metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22⋊C4⋊4D14, (C2×C28).10D4, (C22×C4)⋊2D14, (C2×D4).43D14, C23⋊Dic7⋊6C2, (C2×Dic7).4D4, (C22×D7).4D4, C22.34(D4×D7), C14.50C22≀C2, D4⋊6D14.3C2, (C22×C28)⋊2C22, (C22×C14).21D4, C22.D4⋊1D7, C23.9(C7⋊D4), C7⋊2(C23.7D4), C23.D7⋊5C22, (D4×C14).59C22, C23.1D14⋊6C2, C2.18(C23⋊D14), C23.75(C22×D7), C23.23D14⋊1C2, (C22×C14).114C23, (C2×C14).31(C2×D4), (C2×C4).9(C7⋊D4), (C2×C7⋊D4).6C22, C22.30(C2×C7⋊D4), (C7×C22⋊C4)⋊35C22, (C7×C22.D4)⋊1C2, SmallGroup(448,587)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22⋊C4⋊D14
G = < a,b,c,d,e | a2=b2=c4=d14=e2=1, cac-1=dad-1=ab=ba, ae=ea, bc=cb, bd=db, be=eb, dcd-1=bc-1, ece=abc, ede=d-1 >
Subgroups: 940 in 160 conjugacy classes, 39 normal (23 characteristic)
C1, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C4○D4, Dic7, C28, D14, C2×C14, C2×C14, C2×C14, C23⋊C4, C22.D4, C22.D4, 2+ 1+4, Dic14, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C22×D7, C22×D7, C22×C14, C23.7D4, Dic7⋊C4, D14⋊C4, C23.D7, C7×C22⋊C4, C7×C22⋊C4, C7×C4⋊C4, C4○D28, D4×D7, D4⋊2D7, C2×C7⋊D4, C2×C7⋊D4, C22×C28, D4×C14, C23.1D14, C23⋊Dic7, C23.23D14, C7×C22.D4, D4⋊6D14, C22⋊C4⋊D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C22≀C2, C7⋊D4, C22×D7, C23.7D4, D4×D7, C2×C7⋊D4, C23⋊D14, C22⋊C4⋊D14
►On 112 points
Generators in S112
(1 84)(2 32)(3 72)(4 34)(5 74)(6 36)(7 76)(8 38)(9 78)(10 40)(11 80)(12 42)(13 82)(14 30)(15 85)(16 103)(17 87)(18 105)(19 89)(20 107)(21 91)(22 109)(23 93)(24 111)(25 95)(26 99)(27 97)(28 101)(29 66)(31 68)(33 70)(35 58)(37 60)(39 62)(41 64)(43 92)(44 110)(45 94)(46 112)(47 96)(48 100)(49 98)(50 102)(51 86)(52 104)(53 88)(54 106)(55 90)(56 108)(57 73)(59 75)(61 77)(63 79)(65 81)(67 83)(69 71)
(1 68)(2 69)(3 70)(4 57)(5 58)(6 59)(7 60)(8 61)(9 62)(10 63)(11 64)(12 65)(13 66)(14 67)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 56)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(29 82)(30 83)(31 84)(32 71)(33 72)(34 73)(35 74)(36 75)(37 76)(38 77)(39 78)(40 79)(41 80)(42 81)(85 102)(86 103)(87 104)(88 105)(89 106)(90 107)(91 108)(92 109)(93 110)(94 111)(95 112)(96 99)(97 100)(98 101)
(2 69)(4 57)(6 59)(8 61)(10 63)(12 65)(14 67)(15 102 50 85)(16 103 51 86)(17 104 52 87)(18 105 53 88)(19 106 54 89)(20 107 55 90)(21 108 56 91)(22 109 43 92)(23 110 44 93)(24 111 45 94)(25 112 46 95)(26 99 47 96)(27 100 48 97)(28 101 49 98)(29 82)(31 84)(33 72)(35 74)(37 76)(39 78)(41 80)
(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 97 98)(99 100 101 102 103 104 105 106 107 108 109 110 111 112)
(1 108)(2 107)(3 106)(4 105)(5 104)(6 103)(7 102)(8 101)(9 100)(10 99)(11 112)(12 111)(13 110)(14 109)(15 37)(16 36)(17 35)(18 34)(19 33)(20 32)(21 31)(22 30)(23 29)(24 42)(25 41)(26 40)(27 39)(28 38)(43 83)(44 82)(45 81)(46 80)(47 79)(48 78)(49 77)(50 76)(51 75)(52 74)(53 73)(54 72)(55 71)(56 84)(57 88)(58 87)(59 86)(60 85)(61 98)(62 97)(63 96)(64 95)(65 94)(66 93)(67 92)(68 91)(69 90)(70 89)
