direct product, non-abelian, soluble, monomial
Aliases: C3×A4⋊C8, A4⋊C24, C12.18S4, (C3×A4)⋊1C8, (C2×A4).C12, C4.4(C3×S4), (C6×A4).1C4, (C4×A4).2C6, C6.8(A4⋊C4), (C12×A4).4C2, C23.(C3×Dic3), (C22×C12).1S3, (C22×C6).1Dic3, C22⋊(C3×C3⋊C8), (C2×C6)⋊1(C3⋊C8), C2.1(C3×A4⋊C4), (C22×C4).1(C3×S3), SmallGroup(288,398)
Series: Derived ►Chief ►Lower central ►Upper central
| A4 — C3×A4⋊C8 |
Generators and relations for C3×A4⋊C8
G = < a,b,c,d,e | a3=b2=c2=d3=e8=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe-1=bc=cb, dcd-1=b, ce=ec, ede-1=d-1 >
Subgroups: 166 in 58 conjugacy classes, 20 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C23, C32, C12, C12, A4, A4, C2×C6, C2×C6, C2×C8, C22×C4, C3×C6, C3⋊C8, C24, C2×C12, C2×A4, C2×A4, C22×C6, C22⋊C8, C3×C12, C3×A4, C2×C24, C4×A4, C4×A4, C22×C12, C3×C3⋊C8, C6×A4, C3×C22⋊C8, A4⋊C8, C12×A4, C3×A4⋊C8
Quotients: C1, C2, C3, C4, S3, C6, C8, Dic3, C12, C3×S3, C3⋊C8, C24, S4, C3×Dic3, A4⋊C4, C3×C3⋊C8, C3×S4, A4⋊C8, C3×A4⋊C4, C3×A4⋊C8
►On 72 points
Generators in S72
(1 54 71)(2 55 72)(3 56 65)(4 49 66)(5 50 67)(6 51 68)(7 52 69)(8 53 70)(9 60 34)(10 61 35)(11 62 36)(12 63 37)(13 64 38)(14 57 39)(15 58 40)(16 59 33)(17 28 42)(18 29 43)(19 30 44)(20 31 45)(21 32 46)(22 25 47)(23 26 48)(24 27 41)
(2 6)(4 8)(9 13)(10 14)(11 15)(12 16)(18 22)(20 24)(25 29)(27 31)(33 37)(34 38)(35 39)(36 40)(41 45)(43 47)(49 53)(51 55)(57 61)(58 62)(59 63)(60 64)(66 70)(68 72)
(1 5)(2 6)(3 7)(4 8)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(65 69)(66 70)(67 71)(68 72)
(1 61 29)(2 30 62)(3 63 31)(4 32 64)(5 57 25)(6 26 58)(7 59 27)(8 28 60)(9 70 17)(10 18 71)(11 72 19)(12 20 65)(13 66 21)(14 22 67)(15 68 23)(16 24 69)(33 41 52)(34 53 42)(35 43 54)(36 55 44)(37 45 56)(38 49 46)(39 47 50)(40 51 48)
(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)| (1,54,71)(2,55,72)(3,56,65)(4,49,66)(5,50,67)(6,51,68)(7,52,69)(8,53,70)(9,60,34)(10,61,35)(11,62,36)(12,63,37)(13,64,38)(14,57,39)(15,58,40)(16,59,33)(17,28,42)(18,29,43)(19,30,44)(20,31,45)(21,32,46)(22,25,47)(23,26,48)(24,27,41), (2,6)(4,8)(9,13)(10,14)(11,15)(12,16)(18,22)(20,24)(25,29)(27,31)(33,37)(34,38)(35,39)(36,40)(41,45)(43,47)(49,53)(51,55)(57,61)(58,62)(59,63)(60,64)(66,70)(68,72), (1,5)(2,6)(3,7)(4,8)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(65,69)(66,70)(67,71)(68,72), (1,61,29)(2,30,62)(3,63,31)(4,32,64)(5,57,25)(6,26,58)(7,59,27)(8,28,60)(9,70,17)(10,18,71)(11,72,19)(12,20,65)(13,66,21)(14,22,67)(15,68,23)(16,24,69)(33,41,52)(34,53,42)(35,43,54)(36,55,44)(37,45,56)(38,49,46)(39,47,50)(40,51,48), (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( (1,54,71)(2,55,72)(3,56,65)(4,49,66)(5,50,67)(6,51,68)(7,52,69)(8,53,70)(9,60,34)(10,61,35)(11,62,36)(12,63,37)(13,64,38)(14,57,39)(15,58,40)(16,59,33)(17,28,42)(18,29,43)(19,30,44)(20,31,45)(21,32,46)(22,25,47)(23,26,48)(24,27,41), (2,6)(4,8)(9,13)(10,14)(11,15)(12,16)(18,22)(20,24)(25,29)(27,31)(33,37)(34,38)(35,39)(36,40)(41,45)(43,47)(49,53)(51,55)(57,61)(58,62)(59,63)(60,64)(66,70)(68,72), (1,5)(2,6)(3,7)(4,8)