direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D4.Dic3, C12.75C24, C6⋊3(C8○D4), C4○D4.60D6, (C6×D4).12C4, C3⋊C8.37C23, (C6×Q8).12C4, C4.74(S3×C23), C6.48(C23×C4), C4○D4.7Dic3, D4.8(C2×Dic3), C12.97(C22×C4), (C2×Q8).13Dic3, (C2×D4).12Dic3, Q8.14(C2×Dic3), (C22×C4).402D6, (C2×C12).553C23, C4.Dic3⋊34C22, C2.10(C23×Dic3), C23.24(C2×Dic3), C4.40(C22×Dic3), (C22×C12).288C22, C22.33(C22×Dic3), C3⋊4(C2×C8○D4), (C22×C3⋊C8)⋊14C2, (C2×C3⋊C8)⋊41C22, (C6×C4○D4).9C2, (C3×C4○D4).3C4, (C2×C4○D4).19S3, (C3×D4).25(C2×C4), (C3×Q8).27(C2×C4), (C2×C12).135(C2×C4), (C2×C4.Dic3)⋊28C2, (C2×C6).28(C22×C4), (C22×C6).80(C2×C4), (C2×C4).56(C2×Dic3), (C2×C4).831(C22×S3), (C3×C4○D4).48C22, SmallGroup(192,1377)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D4.Dic3
G = < a,b,c,d,e | a2=b4=c2=1, d6=b2, e2=b2d3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d5 >
Subgroups: 392 in 266 conjugacy classes, 191 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C12, C12, C2×C6, C2×C6, C2×C6, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C3⋊C8, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C22×C8, C2×M4(2), C8○D4, C2×C4○D4, C2×C3⋊C8, C2×C3⋊C8, C4.Dic3, C22×C12, C6×D4, C6×Q8, C3×C4○D4, C2×C8○D4, C22×C3⋊C8, C2×C4.Dic3, D4.Dic3, C6×C4○D4, C2×D4.Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, C24, C2×Dic3, C22×S3, C8○D4, C23×C4, C22×Dic3, S3×C23, C2×C8○D4, D4.Dic3, C23×Dic3, C2×D4.Dic3
►On 96 points
Generators in S96
(1 22)(2 23)(3 24)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(10 19)(11 20)(12 21)(25 40)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 48)(34 37)(35 38)(36 39)(49 61)(50 62)(51 63)(52 64)(53 65)(54 66)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)(73 85)(74 86)(75 87)(76 88)(77 89)(78 90)(79 91)(80 92)(81 93)(82 94)(83 95)(84 96)
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 22 19 16)(14 23 20 17)(15 24 21 18)(25 28 31 34)(26 29 32 35)(27 30 33 36)(37 40 43 46)(38 41 44 47)(39 42 45 48)(49 58 55 52)(50 59 56 53)(51 60 57 54)(61 70 67 64)(62 71 68 65)(63 72 69 66)(73 76 79 82)(74 77 80 83)(75 78 81 84)(85 88 91 94)(86 89 92 95)(87 90 93 96)
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 34)(11 35)(12 36)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(49 77)(50 78)(51 79)(52 80)(53 81)(54 82)(55 83)(56 84)(57 73)(58 74)(59 75)(60 76)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 85)(70 86)(71 87)(72 88)
(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 68 10 65 7 62 4 71)(2 61 11 70 8 67 5 64)(3 66 12 63 9 72 6 69)(13 59 22 56 19 53 16 50)(14 52 23 49 20 58 17 55)(15 57 24 54 21 51 18 60)(25 96 34 93 31 90 28 87)(26 89 35 86 32 95 29 92)(27 94 36 91 33 88 30 85)(37 81 46 78 43 75 40 84)(38 74 47 83 44 80 41 77)(39 79 48 76 45 73 42 82)
G:=sub<Sym(96)| (1,22)(2,23)(3,24)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21)(25,40)(26,41)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(33,48)(34,37)(35,38)(36,39)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,22,19,16)(14,23,20,17)(15,24,21,18)(25,28,31,34)(26,29,32,35)(27,30,33,36)(37,40,43,46)(38,41,44,47)(39,42,45,48)(49,58,55,52)(50,59,56,53)(51,60,57,54)(61,70,67,64)(62,71,68,65)(63,72,69,66)(73,76,79,82)(74,77,80,83)(75,78,81,84)(85,88,91,94)(86,89,92,95)(87,90,93,96), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(49,77)(50,78)(51,79)(52,80)(53,81)(54,82)(55,83)(56,84)(57,73)(58,74)(59,75)(60,76)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,85)(70,86)(71,87)(72,88), (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), (1,68,10,65,7,62,4,71)(2,61,11,70,8,67,5,64)(3,66,12,63,9,72,6,69)(13,59,22,56,19,53,16,50)(14,52,23,49,20,58,17,55)(15,57,24,54,21,51,18,60)(25,96,34,93,31,90,28,87)(26,89,35,86,32,95,29,92)(27,94,36,91,33,88,30,85)(37,81,46,78,43,75,40,84)(38,74,47,83,44,80,41,77)(39,79,48,76,45,73,42,82)>;
