direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22×D4.S3, C12.29C24, Dic6⋊7C23, C3⋊C8⋊9C23, C6⋊3(C2×SD16), (C2×C6)⋊12SD16, C3⋊3(C22×SD16), (C2×D4).227D6, C12.250(C2×D4), (C2×C12).208D4, C4.29(S3×C23), (C22×D4).10S3, D4.21(C22×S3), (C3×D4).21C23, (C22×C6).208D4, (C22×C4).392D6, C6.138(C22×D4), (C2×C12).538C23, (C22×Dic6)⋊19C2, (C2×Dic6)⋊66C22, (C6×D4).267C22, C23.112(C3⋊D4), (C22×C12).271C22, (D4×C2×C6).6C2, (C2×C3⋊C8)⋊39C22, (C22×C3⋊C8)⋊12C2, C4.22(C2×C3⋊D4), (C2×C6).578(C2×D4), C2.11(C22×C3⋊D4), (C2×C4).152(C3⋊D4), (C2×C4).621(C22×S3), C22.107(C2×C3⋊D4), SmallGroup(192,1353)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22×D4.S3
G = < a,b,c,d,e,f | a2=b2=c4=d2=e3=1, f2=c2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=fcf-1=c-1, ce=ec, de=ed, fdf-1=cd, fef-1=e-1 >
Subgroups: 680 in 298 conjugacy classes, 127 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C23, Dic3, C12, C12, C2×C6, C2×C6, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, C3⋊C8, Dic6, Dic6, C2×Dic3, C2×C12, C3×D4, C3×D4, C22×C6, C22×C6, C22×C8, C2×SD16, C22×D4, C22×Q8, C2×C3⋊C8, D4.S3, C2×Dic6, C2×Dic6, C22×Dic3, C22×C12, C6×D4, C6×D4, C23×C6, C22×SD16, C22×C3⋊C8, C2×D4.S3, C22×Dic6, D4×C2×C6, C22×D4.S3
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C24, C3⋊D4, C22×S3, C2×SD16, C22×D4, D4.S3, C2×C3⋊D4, S3×C23, C22×SD16, C2×D4.S3, C22×C3⋊D4, C22×D4.S3
►On 96 points
Generators in S96
(1 24)(2 21)(3 22)(4 23)(5 85)(6 86)(7 87)(8 88)(9 89)(10 90)(11 91)(12 92)(13 32)(14 29)(15 30)(16 31)(17 26)(18 27)(19 28)(20 25)(33 47)(34 48)(35 45)(36 46)(37 49)(38 50)(39 51)(40 52)(41 53)(42 54)(43 55)(44 56)(57 71)(58 72)(59 69)(60 70)(61 80)(62 77)(63 78)(64 79)(65 74)(66 75)(67 76)(68 73)(81 95)(82 96)(83 93)(84 94)
(1 34)(2 35)(3 36)(4 33)(5 80)(6 77)(7 78)(8 79)(9 74)(10 75)(11 76)(12 73)(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 56)(26 53)(27 54)(28 55)(29 50)(30 51)(31 52)(32 49)(57 81)(58 82)(59 83)(60 84)(61 85)(62 86)(63 87)(64 88)(65 89)(66 90)(67 91)(68 92)(69 93)(70 94)(71 95)(72 96)
(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 33)(2 36)(3 35)(4 34)(5 80)(6 79)(7 78)(8 77)(9 76)(10 75)(11 74)(12 73)(13 38)(14 37)(15 40)(16 39)(17 44)(18 43)(19 42)(20 41)(21 46)(22 45)(23 48)(24 47)(25 53)(26 56)(27 55)(28 54)(29 49)(30 52)(31 51)(32 50)(57 81)(58 84)(59 83)(60 82)(61 85)(62 88)(63 87)(64 86)(65 91)(66 90)(67 89)(68 92)(69 93)(70 96)(71 95)(72 94)
(1 14 19)(2 15 20)(3 16 17)(4 13 18)(5 10 95)(6 11 96)(7 12 93)(8 9 94)(21 30 25)(22 31 26)(23 32 27)(24 29 28)(33 37 42)(34 38 43)(35 39 44)(36 40 41)(45 51 56)(46 52 53)(47 49 54)(48 50 55)(57 61 66)(58 62 67)(59 63 68)(60 64 65)(69 78 73)(70 79 74)(71 80 75)(72 77 76)(81 85 90)(82 86 91)(83 87 92)(84 88 89)
(1 72 3 70)(2 71 4 69)(5 42 7 44)(6 41 8 43)(9 38 11 40)(10 37 12 39)(13 73 15 75)(14 76 16 74)(17 79 19 77)(18 78 20 80)(21 57 23 59)(22 60 24 58)(25 61 27 63)(26 64 28 62)(29 67 31 65)(30 66 32 68)(33 93 35 95)(34 96 36 94)(45 81 47 83)(46 84 48 82)(49 92 51 90)(50 91 52 89)(53 88 55 86)(54 87 56 85)
G:=sub<Sym(96)| (1,24)(2,21)(3,22)(4,23)(5,85)(6,86)(7,87)(8,88)(9,89)(10,90)(11,91)(12,92)(13,32)(14,29)(15,30)(16,31)(17,26)(18,27)(19,28)(20,25)(33,47)(34,48)(35,45)(36,46)(37,49)(38,50)(39,51)(40,52)(41,53)(42,54)(43,55)(44,56)(57,71)(58,72)(59,69)(60,70)(61,80)(62,77)(63,78)(64,79)(65,74)(66,75)(67,76)(68,73)(81,95)(82,96)(83,93)(84,94), (1,34)(2,35)(3,36)(4,33)(5,80)(6,77)(7,78)(8,79)(9,74)(10,75)(11,76)(12,73)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,56)(26,53)(27,54)(28,55)(29,50)(30,51)(31,52)(32,49)(57,81)(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96), (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), (1,33)(2,36)(3,35)(4,34)(5,80)(6,79)(7,78)(8,77)(9,76)(10,75)(11,74)(12,73)(13,38)(14,37)(15,40)(16,39)(17,44)(18,43)(19,42)(20,41)(21,46)(22,45)(23,48)(24,47)(25,53)(26,56)(27,55)(28,54)(29,49)(30,52)(31,51)(32,50)(57,81)(58,84)(59,83)(60,82)(61,85)(62,88)(63,87)(64,86)(65,91)(66,90)(67,89)(68,92)(69,93)(70,96)(71,95)(72,94), (1,14,19)(2,15,20)(3,16,17)(4,13,18)(5,10,95)(6,11,96)(7,12,93)(8,9,94)(21,30,25)(22,31,26)(23,32,27)(24,29,28)(33,37,42)(34,38,43)(35,39,44)(36,40,41)(45,51,56)(46,52,53)(47,49,54)(48,50,55)(57,61,66)(58,62,67)(59,63,68)(60,64,65)(69,78,73)(70,79,74)(71,80,75)(72,77,76)(81,85,90)(82,86,91)(83,87,92)(84,88,89), (1,72,3,70)(2,71,4,69)(5,42,7,44)(6,41,8,43)(9,38,11,40)(10,37,12,39)(13,73,15,75)(14,76,16,74)(17,79,19,77)(18,78,20,80)(21,57,23,59)(22,60,24,58)(25,61,27,63)(26,64,28,62)(29,67,31,65)(30,66,32,68)(33,93,35,95)(34,96,36,94)(45,81,47,83)(46,84,48,82)(49,92,51,90)(50,91,52,89)(53,88,55,86)(54,87,56,85)>;
G:=Group( (1,24)(2,21)(3,22)(4,23)(5,85)(6,86)(7,87)(8,88)(9,89)(10,90)(11,91)(12,92)(13,32)(14,29)(15,30)(16,31)(17,26)(18,27)(19,28)(20,25)(33,47)(34,48)(35,45)(36,46)(37,49)(38,50)(39,51)(40,52)(41,53)(42,54)(43,55)(44,56)(57,71)(58,72)(59,69)(60,70)(61,80)(62,77)(63,78)(64,79)(65,74)(66,75)(67,76)(68,73)(81,95)(82,96)(83,93)(84,94), (1,34)(2,35)(3,36)(4,33)(5,80)(6,77)(7,78)(8,79)(9,74)(10,75)(11,76)(12,73)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,56)(26,53)(27,54)(28,55)(29,50)(30,51)(31,52)(32,49)(57,81)(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96), (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), (1,33)(2,36)(3,35)(4,34)(5,80)(6,79)(7,78)(8,77)(9,76)(10,75)(11,74)(12,73)(13,38)(14,37)(15,40)(16,39)(17,44)(18,43)(19,42)(20,41)(21,46)(22,45)(23,48)(24,47)(25,53)(26,56)(27,55)(28,54)(29,49)(30,52)(31,51)(32,50)(57,81)(58,84)(59,83)(60,82)(61,85)(62,88)(63,87)(64,86)(65,91)(66,90)(67,89)(68,92)(69,93)(70,96)(71,95)(72,94), (1,14,19)(2,15,20)(3,16,17)(4,13,18)(5,10,95)(6,11,96)(7,12,93)(8,9,94)(21,30,25)(22,31,26)(23,32,27)(24,29,28)(33,37,42)(34,38,43)(35,39,44)(36,40,41)(45,51,56)(46,52,53)(47,49,54)(48,50,55)(57,61,66)(58,62,67)(59,63,68)(60,64,65)(69,78,73)(70,79,74)(71,80,75)(72,77,76)(81,85,90)(82,86,91)(83,87,92)(84,88,89), (1,72,3,70)(2,71,4,69)(5,42,7,44)(6,41,8,43)(9,38,11,40)(10,37,12,39)(13,73,15,75)(14,76,16,74)(17,79,19,77)(18,78,20,80)(21,57,23,59)(22,60,24,58)(25,61,27,63)(26,64,28,62)(29,67,31,65)(30,66,32,68)(33,93,35,95)(34,96,36,94)(45,81,47,83)(46,84,48,82)(49,92,51,90)(50,91,52,89)(53,88,55,86)(54,87,56,85) );