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(65,69)(66,70)(67,71)(68,72), (1,61,29)(2,30,62)(3,63,31)(4,32,64)(5,57,25)(6,26,58)(7,59,27)(8,28,60)(9,70,17)(10,18,71)(11,72,19)(12,20,65)(13,66,21)(14,22,67)(15,68,23)(16,24,69)(33,41,52)(34,53,42)(35,43,54)(36,55,44)(37,45,56)(38,49,46)(39,47,50)(40,51,48), (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([[(1,54,71),(2,55,72),(3,56,65),(4,49,66),(5,50,67),(6,51,68),(7,52,69),(8,53,70),(9,60,34),(10,61,35),(11,62,36),(12,63,37),(13,64,38),(14,57,39),(15,58,40),(16,59,33),(17,28,42),(18,29,43),(19,30,44),(20,31,45),(21,32,46),(22,25,47),(23,26,48),(24,27,41)], [(2,6),(4,8),(9,13),(10,14),(11,15),(12,16),(18,22),(20,24),(25,29),(27,31),(33,37),(34,38),(35,39),(36,40),(41,45),(43,47),(49,53),(51,55),(57,61),(58,62),(59,63),(60,64),(66,70),(68,72)], [(1,5),(2,6),(3,7),(4,8),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(65,69),(66,70),(67,71),(68,72)], [(1,61,29),(2,30,62),(3,63,31),(4,32,64),(5,57,25),(6,26,58),(7,59,27),(8,28,60),(9,70,17),(10,18,71),(11,72,19),(12,20,65),(13,66,21),(14,22,67),(15,68,23),(16,24,69),(33,41,52),(34,53,42),(35,43,54),(36,55,44),(37,45,56),(38,49,46),(39,47,50),(40,51,48)], [(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)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 8A | ··· | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | ··· | 12N | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 3 | 3 | 1 | 1 | 8 | 8 | 8 | 1 | 1 | 3 | 3 | 1 | 1 | 3 | 3 | 3 | 3 | 8 | 8 | 8 | 6 | ··· | 6 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 8 | ··· | 8 | 6 | ··· | 6 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | + | - | + | |||||||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C12 | C24 | S3 | Dic3 | C3×S3 | C3⋊C8 | C3×Dic3 | C3×C3⋊C8 | S4 | A4⋊C4 | C3×S4 | A4⋊C8 | C3×A4⋊C4 | C3×A4⋊C8 |
| kernel | C3×A4⋊C8 | C12×A4 | A4⋊C8 | C6×A4 | C4×A4 | C3×A4 | C2×A4 | A4 | C22×C12 | C22×C6 | C22×C4 | C2×C6 | C23 | C22 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 4 | 4 | 4 | 8 |
Matrix representation of C3×A4⋊C8 ►in GL5(𝔽73)
| 64 | 0 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 72 |
| 72 | 72 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 51 | 0 | 0 | 0 | 0 |
| 22 | 22 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 63 |
| 0 | 0 | 0 | 63 | 0 |
| 0 | 0 | 63 | 0 | 0 |
G:=sub<GL(5,GF(73))| [64,0,0,0,0,0,64,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,72],[1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,72],[72,1,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[51,22,0,0,0,0,22,0,0,0,0,0,0,0,63,0,0,0,63,0,0,0,63,0,0] >;
C3×A4⋊C8 in GAP, Magma, Sage, TeX
C_3\times A_4\rtimes C_8
% in TeX
G:=Group("C3xA4:C8"); // GroupNames label
G:=SmallGroup(288,398);
// by ID
G=gap.SmallGroup(288,398);
# by ID
G:=PCGroup([7,-2,-3,-2,-2,-3,-2,2,42,58,1684,6053,285,3534,475]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^2=d^3=e^8=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e^-1=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations