direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C3xC4:C8, C4:C24, C12:3C8, C12.67D4, C42.2C6, C12.12Q8, C6.9M4(2), (C2xC8).2C6, C4.4(C3xQ8), C2.2(C2xC24), (C2xC24).4C2, (C4xC12).8C2, C6.12(C2xC8), (C2xC4).4C12, C4.18(C3xD4), C6.11(C4:C4), (C2xC12).13C4, C2.3(C3xM4(2)), C22.10(C2xC12), (C2xC12).136C22, C2.2(C3xC4:C4), (C2xC6).39(C2xC4), (C2xC4).32(C2xC6), SmallGroup(96,55)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3xC4:C8
G = < a,b,c | a3=b4=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Quotients: C1, C2, C3, C4, C22, C6, C8, C2xC4, D4, Q8, C12, C2xC6, C4:C4, C2xC8, M4(2), C24, C2xC12, C3xD4, C3xQ8, C4:C8, C3xC4:C4, C2xC24, C3xM4(2), C3xC4:C8
Smallest permutation representation of C3xC4:C8
►Regular action on 96 points
Generators in S96
(1 67 19)(2 68 20)(3 69 21)(4 70 22)(5 71 23)(6 72 24)(7 65 17)(8 66 18)(9 30 60)(10 31 61)(11 32 62)(12 25 63)(13 26 64)(14 27 57)(15 28 58)(16 29 59)(33 73 81)(34 74 82)(35 75 83)(36 76 84)(37 77 85)(38 78 86)(39 79 87)(40 80 88)(41 53 89)(42 54 90)(43 55 91)(44 56 92)(45 49 93)(46 50 94)(47 51 95)(48 52 96)
(1 43 79 31)(2 32 80 44)(3 45 73 25)(4 26 74 46)(5 47 75 27)(6 28 76 48)(7 41 77 29)(8 30 78 42)(9 38 90 18)(10 19 91 39)(11 40 92 20)(12 21 93 33)(13 34 94 22)(14 23 95 35)(15 36 96 24)(16 17 89 37)(49 81 63 69)(50 70 64 82)(51 83 57 71)(52 72 58 84)(53 85 59 65)(54 66 60 86)(55 87 61 67)(56 68 62 88)
(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)(73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)
G:=sub<Sym(96)| (1,67,19)(2,68,20)(3,69,21)(4,70,22)(5,71,23)(6,72,24)(7,65,17)(8,66,18)(9,30,60)(10,31,61)(11,32,62)(12,25,63)(13,26,64)(14,27,57)(15,28,58)(16,29,59)(33,73,81)(34,74,82)(35,75,83)(36,76,84)(37,77,85)(38,78,86)(39,79,87)(40,80,88)(41,53,89)(42,54,90)(43,55,91)(44,56,92)(45,49,93)(46,50,94)(47,51,95)(48,52,96), (1,43,79,31)(2,32,80,44)(3,45,73,25)(4,26,74,46)(5,47,75,27)(6,28,76,48)(7,41,77,29)(8,30,78,42)(9,38,90,18)(10,19,91,39)(11,40,92,20)(12,21,93,33)(13,34,94,22)(14,23,95,35)(15,36,96,24)(16,17,89,37)(49,81,63,69)(50,70,64,82)(51,83,57,71)(52,72,58,84)(53,85,59,65)(54,66,60,86)(55,87,61,67)(56,68,62,88), (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)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)>;