G:=Group( (1,22)(2,23)(3,24)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21)(25,40)(26,41)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(33,48)(34,37)(35,38)(36,39)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,22,19,16)(14,23,20,17)(15,24,21,18)(25,28,31,34)(26,29,32,35)(27,30,33,36)(37,40,43,46)(38,41,44,47)(39,42,45,48)(49,58,55,52)(50,59,56,53)(51,60,57,54)(61,70,67,64)(62,71,68,65)(63,72,69,66)(73,76,79,82)(74,77,80,83)(75,78,81,84)(85,88,91,94)(86,89,92,95)(87,90,93,96), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(49,77)(50,78)(51,79)(52,80)(53,81)(54,82)(55,83)(56,84)(57,73)(58,74)(59,75)(60,76)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,85)(70,86)(71,87)(72,88), (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), (1,68,10,65,7,62,4,71)(2,61,11,70,8,67,5,64)(3,66,12,63,9,72,6,69)(13,59,22,56,19,53,16,50)(14,52,23,49,20,58,17,55)(15,57,24,54,21,51,18,60)(25,96,34,93,31,90,28,87)(26,89,35,86,32,95,29,92)(27,94,36,91,33,88,30,85)(37,81,46,78,43,75,40,84)(38,74,47,83,44,80,41,77)(39,79,48,76,45,73,42,82) );
G=PermutationGroup([[(1,22),(2,23),(3,24),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18),(10,19),(11,20),(12,21),(25,40),(26,41),(27,42),(28,43),(29,44),(30,45),(31,46),(32,47),(33,48),(34,37),(35,38),(36,39),(49,61),(50,62),(51,63),(52,64),(53,65),(54,66),(55,67),(56,68),(57,69),(58,70),(59,71),(60,72),(73,85),(74,86),(75,87),(76,88),(77,89),(78,90),(79,91),(80,92),(81,93),(82,94),(83,95),(84,96)], [(1,10,7,4),(2,11,8,5),(3,12,9,6),(13,22,19,16),(14,23,20,17),(15,24,21,18),(25,28,31,34),(26,29,32,35),(27,30,33,36),(37,40,43,46),(38,41,44,47),(39,42,45,48),(49,58,55,52),(50,59,56,53),(51,60,57,54),(61,70,67,64),(62,71,68,65),(63,72,69,66),(73,76,79,82),(74,77,80,83),(75,78,81,84),(85,88,91,94),(86,89,92,95),(87,90,93,96)], [(1,25),(2,26),(3,27),(4,28),(5,29),(6,30),(7,31),(8,32),(9,33),(10,34),(11,35),(12,36),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,37),(20,38),(21,39),(22,40),(23,41),(24,42),(49,77),(50,78),(51,79),(52,80),(53,81),(54,82),(55,83),(56,84),(57,73),(58,74),(59,75),(60,76),(61,89),(62,90),(63,91),(64,92),(65,93),(66,94),(67,95),(68,96),(69,85),(70,86),(71,87),(72,88)], [(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)], [(1,68,10,65,7,62,4,71),(2,61,11,70,8,67,5,64),(3,66,12,63,9,72,6,69),(13,59,22,56,19,53,16,50),(14,52,23,49,20,58,17,55),(15,57,24,54,21,51,18,60),(25,96,34,93,31,90,28,87),(26,89,35,86,32,95,29,92),(27,94,36,91,33,88,30,85),(37,81,46,78,43,75,40,84),(38,74,47,83,44,80,41,77),(39,79,48,76,45,73,42,82)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 6A | 6B | 6C | 6D | ··· | 6I | 8A | ··· | 8H | 8I | ··· | 8T | 12A | 12B | 12C | 12D | 12E | ··· | 12J |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | ··· | 6 | 8 | ··· | 8 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 3 | ··· | 3 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | - | - | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D6 | Dic3 | Dic3 | Dic3 | D6 | C8○D4 | D4.Dic3 |
| kernel | C2×D4.Dic3 | C22×C3⋊C8 | C2×C4.Dic3 | D4.Dic3 | C6×C4○D4 | C6×D4 | C6×Q8 | C3×C4○D4 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C4○D4 | C6 | C2 |
| # reps | 1 | 3 | 3 | 8 | 1 | 6 | 2 | 8 | 1 | 3 | 3 | 1 | 4 | 4 | 8 | 4 |
Matrix representation of C2×D4.Dic3 ►in GL5(𝔽73)
| 72 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 0 | 46 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 46 |
| 0 | 0 | 0 | 27 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 0 | 27 |
| 72 | 0 | 0 | 0 | 0 |
| 0 | 30 | 13 | 0 | 0 |
| 0 | 43 | 43 | 0 | 0 |
| 0 | 0 | 0 | 63 | 0 |
| 0 | 0 | 0 | 0 | 63 |
G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,27,0,0,0,0,0,46],[1,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,0,27,0,0,0,46,0],[1,0,0,0,0,0,0,72,0,0,0,1,1,0,0,0,0,0,27,0,0,0,0,0,27],[72,0,0,0,0,0,30,43,0,0,0,13,43,0,0,0,0,0,63,0,0,0,0,0,63] >;
C2×D4.Dic3 in GAP, Magma, Sage, TeX
C_2\times D_4.{\rm Dic}_3 % in TeX
G:=Group("C2xD4.Dic3"); // GroupNames label
G:=SmallGroup(192,1377);
// by ID
G=gap.SmallGroup(192,1377);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,297,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=1,d^6=b^2,e^2=b^2*d^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^5>;
// generators/relations