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G = C8×C3.A4order 288 = 25·32

Direct product of C8 and C3.A4

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

Aliases: C8×C3.A4, C22⋊C72, C24.1A4, C23.2C36, C3.(C8×A4), (C22×C8)⋊C9, (C2×C6).C24, C6.5(C4×A4), (C22×C24).C3, C12.14(C2×A4), (C22×C12).6C6, (C22×C4).2C18, (C22×C6).6C12, C2.1(C4×C3.A4), C4.4(C2×C3.A4), (C4×C3.A4).4C2, (C2×C3.A4).2C4, SmallGroup(288,76)

Series: Derived Chief Lower central Upper central

C1C22 — C8×C3.A4
C1C22C23C22×C6C22×C12C4×C3.A4 — C8×C3.A4
C22 — C8×C3.A4
C1C24

Generators and relations for C8×C3.A4
 G = < a,b,c,d,e | a8=b3=c2=d2=1, e3=b, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

3C2
3C2
3C22
3C4
3C22
3C6
3C6
4C9
3C8
3C2×C4
3C2×C4
3C12
3C2×C6
3C2×C6
4C18
3C2×C8
3C2×C8
3C2×C12
3C2×C12
3C24
4C36
3C2×C24
3C2×C24
4C72

Smallest permutation representation of C8×C3.A4
On 72 points
Generators in S72
(1 28 59 25 53 66 41 13)(2 29 60 26 54 67 42 14)(3 30 61 27 46 68 43 15)(4 31 62 19 47 69 44 16)(5 32 63 20 48 70 45 17)(6 33 55 21 49 71 37 18)(7 34 56 22 50 72 38 10)(8 35 57 23 51 64 39 11)(9 36 58 24 52 65 40 12)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)
(2 54)(3 46)(5 48)(6 49)(8 51)(9 52)(11 23)(12 24)(14 26)(15 27)(17 20)(18 21)(29 67)(30 68)(32 70)(33 71)(35 64)(36 65)(37 55)(39 57)(40 58)(42 60)(43 61)(45 63)
(1 53)(3 46)(4 47)(6 49)(7 50)(9 52)(10 22)(12 24)(13 25)(15 27)(16 19)(18 21)(28 66)(30 68)(31 69)(33 71)(34 72)(36 65)(37 55)(38 56)(40 58)(41 59)(43 61)(44 62)
(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)

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

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

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

96 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D6A6B6C6D6E6F8A8B8C8D8E8F8G8H9A···9F12A12B12C12D12E12F12G12H18A···18F24A···24H24I···24P36A···36L72A···72X
order1222334444666666888888889···9121212121212121218···1824···2424···2436···3672···72
size1133111133113333111133334···4111133334···41···13···34···44···4

96 irreducible representations

dim11111111111133333333
type++++
imageC1C2C3C4C6C8C9C12C18C24C36C72A4C2×A4C3.A4C4×A4C2×C3.A4C8×A4C4×C3.A4C8×C3.A4
kernelC8×C3.A4C4×C3.A4C22×C24C2×C3.A4C22×C12C3.A4C22×C8C22×C6C22×C4C2×C6C23C22C24C12C8C6C4C3C2C1
# reps1122246468122411222448

Matrix representation of C8×C3.A4 in GL3(𝔽73) generated by

6300
0630
0063
,
800
080
008
,
100
0720
0072
,
7200
0720
001
,
010
001
800
G:=sub<GL(3,GF(73))| [63,0,0,0,63,0,0,0,63],[8,0,0,0,8,0,0,0,8],[1,0,0,0,72,0,0,0,72],[72,0,0,0,72,0,0,0,1],[0,0,8,1,0,0,0,1,0] >;

C8×C3.A4 in GAP, Magma, Sage, TeX

C_8\times C_3.A_4
% in TeX

G:=Group("C8xC3.A4");
// GroupNames label

G:=SmallGroup(288,76);
// by ID

G=gap.SmallGroup(288,76);
# by ID

G:=PCGroup([7,-2,-3,-2,-3,-2,-2,2,42,92,142,3036,5305]);
// Polycyclic

G:=Group<a,b,c,d,e|a^8=b^3=c^2=d^2=1,e^3=b,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

Export

Subgroup lattice of C8×C3.A4 in TeX

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