direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C12.23D4, (C2×Q8)⋊32D6, D6⋊C4⋊74C22, C12.259(C2×D4), (C2×C12).215D4, C6⋊4(C4.4D4), (C22×Q8)⋊13S3, (C6×Q8)⋊36C22, (C2×C6).306C24, (C22×C4).401D6, C6.154(C22×D4), (C2×C12).646C23, (C4×Dic3)⋊69C22, (C22×D12).19C2, (C2×D12).278C22, (S3×C23).78C22, C22.317(S3×C23), C23.352(C22×S3), (C22×C6).424C23, (C22×S3).133C23, (C22×C12).439C22, C22.40(Q8⋊3S3), (C2×Dic3).289C23, (C22×Dic3).234C22, (Q8×C2×C6)⋊5C2, C3⋊5(C2×C4.4D4), (C2×D6⋊C4)⋊43C2, (C2×C4×Dic3)⋊13C2, C4.28(C2×C3⋊D4), C6.128(C2×C4○D4), (C2×C6).589(C2×D4), C2.35(C2×Q8⋊3S3), C2.27(C22×C3⋊D4), (C2×C6).201(C4○D4), (C2×C4).157(C3⋊D4), (C2×C4).243(C22×S3), C22.117(C2×C3⋊D4), SmallGroup(192,1373)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 904 in 330 conjugacy classes, 127 normal (15 characteristic)
C1, C2, C2 [×6], C2 [×4], C3, C4 [×4], C4 [×8], C22, C22 [×6], C22 [×20], S3 [×4], C6, C6 [×6], C2×C4 [×10], C2×C4 [×12], D4 [×8], Q8 [×8], C23, C23 [×16], Dic3 [×4], C12 [×4], C12 [×4], D6 [×20], C2×C6, C2×C6 [×6], C42 [×4], C22⋊C4 [×16], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×8], C2×Q8 [×4], C2×Q8 [×4], C24 [×2], D12 [×8], C2×Dic3 [×4], C2×Dic3 [×4], C2×C12 [×10], C2×C12 [×4], C3×Q8 [×8], C22×S3 [×4], C22×S3 [×12], C22×C6, C2×C42, C2×C22⋊C4 [×4], C4.4D4 [×8], C22×D4, C22×Q8, C4×Dic3 [×4], D6⋊C4 [×16], C2×D12 [×4], C2×D12 [×4], C22×Dic3 [×2], C22×C12, C22×C12 [×2], C6×Q8 [×4], C6×Q8 [×4], S3×C23 [×2], C2×C4.4D4, C2×C4×Dic3, C2×D6⋊C4 [×4], C12.23D4 [×8], C22×D12, Q8×C2×C6, C2×C12.23D4
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×4], C24, C3⋊D4 [×4], C22×S3 [×7], C4.4D4 [×4], C22×D4, C2×C4○D4 [×2], Q8⋊3S3 [×4], C2×C3⋊D4 [×6], S3×C23, C2×C4.4D4, C12.23D4 [×4], C2×Q8⋊3S3 [×2], C22×C3⋊D4, C2×C12.23D4
Generators and relations
G = < a,b,c,d | a2=b12=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd=b-1, dcd=b6c-1 >
►On 96 points
(1 39)(2 40)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 37)(12 38)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(49 95)(50 96)(51 85)(52 86)(53 87)(54 88)(55 89)(56 90)(57 91)(58 92)(59 93)(60 94)(61 82)(62 83)(63 84)(64 73)(65 74)(66 75)(67 76)(68 77)(69 78)(70 79)(71 80)(72 81)
(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 61 53 23)(2 66 54 16)(3 71 55 21)(4 64 56 14)(5 69 57 19)(6 62 58 24)(7 67 59 17)(8 72 60 22)(9 65 49 15)(10 70 50 20)(11 63 51 13)(12 68 52 18)(25 43 78 91)(26 48 79 96)(27 41 80 89)(28 46 81 94)(29 39 82 87)(30 44 83 92)(31 37 84 85)(32 42 73 90)(33 47 74 95)(34 40 75 88)(35 45 76 93)(36 38 77 86)
(1 45)(2 44)(3 43)(4 42)(5 41)(6 40)(7 39)(8 38)(9 37)(10 48)(11 47)(12 46)(13 80)(14 79)(15 78)(16 77)(17 76)(18 75)(19 74)(20 73)(21 84)(22 83)(23 82)(24 81)(25 65)(26 64)(27 63)(28 62)(29 61)(30 72)(31 71)(32 70)(33 69)(34 68)(35 67)(36 66)(49 85)(50 96)(51 95)(52 94)(53 93)(54 92)(55 91)(56 90)(57 89)(58 88)(59 87)(60 86)
G:=sub<Sym(96)| (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,37)(12,38)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(49,95)(50,96)(51,85)(52,86)(53,87)(54,88)(55,89)(56,90)(57,91)(58,92)(59,93)(60,94)(61,82)(62,83)(63,84)(64,73)(65,74)(66,75)(67,76)(68,77)(69,78)(70,79)(71,80)(72,81), (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), (1,61,53,23)(2,66,54,16)(3,71,55,21)(4,64,56,14)(5,69,57,19)(6,62,58,24)(7,67,59,17)(8,72,60,22)(9,65,49,15)(10,70,50,20)(11,63,51,13)(12,68,52,18)(25,43,78,91)(26,48,79,96)(27,41,80,89)(28,46,81,94)(29,39,82,87)(30,44,83,92)(31,37,84,85)(32,42,73,90)(33,47,74,95)(34,40,75,88)(35,45,76,93)(36,38,77,86), (1,45)(2,44)(3,43)(4,42)(5,41)(6,40)(7,39)(8,38)(9,37)(10,48)(11,47)(12,46)(13,80)(14,79)(15,78)(16,77)(17,76)(18,75)(19,74)(20,73)(21,84)(22,83)(23,82)(24,81)(25,65)(26,64)(27,63)(28,62)(29,61)(30,72)(31,71)(32,70)(33,69)(34,68)(35,67)(36,66)(49,85)(50,96)(51,95)(52,94)(53,93)(54,92)(55,91)(56,90)(57,89)(58,88)(59,87)(60,86)>;