G=PermutationGroup([[(1,24),(2,21),(3,22),(4,23),(5,85),(6,86),(7,87),(8,88),(9,89),(10,90),(11,91),(12,92),(13,32),(14,29),(15,30),(16,31),(17,26),(18,27),(19,28),(20,25),(33,47),(34,48),(35,45),(36,46),(37,49),(38,50),(39,51),(40,52),(41,53),(42,54),(43,55),(44,56),(57,71),(58,72),(59,69),(60,70),(61,80),(62,77),(63,78),(64,79),(65,74),(66,75),(67,76),(68,73),(81,95),(82,96),(83,93),(84,94)], [(1,34),(2,35),(3,36),(4,33),(5,80),(6,77),(7,78),(8,79),(9,74),(10,75),(11,76),(12,73),(13,37),(14,38),(15,39),(16,40),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48),(25,56),(26,53),(27,54),(28,55),(29,50),(30,51),(31,52),(32,49),(57,81),(58,82),(59,83),(60,84),(61,85),(62,86),(63,87),(64,88),(65,89),(66,90),(67,91),(68,92),(69,93),(70,94),(71,95),(72,96)], [(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)], [(1,33),(2,36),(3,35),(4,34),(5,80),(6,79),(7,78),(8,77),(9,76),(10,75),(11,74),(12,73),(13,38),(14,37),(15,40),(16,39),(17,44),(18,43),(19,42),(20,41),(21,46),(22,45),(23,48),(24,47),(25,53),(26,56),(27,55),(28,54),(29,49),(30,52),(31,51),(32,50),(57,81),(58,84),(59,83),(60,82),(61,85),(62,88),(63,87),(64,86),(65,91),(66,90),(67,89),(68,92),(69,93),(70,96),(71,95),(72,94)], [(1,14,19),(2,15,20),(3,16,17),(4,13,18),(5,10,95),(6,11,96),(7,12,93),(8,9,94),(21,30,25),(22,31,26),(23,32,27),(24,29,28),(33,37,42),(34,38,43),(35,39,44),(36,40,41),(45,51,56),(46,52,53),(47,49,54),(48,50,55),(57,61,66),(58,62,67),(59,63,68),(60,64,65),(69,78,73),(70,79,74),(71,80,75),(72,77,76),(81,85,90),(82,86,91),(83,87,92),(84,88,89)], [(1,72,3,70),(2,71,4,69),(5,42,7,44),(6,41,8,43),(9,38,11,40),(10,37,12,39),(13,73,15,75),(14,76,16,74),(17,79,19,77),(18,78,20,80),(21,57,23,59),(22,60,24,58),(25,61,27,63),(26,64,28,62),(29,67,31,65),(30,66,32,68),(33,93,35,95),(34,96,36,94),(45,81,47,83),(46,84,48,82),(49,92,51,90),(50,91,52,89),(53,88,55,86),(54,87,56,85)]])
48 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6G | 6H | ··· | 6O | 8A | ··· | 8H | 12A | 12B | 12C | 12D |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | 4 | 4 | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | SD16 | C3⋊D4 | C3⋊D4 | D4.S3 |
| kernel | C22×D4.S3 | C22×C3⋊C8 | C2×D4.S3 | C22×Dic6 | D4×C2×C6 | C22×D4 | C2×C12 | C22×C6 | C22×C4 | C2×D4 | C2×C6 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 12 | 1 | 1 | 1 | 3 | 1 | 1 | 6 | 8 | 6 | 2 | 4 |
Matrix representation of C22×D4.S3 ►in GL7(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 48 | 0 | 0 | 0 | 0 |
| 0 | 3 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 7 | 0 | 0 |
| 0 | 0 | 0 | 31 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 48 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 31 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 0 | 0 | 64 | 8 |
| 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 42 | 0 | 0 |
| 0 | 0 | 0 | 33 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 29 | 2 |
| 0 | 0 | 0 | 0 | 0 | 18 | 44 |
G:=sub<GL(7,GF(73))| [72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,3,0,0,0,0,0,48,72,0,0,0,0,0,0,0,72,31,0,0,0,0,0,7,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,48,72,0,0,0,0,0,0,0,72,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,64,64,0,0,0,0,0,0,8],[72,0,0,0,0,0,0,0,0,18,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,33,0,0,0,0,0,42,0,0,0,0,0,0,0,0,29,18,0,0,0,0,0,2,44] >;
C22×D4.S3 in GAP, Magma, Sage, TeX
C_2^2\times D_4.S_3
% in TeX
G:=Group("C2^2xD4.S3"); // GroupNames label
G:=SmallGroup(192,1353);
// by ID
G=gap.SmallGroup(192,1353);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,675,1684,235,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^4=d^2=e^3=1,f^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,d*c*d=f*c*f^-1=c^-1,c*e=e*c,d*e=e*d,f*d*f^-1=c*d,f*e*f^-1=e^-1>;
// generators/relations