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
| C22 — C8×C3.A4 |
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 >
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 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 9A | ··· | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 18A | ··· | 18F | 24A | ··· | 24H | 24I | ··· | 24P | 36A | ··· | 36L | 72A | ··· | 72X |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 24 | ··· | 24 | 24 | ··· | 24 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 3 | 3 | 1 | 1 | 1 | 1 | 3 | 3 | 1 | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 1 | ··· | 1 | 3 | ··· | 3 | 4 | ··· | 4 | 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 | C4 | C6 | C8 | C9 | C12 | C18 | C24 | C36 | C72 | A4 | C2×A4 | C3.A4 | C4×A4 | C2×C3.A4 | C8×A4 | C4×C3.A4 | C8×C3.A4 |
| kernel | C8×C3.A4 | C4×C3.A4 | C22×C24 | C2×C3.A4 | C22×C12 | C3.A4 | C22×C8 | C22×C6 | C22×C4 | C2×C6 | C23 | C22 | C24 | C12 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 6 | 4 | 6 | 8 | 12 | 24 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C8×C3.A4 ►in GL3(𝔽73) generated by
| 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 | 1 | 0 |
| 0 | 0 | 1 |
| 8 | 0 | 0 |
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
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