direct product, metabelian, supersoluble, monomial
Aliases: C3×C12.23D4, C62.209C23, (C6×Q8)⋊8C6, D6⋊C4⋊16C6, (C6×Q8)⋊15S3, C6.58(C6×D4), (C4×Dic3)⋊7C6, (C2×D12).9C6, (C3×C12).93D4, C12.23(C3×D4), (C6×D12).14C2, (C2×C12).246D6, (Dic3×C12)⋊17C2, C12.93(C3⋊D4), (C6×C12).129C22, C6.64(Q8⋊3S3), C32⋊17(C4.4D4), (C6×Dic3).140C22, (Q8×C3×C6)⋊4C2, (C2×Q8)⋊8(C3×S3), (C3×D6⋊C4)⋊37C2, (C2×C4).57(S3×C6), C6.37(C3×C4○D4), C3⋊4(C3×C4.4D4), C2.22(C6×C3⋊D4), C4.11(C3×C3⋊D4), C22.65(S3×C2×C6), (C2×C12).40(C2×C6), (C3×C6).266(C2×D4), C6.159(C2×C3⋊D4), (S3×C2×C6).63C22, C2.9(C3×Q8⋊3S3), (C2×C6).64(C22×C6), (C3×C6).159(C4○D4), (C22×S3).13(C2×C6), (C2×C6).342(C22×S3), (C2×Dic3).42(C2×C6), SmallGroup(288,718)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C12.23D4
G = < a,b,c,d | a3=b12=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd=b-1, dcd=b6c-1 >
Subgroups: 426 in 171 conjugacy classes, 66 normal (26 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C2×D4, C2×Q8, C3×S3, C3×C6, C3×C6, D12, C2×Dic3, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×C6, C4.4D4, C3×Dic3, C3×C12, C3×C12, S3×C6, C62, C4×Dic3, D6⋊C4, C4×C12, C3×C22⋊C4, C2×D12, C6×D4, C6×Q8, C6×Q8, C3×D12, C6×Dic3, C6×C12, C6×C12, Q8×C32, S3×C2×C6, C12.23D4, C3×C4.4D4, Dic3×C12, C3×D6⋊C4, C6×D12, Q8×C3×C6, C3×C12.23D4
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C4○D4, C3×S3, C3⋊D4, C3×D4, C22×S3, C22×C6, C4.4D4, S3×C6, Q8⋊3S3, C2×C3⋊D4, C6×D4, C3×C4○D4, C3×C3⋊D4, S3×C2×C6, C12.23D4, C3×C4.4D4, C3×Q8⋊3S3, C6×C3⋊D4, C3×C12.23D4
►On 96 points
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 41 45)(38 42 46)(39 43 47)(40 44 48)(49 57 53)(50 58 54)(51 59 55)(52 60 56)(61 65 69)(62 66 70)(63 67 71)(64 68 72)(73 77 81)(74 78 82)(75 79 83)(76 80 84)(85 93 89)(86 94 90)(87 95 91)(88 96 92)
(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 14 41 35)(2 19 42 28)(3 24 43 33)(4 17 44 26)(5 22 45 31)(6 15 46 36)(7 20 47 29)(8 13 48 34)(9 18 37 27)(10 23 38 32)(11 16 39 25)(12 21 40 30)(49 82 91 69)(50 75 92 62)(51 80 93 67)(52 73 94 72)(53 78 95 65)(54 83 96 70)(55 76 85 63)(56 81 86 68)(57 74 87 61)(58 79 88 66)(59 84 89 71)(60 77 90 64)
(1 51)(2 50)(3 49)(4 60)(5 59)(6 58)(7 57)(8 56)(9 55)(10 54)(11 53)(12 52)(13 62)(14 61)(15 72)(16 71)(17 70)(18 69)(19 68)(20 67)(21 66)(22 65)(23 64)(24 63)(25 84)(26 83)(27 82)(28 81)(29 80)(30 79)(31 78)(32 77)(33 76)(34 75)(35 74)(36 73)(37 85)(38 96)(39 95)(40 94)(41 93)(42 92)(43 91)(44 90)(45 89)(46 88)(47 87)(48 86)
G:=sub<Sym(96)| (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,41,45)(38,42,46)(39,43,47)(40,44,48)(49,57,53)(50,58,54)(51,59,55)(52,60,56)(61,65,69)(62,66,70)(63,67,71)(64,68,72)(73,77,81)(74,78,82)(75,79,83)(76,80,84)(85,93,89)(86,94,90)(87,95,91)(88,96,92), (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), (1,14,41,35)(2,19,42,28)(3,24,43,33)(4,17,44,26)(5,22,45,31)(6,15,46,36)(7,20,47,29)(8,13,48,34)(9,18,37,27)(10,23,38,32)(11,16,39,25)(12,21,40,30)(49,82,91,69)(50,75,92,62)(51,80,93,67)(52,73,94,72)(53,78,95,65)(54,83,96,70)(55,76,85,63)(56,81,86,68)(57,74,87,61)(58,79,88,66)(59,84,89,71)(60,77,90,64), (1,51)(2,50)(3,49)(4,60)(5,59)(6,58)(7,57)(8,56)(9,55)(10,54)(11,53)(12,52)(13,62)(14,61)(15,72)(16,71)(17,70)(18,69)(19,68)(20,67)(21,66)(22,65)(23,64)(24,63)(25,84)(26,83)(27,82)(28,81)(29,80)(30,79)(31,78)(32,77)(33,76)(34,75)(35,74)(36,73)(37,85)(38,96)(39,95)(40,94)(41,93)(42,92)(43,91)(44,90)(45,89)(46,88)(47,87)(48,86)>;
