC₄, C₂²xC₄, C₂⁴, C₃xC₆, C₂xC₁₂, C₂xC₁₂, C₂xA₄, C₂²xC₆, C₂²xC₆, C₂²xC₆, C₂³xC₄, C₃xC₁₂, C₃xA₄, C₆², C₄xA₄, C₂²xC₁₂, C₂²xC₁₂, C₂²xA₄, C₂³xC₆, C₆xC₁₂, C₆xA₄, C₆xA₄, C₂xC₄xA₄, C₂³xC₁₂, C₁₂xA₄, A₄xC₂xC₆, A₄xC₂xC₁₂
Quotients: C₁, C₂, C₃, C₄, C₂², C₆, C₂xC₄, C₃², C₁₂, A₄, C₂xC₆, C₃xC₆, C₂xC₁₂, C₂xA₄, C₃xC₁₂, C₃xA₄, C₆², C₄xA₄, C₂²xA₄, C₆xC₁₂, C₆xA₄, C₂xC₄xA₄, C₁₂xA₄, A₄xC₂xC₆, A₄xC₂xC₁₂

Smallest permutation representation of A₄xC₂xC₁₂
►On 72 points

Generators in S₇₂

(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 37)(10 38)(11 39)(12 40)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 59)(22 60)(23 49)(24 50)(25 66)(26 67)(27 68)(28 69)(29 70)(30 71)(31 72)(32 61)(33 62)(34 63)(35 64)(36 65)

(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 65 66 67 68 69 70 71 72)

(13 57)(14 58)(15 59)(16 60)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(25 72)(26 61)(27 62)(28 63)(29 64)(30 65)(31 66)(32 67)(33 68)(34 69)(35 70)(36 71)

(1 47)(2 48)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 43)(10 44)(11 45)(12 46)(13 57)(14 58)(15 59)(16 60)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)

(1 72 17)(2 61 18)(3 62 19)(4 63 20)(5 64 21)(6 65 22)(7 66 23)(8 67 24)(9 68 13)(10 69 14)(11 70 15)(12 71 16)(25 49 47)(26 50 48)(27 51 37)(28 52 38)(29 53 39)(30 54 40)(31 55 41)(32 56 42)(33 57 43)(34 58 44)(35 59 45)(36 60 46)

G:=sub<Sym(72)| (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,37)(10,38)(11,39)(12,40)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,49)(24,50)(25,66)(26,67)(27,68)(28,69)(29,70)(30,71)(31,72)(32,61)(33,62)(34,63)(35,64)(36,65), (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)(61,62,63,64,65,66,67,68,69,70,71,72), (13,57)(14,58)(15,59)(16,60)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,72)(26,61)(27,62)(28,63)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,70)(36,71), (1,47)(2,48)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,43)(10,44)(11,45)(12,46)(13,57)(14,58)(15,59)(16,60)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56), (1,72,17)(2,61,18)(3,62,19)(4,63,20)(5,64,21)(6,65,22)(7,66,23)(8,67,24)(9,68,13)(10,69,14)(11,70,15)(12,71,16)(25,49,47)(26,50,48)(27,51,37)(28,52,38)(29,53,39)(30,54,40)(31,55,41)(32,56,42)(33,57,43)(34,58,44)(35,59,45)(36,60,46)>;

G:=Group( (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,37)(10,38)(11,39)(12,40)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,49)(24,50)(25,66)(26,67)(27,68)(28,69)(29,70)(30,71)(31,72)(32,61)(33,62)(34,63)(35,64)(36,65), (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)(61,62,63,64,65,66,67,68,69,70,71,72), (13,57)(14,58)(15,59)(16,60)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,72)(26,61)(27,62)(28,63)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,70)(36,71), (1,47)(2,48)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,43)(10,44)(11,45)(12,46)(13,57)(14,58)(15,59)(16,60)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56), (1,72,17)(2,61,18)(3,62,19)(4,63,20)(5,64,21)(6,65,22)(7,66,23)(8,67,24)(9,68,13)(10,69,14)(11,70,15)(12,71,16)(25,49,47)(26,50,48)(27,51,37)(28,52,38)(29,53,39)(30,54,40)(31,55,41)(32,56,42)(33,57,43)(34,58,44)(35,59,45)(36,60,46) );

