direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×C3⋊C8, C12⋊2C8, C42.6S3, C6.1C42, C3⋊1(C4×C8), C6.6(C2×C8), C4.18(C4×S3), (C4×C12).6C2, (C2×C4).87D6, C12.23(C2×C4), (C2×C12).10C4, C2.1(C4×Dic3), (C2×C4).7Dic3, C22.6(C2×Dic3), (C2×C12).101C22, C2.1(C2×C3⋊C8), C42○(C2×C3⋊C8), (C2×C3⋊C8).11C2, (C2×C6).24(C2×C4), SmallGroup(96,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C4×C3⋊C8 |
Generators and relations for C4×C3⋊C8
G = < a,b,c | a4=b3=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C4×C3⋊C8
►Regular action on 96 points
(1 76 23 10)(2 77 24 11)(3 78 17 12)(4 79 18 13)(5 80 19 14)(6 73 20 15)(7 74 21 16)(8 75 22 9)(25 72 53 42)(26 65 54 43)(27 66 55 44)(28 67 56 45)(29 68 49 46)(30 69 50 47)(31 70 51 48)(32 71 52 41)(33 81 64 90)(34 82 57 91)(35 83 58 92)(36 84 59 93)(37 85 60 94)(38 86 61 95)(39 87 62 96)(40 88 63 89)
(1 71 82)(2 83 72)(3 65 84)(4 85 66)(5 67 86)(6 87 68)(7 69 88)(8 81 70)(9 33 31)(10 32 34)(11 35 25)(12 26 36)(13 37 27)(14 28 38)(15 39 29)(16 30 40)(17 43 93)(18 94 44)(19 45 95)(20 96 46)(21 47 89)(22 90 48)(23 41 91)(24 92 42)(49 73 62)(50 63 74)(51 75 64)(52 57 76)(53 77 58)(54 59 78)(55 79 60)(56 61 80)
(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)| (1,76,23,10)(2,77,24,11)(3,78,17,12)(4,79,18,13)(5,80,19,14)(6,73,20,15)(7,74,21,16)(8,75,22,9)(25,72,53,42)(26,65,54,43)(27,66,55,44)(28,67,56,45)(29,68,49,46)(30,69,50,47)(31,70,51,48)(32,71,52,41)(33,81,64,90)(34,82,57,91)(35,83,58,92)(36,84,59,93)(37,85,60,94)(38,86,61,95)(39,87,62,96)(40,88,63,89), (1,71,82)(2,83,72)(3,65,84)(4,85,66)(5,67,86)(6,87,68)(7,69,88)(8,81,70)(9,33,31)(10,32,34)(11,35,25)(12,26,36)(13,37,27)(14,28,38)(15,39,29)(16,30,40)(17,43,93)(18,94,44)(19,45,95)(20,96,46)(21,47,89)(22,90,48)(23,41,91)(24,92,42)(49,73,62)(50,63,74)(51,75,64)(52,57,76)(53,77,58)(54,59,78)(55,79,60)(56,61,80), (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( (1,76,23,10)(2,77,24,11)(3,78,17,12)(4,79,18,13)(5,80,19,14)(6,73,20,15)(7,74,21,16)(8,75,22,9)(25,72,53,42)(26,65,54,43)(27,66,55,44)(28,67,56,45)(29,68,49,46)(30,69,50,47)(31,70,51,48)(32,71,52,41)(33,81,64,90)(34,82,57,91)(35,83,58,92)(36,84,59,93)(37,85,60,94)(38,86,61,95)(39,87,62,96)(40,88,63,89), (1,71,82)(2,83,72)(3,65,84)(4,85,66)(5,67,86)(6,87,68)(7,69,88)(8,81,70)(9,33,31)(10,32,34)(11,35,25)(12,26,36)(13,37,27)(14,28,38)(15,39,29)(16,30,40)(17,43,93)(18,94,44)(19,45,95)(20,96,46)(21,47,89)(22,90,48)(23,41,91)(24,92,42)(49,73,62)(50,63,74)(51,75,64)(52,57,76)(53,77,58)(54,59,78)(55,79,60)(56,61,80), (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([[(1,76,23,10),(2,77,24,11),(3,78,17,12),(4,79,18,13),(5,80,19,14),(6,73,20,15),(7,74,21,16),(8,75,22,9),(25,72,53,42),(26,65,54,43),(27,66,55,44),(28,67,56,45),(29,68,49,46),(30,69,50,47),(31,70,51,48),(32,71,52,41),(33,81,64,90),(34,82,57,91),(35,83,58,92),(36,84,59,93),(37,85,60,94),(38,86,61,95),(39,87,62,96),(40,88,63,89)], [(1,71,82),(2,83,72),(3,65,84),(4,85,66),(5,67,86),(6,87,68),(7,69,88),(8,81,70),(9,33,31),(10,32,34),(11,35,25),(12,26,36),(13,37,27),(14,28,38),(15,39,29),(16,30,40),(17,43,93),(18,94,44),(19,45,95),(20,96,46),(21,47,89),(22,90,48),(23,41,91),(24,92,42),(49,73,62),(50,63,74),(51,75,64),(52,57,76),(53,77,58),(54,59,78),(55,79,60),(56,61,80)], [(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)]])
C4×C3⋊C8 is a maximal subgroup of
C42.279D6 C24⋊C8 C12.53D8 C12.39SD16 D12⋊2C8 Dic6⋊2C8 C12.57D8 C12.26Q16 S3×C4×C8 C42.282D6 D6.4C42 C42.185D6 C42.196D6 Dic6⋊C8 C42.198D6 D12⋊C8 C12⋊2M4(2) C42.285D6 C12.5C42 C42.187D6 C12⋊3M4(2) C42.210D6 C42.213D6 C42.214D6 C42.215D6 C42.216D6 C12.16D8 C12⋊D8 C12⋊4SD16 C12.17D8 C12.9Q16 C12.SD16 C12⋊6SD16 C12.D8 C12⋊3Q16 C12.Q16
C4×C3⋊C8 is a maximal quotient of
C24⋊C8 C24.C8 (C2×C12)⋊3C8
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | ··· | 4L | 6A | 6B | 6C | 8A | ··· | 8P | 12A | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 2 | ··· | 2 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | |||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | S3 | Dic3 | D6 | C3⋊C8 | C4×S3 |
| kernel | C4×C3⋊C8 | C2×C3⋊C8 | C4×C12 | C3⋊C8 | C2×C12 | C12 | C42 | C2×C4 | C2×C4 | C4 | C4 |
| # reps | 1 | 2 | 1 | 8 | 4 | 16 | 1 | 2 | 1 | 8 | 4 |
Matrix representation of C4×C3⋊C8 ►in GL3(𝔽73) generated by
| 46 | 0 | 0 |
| 0 | 27 | 0 |
| 0 | 0 | 27 |
| 1 | 0 | 0 |
| 0 | 0 | 72 |
| 0 | 1 | 72 |
| 27 | 0 | 0 |
| 0 | 23 | 6 |
| 0 | 29 | 50 |
G:=sub<GL(3,GF(73))| [46,0,0,0,27,0,0,0,27],[1,0,0,0,0,1,0,72,72],[27,0,0,0,23,29,0,6,50] >;
C4×C3⋊C8 in GAP, Magma, Sage, TeX
C_4\times C_3\rtimes C_8
% in TeX
G:=Group("C4xC3:C8"); // GroupNames label
G:=SmallGroup(96,9);
// by ID
G=gap.SmallGroup(96,9);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,55,86,2309]);
// Polycyclic
G:=Group<a,b,c|a^4=b^3=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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