direct product, abelian, monomial, 2-elementary
Aliases: C23×C12, SmallGroup(96,220)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C23×C12 |
| C1 — C23×C12 |
| C1 — C23×C12 |
Generators and relations for C23×C12
G = < a,b,c,d | a2=b2=c2=d12=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >
Subgroups: 236, all normal (8 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C2×C4, C23, C12, C2×C6, C22×C4, C24, C2×C12, C22×C6, C23×C4, C22×C12, C23×C6, C23×C12
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, C22×C4, C24, C2×C12, C22×C6, C23×C4, C22×C12, C23×C6, C23×C12
►Regular action on 96 points
Generators in S96
(1 70)(2 71)(3 72)(4 61)(5 62)(6 63)(7 64)(8 65)(9 66)(10 67)(11 68)(12 69)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 74)(26 75)(27 76)(28 77)(29 78)(30 79)(31 80)(32 81)(33 82)(34 83)(35 84)(36 73)(49 89)(50 90)(51 91)(52 92)(53 93)(54 94)(55 95)(56 96)(57 85)(58 86)(59 87)(60 88)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 25)(11 26)(12 27)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(37 57)(38 58)(39 59)(40 60)(41 49)(42 50)(43 51)(44 52)(45 53)(46 54)(47 55)(48 56)(61 80)(62 81)(63 82)(64 83)(65 84)(66 73)(67 74)(68 75)(69 76)(70 77)(71 78)(72 79)
(1 89)(2 90)(3 91)(4 92)(5 93)(6 94)(7 95)(8 96)(9 85)(10 86)(11 87)(12 88)(13 36)(14 25)(15 26)(16 27)(17 28)(18 29)(19 30)(20 31)(21 32)(22 33)(23 34)(24 35)(37 73)(38 74)(39 75)(40 76)(41 77)(42 78)(43 79)(44 80)(45 81)(46 82)(47 83)(48 84)(49 70)(50 71)(51 72)(52 61)(53 62)(54 63)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)
(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)| (1,70)(2,71)(3,72)(4,61)(5,62)(6,63)(7,64)(8,65)(9,66)(10,67)(11,68)(12,69)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,81)(33,82)(34,83)(35,84)(36,73)(49,89)(50,90)(51,91)(52,92)(53,93)(54,94)(55,95)(56,96)(57,85)(58,86)(59,87)(60,88), (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,25)(11,26)(12,27)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(37,57)(38,58)(39,59)(40,60)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56)(61,80)(62,81)(63,82)(64,83)(65,84)(66,73)(67,74)(68,75)(69,76)(70,77)(71,78)(72,79), (1,89)(2,90)(3,91)(4,92)(5,93)(6,94)(7,95)(8,96)(9,85)(10,86)(11,87)(12,88)(13,36)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,31)(21,32)(22,33)(23,34)(24,35)(37,73)(38,74)(39,75)(40,76)(41,77)(42,78)(43,79)(44,80)(45,81)(46,82)(47,83)(48,84)(49,70)(50,71)(51,72)(52,61)(53,62)(54,63)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69), (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( (1,70)(2,71)(3,72)(4,61)(5,62)(6,63)(7,64)(8,65)(9,66)(10,67)(11,68)(12,69)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,81)(33,82)(34,83)(35,84)(36,73)(49,89)(50,90)(51,91)(52,92)(53,93)(54,94)(55,95)(56,96)(57,85)(58,86)(59,87)(60,88), (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,25)(11,26)(12,27)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(37,57)(38,58)(39,59)(40,60)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56)(61,80)(62,81)(63,82)(64,83)(65,84)(66,73)(67,74)(68,75)(69,76)(70,77)(71,78)(72,79), (1,89)(2,90)(3,91)(4,92)(5,93)(6,94)(7,95)(8,96)(9,85)(10,86)(11,87)(12,88)(13,36)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,31)(21,32)(22,33)(23,34)(24,35)(37,73)(38,74)(39,75)(40,76)(41,77)(42,78)(43,79)(44,80)(45,81)(46,82)(47,83)(48,84)(49,70)(50,71)(51,72)(52,61)(53,62)(54,63)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69), (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([[(1,70),(2,71),(3,72),(4,61),(5,62),(6,63),(7,64),(8,65),(9,66),(10,67),(11,68),(12,69),(13,37),(14,38),(15,39),(16,40),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48),(25,74),(26,75),(27,76),(28,77),(29,78),(30,79),(31,80),(32,81),(33,82),(34,83),(35,84),(36,73),(49,89),(50,90),(51,91),(52,92),(53,93),(54,94),(55,95),(56,96),(57,85),(58,86),(59,87),(60,88)], [(1,28),(2,29),(3,30),(4,31),(5,32),(6,33),(7,34),(8,35),(9,36),(10,25),(11,26),(12,27),(13,85),(14,86),(15,87),(16,88),(17,89),(18,90),(19,91),(20,92),(21,93),(22,94),(23,95),(24,96),(37,57),(38,58),(39,59),(40,60),(41,49),(42,50),(43,51),(44,52),(45,53),(46,54),(47,55),(48,56),(61,80),(62,81),(63,82),(64,83),(65,84),(66,73),(67,74),(68,75),(69,76),(70,77),(71,78),(72,79)], [(1,89),(2,90),(3,91),(4,92),(5,93),(6,94),(7,95),(8,96),(9,85),(10,86),(11,87),(12,88),(13,36),(14,25),(15,26),(16,27),(17,28),(18,29),(19,30),(20,31),(21,32),(22,33),(23,34),(24,35),(37,73),(38,74),(39,75),(40,76),(41,77),(42,78),(43,79),(44,80),(45,81),(46,82),(47,83),(48,84),(49,70),(50,71),(51,72),(52,61),(53,62),(54,63),(55,64),(56,65),(57,66),(58,67),(59,68),(60,69)], [(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)]])
C23×C12 is a maximal subgroup of
C24.6Dic3 C24.73D6 C24.74D6 C24.75D6 C24.76D6 C24.83D6
96 conjugacy classes
| class | 1 | 2A | ··· | 2O | 3A | 3B | 4A | ··· | 4P | 6A | ··· | 6AD | 12A | ··· | 12AF |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 |
| kernel | C23×C12 | C22×C12 | C23×C6 | C23×C4 | C22×C6 | C22×C4 | C24 | C23 |
| # reps | 1 | 14 | 1 | 2 | 16 | 28 | 2 | 32 |
Matrix representation of C23×C12 ►in GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 6 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 3 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,1,0,0,0,0,1,0,0,0,0,12],[12,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[6,0,0,0,0,7,0,0,0,0,10,0,0,0,0,3] >;
C23×C12 in GAP, Magma, Sage, TeX
C_2^3\times C_{12} % in TeX
G:=Group("C2^3xC12"); // GroupNames label
G:=SmallGroup(96,220);
// by ID
G=gap.SmallGroup(96,220);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-2,288]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^2=d^12=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,c*d=d*c>;
// generators/relations