metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.1S3, C6.2C42, C6.4M4(2), C3⋊C8⋊3C4, C3⋊1(C8⋊C4), C4.19(C4×S3), (C4×C12).7C2, (C2×C12).3C4, (C2×C4).88D6, C12.24(C2×C4), (C2×C4).2Dic3, C2.3(C4×Dic3), C2.1(C4.Dic3), C22.7(C2×Dic3), (C2×C12).102C22, (C2×C3⋊C8).7C2, (C2×C6).25(C2×C4), SmallGroup(96,10)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.S3
G = < a,b,c | a6=c4=1, b4=a3, bab-1=a-1, ac=ca, cbc-1=a3b >
Smallest permutation representation of C42.S3
►Regular action on 96 points
(1 18 67 5 22 71)(2 72 23 6 68 19)(3 20 69 7 24 65)(4 66 17 8 70 21)(9 30 33 13 26 37)(10 38 27 14 34 31)(11 32 35 15 28 39)(12 40 29 16 36 25)(41 83 89 45 87 93)(42 94 88 46 90 84)(43 85 91 47 81 95)(44 96 82 48 92 86)(49 63 73 53 59 77)(50 78 60 54 74 64)(51 57 75 55 61 79)(52 80 62 56 76 58)
(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 77 47 15)(2 74 48 12)(3 79 41 9)(4 76 42 14)(5 73 43 11)(6 78 44 16)(7 75 45 13)(8 80 46 10)(17 52 88 31)(18 49 81 28)(19 54 82 25)(20 51 83 30)(21 56 84 27)(22 53 85 32)(23 50 86 29)(24 55 87 26)(33 69 57 89)(34 66 58 94)(35 71 59 91)(36 68 60 96)(37 65 61 93)(38 70 62 90)(39 67 63 95)(40 72 64 92)
G:=sub<Sym(96)| (1,18,67,5,22,71)(2,72,23,6,68,19)(3,20,69,7,24,65)(4,66,17,8,70,21)(9,30,33,13,26,37)(10,38,27,14,34,31)(11,32,35,15,28,39)(12,40,29,16,36,25)(41,83,89,45,87,93)(42,94,88,46,90,84)(43,85,91,47,81,95)(44,96,82,48,92,86)(49,63,73,53,59,77)(50,78,60,54,74,64)(51,57,75,55,61,79)(52,80,62,56,76,58), (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), (1,77,47,15)(2,74,48,12)(3,79,41,9)(4,76,42,14)(5,73,43,11)(6,78,44,16)(7,75,45,13)(8,80,46,10)(17,52,88,31)(18,49,81,28)(19,54,82,25)(20,51,83,30)(21,56,84,27)(22,53,85,32)(23,50,86,29)(24,55,87,26)(33,69,57,89)(34,66,58,94)(35,71,59,91)(36,68,60,96)(37,65,61,93)(38,70,62,90)(39,67,63,95)(40,72,64,92)>;
G:=Group( (1,18,67,5,22,71)(2,72,23,6,68,19)(3,20,69,7,24,65)(4,66,17,8,70,21)(9,30,33,13,26,37)(10,38,27,14,34,31)(11,32,35,15,28,39)(12,40,29,16,36,25)(41,83,89,45,87,93)(42,94,88,46,90,84)(43,85,91,47,81,95)(44,96,82,48,92,86)(49,63,73,53,59,77)(50,78,60,54,74,64)(51,57,75,55,61,79)(52,80,62,56,76,58), (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), (1,77,47,15)(2,74,48,12)(3,79,41,9)(4,76,42,14)(5,73,43,11)(6,78,44,16)(7,75,45,13)(8,80,46,10)(17,52,88,31)(18,49,81,28)(19,54,82,25)(20,51,83,30)(21,56,84,27)(22,53,85,32)(23,50,86,29)(24,55,87,26)(33,69,57,89)(34,66,58,94)(35,71,59,91)(36,68,60,96)(37,65,61,93)(38,70,62,90)(39,67,63,95)(40,72,64,92) );