G:=Group( (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,37)(12,38)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(49,95)(50,96)(51,85)(52,86)(53,87)(54,88)(55,89)(56,90)(57,91)(58,92)(59,93)(60,94)(61,82)(62,83)(63,84)(64,73)(65,74)(66,75)(67,76)(68,77)(69,78)(70,79)(71,80)(72,81), (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), (1,61,53,23)(2,66,54,16)(3,71,55,21)(4,64,56,14)(5,69,57,19)(6,62,58,24)(7,67,59,17)(8,72,60,22)(9,65,49,15)(10,70,50,20)(11,63,51,13)(12,68,52,18)(25,43,78,91)(26,48,79,96)(27,41,80,89)(28,46,81,94)(29,39,82,87)(30,44,83,92)(31,37,84,85)(32,42,73,90)(33,47,74,95)(34,40,75,88)(35,45,76,93)(36,38,77,86), (1,45)(2,44)(3,43)(4,42)(5,41)(6,40)(7,39)(8,38)(9,37)(10,48)(11,47)(12,46)(13,80)(14,79)(15,78)(16,77)(17,76)(18,75)(19,74)(20,73)(21,84)(22,83)(23,82)(24,81)(25,65)(26,64)(27,63)(28,62)(29,61)(30,72)(31,71)(32,70)(33,69)(34,68)(35,67)(36,66)(49,85)(50,96)(51,95)(52,94)(53,93)(54,92)(55,91)(56,90)(57,89)(58,88)(59,87)(60,86) );
G=PermutationGroup([(1,39),(2,40),(3,41),(4,42),(5,43),(6,44),(7,45),(8,46),(9,47),(10,48),(11,37),(12,38),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36),(19,25),(20,26),(21,27),(22,28),(23,29),(24,30),(49,95),(50,96),(51,85),(52,86),(53,87),(54,88),(55,89),(56,90),(57,91),(58,92),(59,93),(60,94),(61,82),(62,83),(63,84),(64,73),(65,74),(66,75),(67,76),(68,77),(69,78),(70,79),(71,80),(72,81)], [(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)], [(1,61,53,23),(2,66,54,16),(3,71,55,21),(4,64,56,14),(5,69,57,19),(6,62,58,24),(7,67,59,17),(8,72,60,22),(9,65,49,15),(10,70,50,20),(11,63,51,13),(12,68,52,18),(25,43,78,91),(26,48,79,96),(27,41,80,89),(28,46,81,94),(29,39,82,87),(30,44,83,92),(31,37,84,85),(32,42,73,90),(33,47,74,95),(34,40,75,88),(35,45,76,93),(36,38,77,86)], [(1,45),(2,44),(3,43),(4,42),(5,41),(6,40),(7,39),(8,38),(9,37),(10,48),(11,47),(12,46),(13,80),(14,79),(15,78),(16,77),(17,76),(18,75),(19,74),(20,73),(21,84),(22,83),(23,82),(24,81),(25,65),(26,64),(27,63),(28,62),(29,61),(30,72),(31,71),(32,70),(33,69),(34,68),(35,67),(36,66),(49,85),(50,96),(51,95),(52,94),(53,93),(54,92),(55,91),(56,90),(57,89),(58,88),(59,87),(60,86)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 11 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 8 | 5 | 0 | 0 | 0 | 0 |
| 3 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 2 |
| 0 | 0 | 0 | 0 | 11 | 4 |
| 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 | 0 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,11,0,0,0,0,1,1,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,1],[8,3,0,0,0,0,5,5,0,0,0,0,0,0,0,8,0,0,0,0,5,0,0,0,0,0,0,0,9,11,0,0,0,0,2,4],[1,0,0,0,0,0,12,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,12,0] >;
48 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 6A | ··· | 6G | 12A | ··· | 12L |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 2 | ··· | 2 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | C4○D4 | C3⋊D4 | Q8⋊3S3 |
| kernel | C2×C12.23D4 | C2×C4×Dic3 | C2×D6⋊C4 | C12.23D4 | C22×D12 | Q8×C2×C6 | C22×Q8 | C2×C12 | C22×C4 | C2×Q8 | C2×C6 | C2×C4 | C22 |
| # reps | 1 | 1 | 4 | 8 | 1 | 1 | 1 | 4 | 3 | 4 | 8 | 8 | 4 |
In GAP, Magma, Sage, TeX
C_2\times C_{12}._{23}D_4 % in TeX
G:=Group("C2xC12.23D4"); // GroupNames label
G:=SmallGroup(192,1373);
// by ID
G=gap.SmallGroup(192,1373);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,184,675,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=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