A4xC24 - GroupNames

G = A₄xC₂₄ order 288 = 2⁵·3²

Direct product of C₂₄ and A₄

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

Aliases: A₄xC₂₄, (C₂xC₆):₂C₂₄, C₂²:(C₃xC₂₄), (C₂²xC₂₄):C₃, C₄.₄(C₆xA₄), C₆.₉(C₄xA₄), (C₂²xC₈):C₃², (C₄xA₄).₄C₆, (C₆xA₄).₄C₄, C₂.₁(C₁₂xA₄), (C₂xA₄).₂C₁₂, (C₁₂xA₄).₈C₂, C₁₂.₁₈(C₂xA₄), (C₂²xC₆).₈C₁₂, C₂³.₂(C₃xC₁₂), (C₂²xC₁₂).₈C₆, (C₂²xC₄).₂(C₃xC₆), SmallGroup(288,637)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — A₄xC₂₄

Chief series

C₁ — C₂² — C₂³ — C₂²xC₄ — C₂²xC₁₂ — C₁₂xA₄ — A₄xC₂₄

Lower central

C₂² — A₄xC₂₄

Upper central

C₁ — C₂₄

Generators and relations for A₄xC₂₄
G = < a,b,c,d | a²⁴=b²=c²=d³=1, ab=ba, ac=ca, ad=da, dbd^-1=bc=cb, dcd^-1=b >

Subgroups: 156 in 68 conjugacy classes, 32 normal (20 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², C₂², C₆, C₆, C₈, C₈, C₂xC₄, C₂³, C₃², C₁₂, C₁₂, A₄, C₂xC₆, C₂xC₆, C₂xC₈, C₂²xC₄, C₃xC₆, C₂₄, C₂₄, C₂xC₁₂, C₂xA₄, C₂²xC₆, C₂²xC₈, C₃xC₁₂, C₃xA₄, C₂xC₂₄, C₄xA₄, C₂²xC₁₂, C₃xC₂₄, C₆xA₄, C₈xA₄, C₂²xC₂₄, C₁₂xA₄, A₄xC₂₄
Quotients: C₁, C₂, C₃, C₄, C₆, C₈, C₃², C₁₂, A₄, C₃xC₆, C₂₄, C₂xA₄, C₃xC₁₂, C₃xA₄, C₄xA₄, C₃xC₂₄, C₆xA₄, C₈xA₄, C₁₂xA₄, A₄xC₂₄

Smallest permutation representation of A₄xC₂₄
►On 72 points

Generators in S₇₂

(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)

(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(49 61)(50 62)(51 63)(52 64)(53 65)(54 66)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)

(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)

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

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

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

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

96 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	···	3H	4A	4B	4C	4D	6A	6B	6C	6D	6E	6F	6G	···	6L	8A	8B	8C	8D	8E	8F	8G	8H	12A	12B	12C	12D	12E	12F	12G	12H	12I	···	12T	24A	···	24H	24I	···	24P	24Q	···	24AN
order	1	2	2	2	3	3	3	···	3	4	4	4	4	6	6	6	6	6	6	6	···	6	8	8	8	8	8	8	8	8	12	12	12	12	12	12	12	12	12	···	12	24	···	24	24	···	24	24	···	24
size	1	1	3	3	1	1	4	···	4	1	1	3	3	1	1	3	3	3	3	4	···	4	1	1	1	1	3	3	3	3	1	1	1	1	3	3	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	C₃	C₃	C₄	C₆	C₆	C₈	C₁₂	C₁₂	C₂₄	C₂₄	A₄	C₂xA₄	C₃xA₄	C₄xA₄	C₆xA₄	C₈xA₄	C₁₂xA₄	A₄xC₂₄
kernel	A₄xC₂₄	C₁₂xA₄	C₈xA₄	C₂²xC₂₄	C₆xA₄	C₄xA₄	C₂²xC₁₂	C₃xA₄	C₂xA₄	C₂²xC₆	A₄	C₂xC₆	C₂₄	C₁₂	C₈	C₆	C₄	C₃	C₂	C₁
# reps	1	1	6	2	2	6	2	4	12	4	24	8	1	1	2	2	2	4	4	8

Matrix representation of A₄xC₂₄ ►in GL₄(F₇₃) generated by

17	0	0	0
0	63	0	0
0	0	63	0
0	0	0	63

1	0	0	0
0	1	64	68
0	0	72	0
0	0	0	72

1	0	0	0
0	72	0	5
0	0	72	0
0	0	0	1

8	0	0	0
0	68	22	17
0	0	0	1
0	71	9	5

G:=sub<GL(4,GF(73))| [17,0,0,0,0,63,0,0,0,0,63,0,0,0,0,63],[1,0,0,0,0,1,0,0,0,64,72,0,0,68,0,72],[1,0,0,0,0,72,0,0,0,0,72,0,0,5,0,1],[8,0,0,0,0,68,0,71,0,22,0,9,0,17,1,5] >;

A₄xC₂₄ in GAP, Magma, Sage, TeX

A_4\times C_{24}

% in TeX

G:=Group("A4xC24");

// GroupNames label

G:=SmallGroup(288,637);

// by ID

G=gap.SmallGroup(288,637);

# by ID

G:=PCGroup([7,-2,-3,-3,-2,-2,-2,2,126,80,3036,5305]);

// Polycyclic

G:=Group<a,b,c,d|a^24=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;

// generators/relations

G = A4xC24 order 288 = 25·32

Direct product of C24 and A4

G = A₄xC₂₄ order 288 = 2⁵·3²

Direct product of C₂₄ and A₄