G=PermutationGroup([[(1,18,67,5,22,71),(2,72,23,6,68,19),(3,20,69,7,24,65),(4,66,17,8,70,21),(9,30,33,13,26,37),(10,38,27,14,34,31),(11,32,35,15,28,39),(12,40,29,16,36,25),(41,83,89,45,87,93),(42,94,88,46,90,84),(43,85,91,47,81,95),(44,96,82,48,92,86),(49,63,73,53,59,77),(50,78,60,54,74,64),(51,57,75,55,61,79),(52,80,62,56,76,58)], [(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)], [(1,77,47,15),(2,74,48,12),(3,79,41,9),(4,76,42,14),(5,73,43,11),(6,78,44,16),(7,75,45,13),(8,80,46,10),(17,52,88,31),(18,49,81,28),(19,54,82,25),(20,51,83,30),(21,56,84,27),(22,53,85,32),(23,50,86,29),(24,55,87,26),(33,69,57,89),(34,66,58,94),(35,71,59,91),(36,68,60,96),(37,65,61,93),(38,70,62,90),(39,67,63,95),(40,72,64,92)]])
C42.S3 is a maximal subgroup of
C42.D6 C42.2D6 C42.7D6 C42.8D6 D6.C42 C42.243D6 S3×C8⋊C4 C42.182D6 Dic3⋊5M4(2) M4(2).22D6 C42.27D6 D6⋊3M4(2) C4×C4.Dic3 C42.270D6 C12.5C42 C42.187D6 C42.47D6 C42.48D6 C42.51D6 C42.210D6 C42.56D6 C42.59D6 C42.62D6 C42.64D6 C42.65D6 C42.68D6 C42.70D6 C42.71D6 C42.72D6 C42.74D6 C42.76D6 C42.77D6 C42.80D6 C42.82D6 C42.D9 C3⋊C8⋊Dic3 C122.C2 C30.21C42 C42.D15 C30.3C42 C30.11C42
C42.S3 is a maximal quotient of
C42.279D6 C12.15C42 (C2×C12)⋊3C8 C42.D9 C3⋊C8⋊Dic3 C122.C2 C30.21C42 C42.D15 C30.3C42 C30.11C42
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 8A | ··· | 8H | 12A | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 2 | ··· | 2 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | |||||
| image | C1 | C2 | C2 | C4 | C4 | S3 | Dic3 | D6 | M4(2) | C4×S3 | C4.Dic3 |
| kernel | C42.S3 | C2×C3⋊C8 | C4×C12 | C3⋊C8 | C2×C12 | C42 | C2×C4 | C2×C4 | C6 | C4 | C2 |
| # reps | 1 | 2 | 1 | 8 | 4 | 1 | 2 | 1 | 4 | 4 | 8 |
Matrix representation of C42.S3 ►in GL3(𝔽73) generated by
| 1 | 0 | 0 |
| 0 | 1 | 1 |
| 0 | 72 | 0 |
| 27 | 0 | 0 |
| 0 | 44 | 67 |
| 0 | 23 | 29 |
| 27 | 0 | 0 |
| 0 | 30 | 60 |
| 0 | 13 | 43 |
G:=sub<GL(3,GF(73))| [1,0,0,0,1,72,0,1,0],[27,0,0,0,44,23,0,67,29],[27,0,0,0,30,13,0,60,43] >;
C42.S3 in GAP, Magma, Sage, TeX
C_4^2.S_3
% in TeX
G:=Group("C4^2.S3"); // GroupNames label
G:=SmallGroup(96,10);
// by ID
G=gap.SmallGroup(96,10);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,217,55,86,2309]);
// Polycyclic
G:=Group<a,b,c|a^6=c^4=1,b^4=a^3,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^3*b>;
// generators/relations
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