G:=Group( (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,41,45)(38,42,46)(39,43,47)(40,44,48)(49,57,53)(50,58,54)(51,59,55)(52,60,56)(61,65,69)(62,66,70)(63,67,71)(64,68,72)(73,77,81)(74,78,82)(75,79,83)(76,80,84)(85,93,89)(86,94,90)(87,95,91)(88,96,92), (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), (1,14,41,35)(2,19,42,28)(3,24,43,33)(4,17,44,26)(5,22,45,31)(6,15,46,36)(7,20,47,29)(8,13,48,34)(9,18,37,27)(10,23,38,32)(11,16,39,25)(12,21,40,30)(49,82,91,69)(50,75,92,62)(51,80,93,67)(52,73,94,72)(53,78,95,65)(54,83,96,70)(55,76,85,63)(56,81,86,68)(57,74,87,61)(58,79,88,66)(59,84,89,71)(60,77,90,64), (1,51)(2,50)(3,49)(4,60)(5,59)(6,58)(7,57)(8,56)(9,55)(10,54)(11,53)(12,52)(13,62)(14,61)(15,72)(16,71)(17,70)(18,69)(19,68)(20,67)(21,66)(22,65)(23,64)(24,63)(25,84)(26,83)(27,82)(28,81)(29,80)(30,79)(31,78)(32,77)(33,76)(34,75)(35,74)(36,73)(37,85)(38,96)(39,95)(40,94)(41,93)(42,92)(43,91)(44,90)(45,89)(46,88)(47,87)(48,86) );
G=PermutationGroup([[(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,21,17),(14,22,18),(15,23,19),(16,24,20),(25,33,29),(26,34,30),(27,35,31),(28,36,32),(37,41,45),(38,42,46),(39,43,47),(40,44,48),(49,57,53),(50,58,54),(51,59,55),(52,60,56),(61,65,69),(62,66,70),(63,67,71),(64,68,72),(73,77,81),(74,78,82),(75,79,83),(76,80,84),(85,93,89),(86,94,90),(87,95,91),(88,96,92)], [(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)], [(1,14,41,35),(2,19,42,28),(3,24,43,33),(4,17,44,26),(5,22,45,31),(6,15,46,36),(7,20,47,29),(8,13,48,34),(9,18,37,27),(10,23,38,32),(11,16,39,25),(12,21,40,30),(49,82,91,69),(50,75,92,62),(51,80,93,67),(52,73,94,72),(53,78,95,65),(54,83,96,70),(55,76,85,63),(56,81,86,68),(57,74,87,61),(58,79,88,66),(59,84,89,71),(60,77,90,64)], [(1,51),(2,50),(3,49),(4,60),(5,59),(6,58),(7,57),(8,56),(9,55),(10,54),(11,53),(12,52),(13,62),(14,61),(15,72),(16,71),(17,70),(18,69),(19,68),(20,67),(21,66),(22,65),(23,64),(24,63),(25,84),(26,83),(27,82),(28,81),(29,80),(30,79),(31,78),(32,77),(33,76),(34,75),(35,74),(36,73),(37,85),(38,96),(39,95),(40,94),(41,93),(42,92),(43,91),(44,90),(45,89),(46,88),(47,87),(48,86)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 6G | ··· | 6O | 6P | 6Q | 6R | 6S | 12A | 12B | 12C | 12D | 12E | ··· | 12Z | 12AA | ··· | 12AH |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | S3 | D4 | D6 | C4○D4 | C3×S3 | C3⋊D4 | C3×D4 | S3×C6 | C3×C4○D4 | C3×C3⋊D4 | Q8⋊3S3 | C3×Q8⋊3S3 |
| kernel | C3×C12.23D4 | Dic3×C12 | C3×D6⋊C4 | C6×D12 | Q8×C3×C6 | C12.23D4 | C4×Dic3 | D6⋊C4 | C2×D12 | C6×Q8 | C6×Q8 | C3×C12 | C2×C12 | C3×C6 | C2×Q8 | C12 | C12 | C2×C4 | C6 | C4 | C6 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 1 | 2 | 3 | 4 | 2 | 4 | 4 | 6 | 8 | 8 | 2 | 4 |
Matrix representation of C3×C12.23D4 ►in GL4(𝔽13) generated by
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 3 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 8 |
| 0 | 0 | 3 | 12 |
| 0 | 1 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 8 | 3 |
| 0 | 0 | 5 | 5 |
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[3,0,0,0,0,9,0,0,0,0,1,3,0,0,8,12],[0,12,0,0,1,0,0,0,0,0,5,0,0,0,0,5],[0,1,0,0,1,0,0,0,0,0,8,5,0,0,3,5] >;
C3×C12.23D4 in GAP, Magma, Sage, TeX
C_3\times C_{12}._{23}D_4 % in TeX
G:=Group("C3xC12.23D4"); // GroupNames label
G:=SmallGroup(288,718);
// by ID
G=gap.SmallGroup(288,718);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,344,590,555,268,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^12=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d=b^-1,d*c*d=b^6*c^-1>;
// generators/relations