G=PermutationGroup([[(1,41),(2,42),(3,43),(4,44),(5,45),(6,46),(7,47),(8,48),(9,37),(10,38),(11,39),(12,40),(13,51),(14,52),(15,53),(16,54),(17,55),(18,56),(19,57),(20,58),(21,59),(22,60),(23,49),(24,50),(25,66),(26,67),(27,68),(28,69),(29,70),(30,71),(31,72),(32,61),(33,62),(34,63),(35,64),(36,65)], [(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),(61,62,63,64,65,66,67,68,69,70,71,72)], [(13,57),(14,58),(15,59),(16,60),(17,49),(18,50),(19,51),(20,52),(21,53),(22,54),(23,55),(24,56),(25,72),(26,61),(27,62),(28,63),(29,64),(30,65),(31,66),(32,67),(33,68),(34,69),(35,70),(36,71)], [(1,47),(2,48),(3,37),(4,38),(5,39),(6,40),(7,41),(8,42),(9,43),(10,44),(11,45),(12,46),(13,57),(14,58),(15,59),(16,60),(17,49),(18,50),(19,51),(20,52),(21,53),(22,54),(23,55),(24,56)], [(1,72,17),(2,61,18),(3,62,19),(4,63,20),(5,64,21),(6,65,22),(7,66,23),(8,67,24),(9,68,13),(10,69,14),(11,70,15),(12,71,16),(25,49,47),(26,50,48),(27,51,37),(28,52,38),(29,53,39),(30,54,40),(31,55,41),(32,56,42),(33,57,43),(34,58,44),(35,59,45),(36,60,46)]])

96 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	3C	···	3H	4A	4B	4C	4D	4E	4F	4G	4H	6A	···	6F	6G	···	6N	6O	···	6AF	12A	···	12H	12I	···	12P	12Q	···	12AN
order	1	2	2	2	2	2	2	2	3	3	3	···	3	4	4	4	4	4	4	4	4	6	···	6	6	···	6	6	···	6	12	···	12	12	···	12	12	···	12
size	1	1	1	1	3	3	3	3	1	1	4	···	4	1	1	1	1	3	3	3	3	1	···	1	3	···	3	4	···	4	1	···	1	3	···	3	4	···	4

96 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	3	3	3	3	3	3	3	3
type	+	+	+										+	+	+
image	C₁	C₂	C₂	C₃	C₃	C₄	C₆	C₆	C₆	C₆	C₁₂	C₁₂	A₄	C₂xA₄	C₂xA₄	C₃xA₄	C₄xA₄	C₆xA₄	C₆xA₄	C₁₂xA₄
kernel	A₄xC₂xC₁₂	C₁₂xA₄	A₄xC₂xC₆	C₂xC₄xA₄	C₂³xC₁₂	C₆xA₄	C₄xA₄	C₂²xC₁₂	C₂²xA₄	C₂³xC₆	C₂xA₄	C₂²xC₆	C₂xC₁₂	C₁₂	C₂xC₆	C₂xC₄	C₆	C₄	C₂²	C₂
# reps	1	2	1	6	2	4	12	4	6	2	24	8	1	2	1	2	4	4	2	8

Matrix representation of A₄xC₂xC₁₂ ►in GL₄(F₁₃) generated by

1	0	0	0
0	12	0	0
0	0	12	0
0	0	0	12

11	0	0	0
0	3	0	0
0	0	3	0
0	0	0	3

1	0	0	0
0	1	0	0
0	0	12	0
0	4	0	12

1	0	0	0
0	12	0	0
0	0	12	0
0	9	3	1

1	0	0	0
0	0	3	0
0	12	4	7
0	0	0	9

G:=sub<GL(4,GF(13))| [1,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[11,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,1,0,4,0,0,12,0,0,0,0,12],[1,0,0,0,0,12,0,9,0,0,12,3,0,0,0,1],[1,0,0,0,0,0,12,0,0,3,4,0,0,0,7,9] >;

A₄xC₂xC₁₂ in GAP, Magma, Sage, TeX

A_4\times C_2\times C_{12}

% in TeX

G:=Group("A4xC2xC12");

// GroupNames label

G:=SmallGroup(288,979);

// by ID

G=gap.SmallGroup(288,979);

# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-2,2,260,1531,2666]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^12=c^2=d^2=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;

// generators/relations

G = A4xC2xC12 order 288 = 25·32

Direct product of C2xC12 and A4

G = A₄xC₂xC₁₂ order 288 = 2⁵·3²

Direct product of C₂xC₁₂ and A₄