G:=Group( (1,67,19)(2,68,20)(3,69,21)(4,70,22)(5,71,23)(6,72,24)(7,65,17)(8,66,18)(9,30,60)(10,31,61)(11,32,62)(12,25,63)(13,26,64)(14,27,57)(15,28,58)(16,29,59)(33,73,81)(34,74,82)(35,75,83)(36,76,84)(37,77,85)(38,78,86)(39,79,87)(40,80,88)(41,53,89)(42,54,90)(43,55,91)(44,56,92)(45,49,93)(46,50,94)(47,51,95)(48,52,96), (1,43,79,31)(2,32,80,44)(3,45,73,25)(4,26,74,46)(5,47,75,27)(6,28,76,48)(7,41,77,29)(8,30,78,42)(9,38,90,18)(10,19,91,39)(11,40,92,20)(12,21,93,33)(13,34,94,22)(14,23,95,35)(15,36,96,24)(16,17,89,37)(49,81,63,69)(50,70,64,82)(51,83,57,71)(52,72,58,84)(53,85,59,65)(54,66,60,86)(55,87,61,67)(56,68,62,88), (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)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96) );
G=PermutationGroup([[(1,67,19),(2,68,20),(3,69,21),(4,70,22),(5,71,23),(6,72,24),(7,65,17),(8,66,18),(9,30,60),(10,31,61),(11,32,62),(12,25,63),(13,26,64),(14,27,57),(15,28,58),(16,29,59),(33,73,81),(34,74,82),(35,75,83),(36,76,84),(37,77,85),(38,78,86),(39,79,87),(40,80,88),(41,53,89),(42,54,90),(43,55,91),(44,56,92),(45,49,93),(46,50,94),(47,51,95),(48,52,96)], [(1,43,79,31),(2,32,80,44),(3,45,73,25),(4,26,74,46),(5,47,75,27),(6,28,76,48),(7,41,77,29),(8,30,78,42),(9,38,90,18),(10,19,91,39),(11,40,92,20),(12,21,93,33),(13,34,94,22),(14,23,95,35),(15,36,96,24),(16,17,89,37),(49,81,63,69),(50,70,64,82),(51,83,57,71),(52,72,58,84),(53,85,59,65),(54,66,60,86),(55,87,61,67),(56,68,62,88)], [(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),(73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96)]])
C3xC4:C8 is a maximal subgroup of
C12.53D8 C12.39SD16 C4.Dic12 C12.47D8 D12:2C8 Dic6:2C8 C4.D24 C12.2D8 C42.27D6 Dic6.3Q8 Dic6:C8 C42.198D6 C42.200D6 D12:C8 C42.202D6 D6:3M4(2) C12:M4(2) C12:2M4(2) C42.30D6 C42.31D6 C12:SD16 D12:3Q8 C4:D24 D12.19D4 C42.36D6 D12:4Q8 D12.3Q8 Dic6:8D4 C4:Dic12 Dic6:3Q8 Dic6:4Q8 D4xC24 Q8xC24
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 8A | ··· | 8H | 12A | ··· | 12H | 12I | ··· | 12P | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C8 | C12 | C24 | D4 | Q8 | M4(2) | C3xD4 | C3xQ8 | C3xM4(2) |
| kernel | C3xC4:C8 | C4xC12 | C2xC24 | C4:C8 | C2xC12 | C42 | C2xC8 | C12 | C2xC4 | C4 | C12 | C12 | C6 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 8 | 8 | 16 | 1 | 1 | 2 | 2 | 2 | 4 |
Matrix representation of C3xC4:C8 ►in GL3(F73) generated by
| 1 | 0 | 0 |
| 0 | 64 | 0 |
| 0 | 0 | 64 |
| 72 | 0 | 0 |
| 0 | 0 | 72 |
| 0 | 1 | 0 |
| 51 | 0 | 0 |
| 0 | 36 | 8 |
| 0 | 8 | 37 |
G:=sub<GL(3,GF(73))| [1,0,0,0,64,0,0,0,64],[72,0,0,0,0,1,0,72,0],[51,0,0,0,36,8,0,8,37] >;
C3xC4:C8 in GAP, Magma, Sage, TeX
C_3\times C_4\rtimes C_8
% in TeX
G:=Group("C3xC4:C8"); // GroupNames label
G:=SmallGroup(96,55);
// by ID
G=gap.SmallGroup(96,55);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-2,144,169,79,88]);
// Polycyclic
G:=Group<a,b,c|a^3=b^4=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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