direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C8⋊C4, C24⋊7C4, C8⋊3C12, C6.7C42, C42.1C6, C6.7M4(2), (C2×C8).7C6, (C2×C4).2C12, (C2×C12).5C4, (C4×C12).1C2, C2.2(C4×C12), (C2×C24).17C2, C4.11(C2×C12), C12.48(C2×C4), C22.8(C2×C12), C2.1(C3×M4(2)), (C2×C12).134C22, (C2×C6).37(C2×C4), (C2×C4).30(C2×C6), SmallGroup(96,47)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C8⋊C4
G = < a,b,c | a3=b8=c4=1, ab=ba, ac=ca, cbc-1=b5 >
Smallest permutation representation of C3×C8⋊C4
►Regular action on 96 points
Generators in S96
(1 50 26)(2 51 27)(3 52 28)(4 53 29)(5 54 30)(6 55 31)(7 56 32)(8 49 25)(9 67 83)(10 68 84)(11 69 85)(12 70 86)(13 71 87)(14 72 88)(15 65 81)(16 66 82)(17 80 91)(18 73 92)(19 74 93)(20 75 94)(21 76 95)(22 77 96)(23 78 89)(24 79 90)(33 42 61)(34 43 62)(35 44 63)(36 45 64)(37 46 57)(38 47 58)(39 48 59)(40 41 60)
(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)
(1 37 17 67)(2 34 18 72)(3 39 19 69)(4 36 20 66)(5 33 21 71)(6 38 22 68)(7 35 23 65)(8 40 24 70)(9 26 57 91)(10 31 58 96)(11 28 59 93)(12 25 60 90)(13 30 61 95)(14 27 62 92)(15 32 63 89)(16 29 64 94)(41 79 86 49)(42 76 87 54)(43 73 88 51)(44 78 81 56)(45 75 82 53)(46 80 83 50)(47 77 84 55)(48 74 85 52)
G:=sub<Sym(96)| (1,50,26)(2,51,27)(3,52,28)(4,53,29)(5,54,30)(6,55,31)(7,56,32)(8,49,25)(9,67,83)(10,68,84)(11,69,85)(12,70,86)(13,71,87)(14,72,88)(15,65,81)(16,66,82)(17,80,91)(18,73,92)(19,74,93)(20,75,94)(21,76,95)(22,77,96)(23,78,89)(24,79,90)(33,42,61)(34,43,62)(35,44,63)(36,45,64)(37,46,57)(38,47,58)(39,48,59)(40,41,60), (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), (1,37,17,67)(2,34,18,72)(3,39,19,69)(4,36,20,66)(5,33,21,71)(6,38,22,68)(7,35,23,65)(8,40,24,70)(9,26,57,91)(10,31,58,96)(11,28,59,93)(12,25,60,90)(13,30,61,95)(14,27,62,92)(15,32,63,89)(16,29,64,94)(41,79,86,49)(42,76,87,54)(43,73,88,51)(44,78,81,56)(45,75,82,53)(46,80,83,50)(47,77,84,55)(48,74,85,52)>;
G:=Group( (1,50,26)(2,51,27)(3,52,28)(4,53,29)(5,54,30)(6,55,31)(7,56,32)(8,49,25)(9,67,83)(10,68,84)(11,69,85)(12,70,86)(13,71,87)(14,72,88)(15,65,81)(16,66,82)(17,80,91)(18,73,92)(19,74,93)(20,75,94)(21,76,95)(22,77,96)(23,78,89)(24,79,90)(33,42,61)(34,43,62)(35,44,63)(36,45,64)(37,46,57)(38,47,58)(39,48,59)(40,41,60), (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), (1,37,17,67)(2,34,18,72)(3,39,19,69)(4,36,20,66)(5,33,21,71)(6,38,22,68)(7,35,23,65)(8,40,24,70)(9,26,57,91)(10,31,58,96)(11,28,59,93)(12,25,60,90)(13,30,61,95)(14,27,62,92)(15,32,63,89)(16,29,64,94)(41,79,86,49)(42,76,87,54)(43,73,88,51)(44,78,81,56)(45,75,82,53)(46,80,83,50)(47,77,84,55)(48,74,85,52) );
G=PermutationGroup([[(1,50,26),(2,51,27),(3,52,28),(4,53,29),(5,54,30),(6,55,31),(7,56,32),(8,49,25),(9,67,83),(10,68,84),(11,69,85),(12,70,86),(13,71,87),(14,72,88),(15,65,81),(16,66,82),(17,80,91),(18,73,92),(19,74,93),(20,75,94),(21,76,95),(22,77,96),(23,78,89),(24,79,90),(33,42,61),(34,43,62),(35,44,63),(36,45,64),(37,46,57),(38,47,58),(39,48,59),(40,41,60)], [(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)], [(1,37,17,67),(2,34,18,72),(3,39,19,69),(4,36,20,66),(5,33,21,71),(6,38,22,68),(7,35,23,65),(8,40,24,70),(9,26,57,91),(10,31,58,96),(11,28,59,93),(12,25,60,90),(13,30,61,95),(14,27,62,92),(15,32,63,89),(16,29,64,94),(41,79,86,49),(42,76,87,54),(43,73,88,51),(44,78,81,56),(45,75,82,53),(46,80,83,50),(47,77,84,55),(48,74,85,52)]])
C3×C8⋊C4 is a maximal subgroup of
C42.D6 C42.2D6 C12.15C42 C24⋊Q8 C8⋊Dic6 C42.14D6 C42.182D6 C8⋊9D12 Dic3⋊5M4(2) D6.4C42 C42.185D6 C42.16D6 D24⋊C4 C8⋊D12 C42.19D6 C42.20D6 C8.D12 Dic12⋊C4 D24⋊4C4 C12×M4(2)
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 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | M4(2) | C3×M4(2) |
| kernel | C3×C8⋊C4 | C4×C12 | C2×C24 | C8⋊C4 | C24 | C2×C12 | C42 | C2×C8 | C8 | C2×C4 | C6 | C2 |
| # reps | 1 | 1 | 2 | 2 | 8 | 4 | 2 | 4 | 16 | 8 | 4 | 8 |
Matrix representation of C3×C8⋊C4 ►in GL4(𝔽73) generated by
| 64 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 59 | 54 |
| 0 | 0 | 54 | 14 |
| 72 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 0 |
G:=sub<GL(4,GF(73))| [64,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,59,54,0,0,54,14],[72,0,0,0,0,46,0,0,0,0,0,72,0,0,1,0] >;
C3×C8⋊C4 in GAP, Magma, Sage, TeX
C_3\times C_8\rtimes C_4
% in TeX
G:=Group("C3xC8:C4"); // GroupNames label
G:=SmallGroup(96,47);
// by ID
G=gap.SmallGroup(96,47);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-2,72,601,151,117]);
// Polycyclic
G:=Group<a,b,c|a^3=b^8=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations
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