G:=sub<Sym(112)| (1,84)(2,32)(3,72)(4,34)(5,74)(6,36)(7,76)(8,38)(9,78)(10,40)(11,80)(12,42)(13,82)(14,30)(15,85)(16,103)(17,87)(18,105)(19,89)(20,107)(21,91)(22,109)(23,93)(24,111)(25,95)(26,99)(27,97)(28,101)(29,66)(31,68)(33,70)(35,58)(37,60)(39,62)(41,64)(43,92)(44,110)(45,94)(46,112)(47,96)(48,100)(49,98)(50,102)(51,86)(52,104)(53,88)(54,106)(55,90)(56,108)(57,73)(59,75)(61,77)(63,79)(65,81)(67,83)(69,71), (1,68)(2,69)(3,70)(4,57)(5,58)(6,59)(7,60)(8,61)(9,62)(10,63)(11,64)(12,65)(13,66)(14,67)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,82)(30,83)(31,84)(32,71)(33,72)(34,73)(35,74)(36,75)(37,76)(38,77)(39,78)(40,79)(41,80)(42,81)(85,102)(86,103)(87,104)(88,105)(89,106)(90,107)(91,108)(92,109)(93,110)(94,111)(95,112)(96,99)(97,100)(98,101), (2,69)(4,57)(6,59)(8,61)(10,63)(12,65)(14,67)(15,102,50,85)(16,103,51,86)(17,104,52,87)(18,105,53,88)(19,106,54,89)(20,107,55,90)(21,108,56,91)(22,109,43,92)(23,110,44,93)(24,111,45,94)(25,112,46,95)(26,99,47,96)(27,100,48,97)(28,101,49,98)(29,82)(31,84)(33,72)(35,74)(37,76)(39,78)(41,80), (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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98)(99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,108)(2,107)(3,106)(4,105)(5,104)(6,103)(7,102)(8,101)(9,100)(10,99)(11,112)(12,111)(13,110)(14,109)(15,37)(16,36)(17,35)(18,34)(19,33)(20,32)(21,31)(22,30)(23,29)(24,42)(25,41)(26,40)(27,39)(28,38)(43,83)(44,82)(45,81)(46,80)(47,79)(48,78)(49,77)(50,76)(51,75)(52,74)(53,73)(54,72)(55,71)(56,84)(57,88)(58,87)(59,86)(60,85)(61,98)(62,97)(63,96)(64,95)(65,94)(66,93)(67,92)(68,91)(69,90)(70,89)>;
G:=Group( (1,84)(2,32)(3,72)(4,34)(5,74)(6,36)(7,76)(8,38)(9,78)(10,40)(11,80)(12,42)(13,82)(14,30)(15,85)(16,103)(17,87)(18,105)(19,89)(20,107)(21,91)(22,109)(23,93)(24,111)(25,95)(26,99)(27,97)(28,101)(29,66)(31,68)(33,70)(35,58)(37,60)(39,62)(41,64)(43,92)(44,110)(45,94)(46,112)(47,96)(48,100)(49,98)(50,102)(51,86)(52,104)(53,88)(54,106)(55,90)(56,108)(57,73)(59,75)(61,77)(63,79)(65,81)(67,83)(69,71), (1,68)(2,69)(3,70)(4,57)(5,58)(6,59)(7,60)(8,61)(9,62)(10,63)(11,64)(12,65)(13,66)(14,67)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,82)(30,83)(31,84)(32,71)(33,72)(34,73)(35,74)(36,75)(37,76)(38,77)(39,78)(40,79)(41,80)(42,81)(85,102)(86,103)(87,104)(88,105)(89,106)(90,107)(91,108)(92,109)(93,110)(94,111)(95,112)(96,99)(97,100)(98,101), (2,69)(4,57)(6,59)(8,61)(10,63)(12,65)(14,67)(15,102,50,85)(16,103,51,86)(17,104,52,87)(18,105,53,88)(19,106,54,89)(20,107,55,90)(21,108,56,91)(22,109,43,92)(23,110,44,93)(24,111,45,94)(25,112,46,95)(26,99,47,96)(27,100,48,97)(28,101,49,98)(29,82)(31,84)(33,72)(35,74)(37,76)(39,78)(41,80), (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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98)(99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,108)(2,107)(3,106)(4,105)(5,104)(6,103)(7,102)(8,101)(9,100)(10,99)(11,112)(12,111)(13,110)(14,109)(15,37)(16,36)(17,35)(18,34)(19,33)(20,32)(21,31)(22,30)(23,29)(24,42)(25,41)(26,40)(27,39)(28,38)(43,83)(44,82)(45,81)(46,80)(47,79)(48,78)(49,77)(50,76)(51,75)(52,74)(53,73)(54,72)(55,71)(56,84)(57,88)(58,87)(59,86)(60,85)(61,98)(62,97)(63,96)(64,95)(65,94)(66,93)(67,92)(68,91)(69,90)(70,89) );
G=PermutationGroup([[(1,84),(2,32),(3,72),(4,34),(5,74),(6,36),(7,76),(8,38),(9,78),(10,40),(11,80),(12,42),(13,82),(14,30),(15,85),(16,103),(17,87),(18,105),(19,89),(20,107),(21,91),(22,109),(23,93),(24,111),(25,95),(26,99),(27,97),(28,101),(29,66),(31,68),(33,70),(35,58),(37,60),(39,62),(41,64),(43,92),(44,110),(45,94),(46,112),(47,96),(48,100),(49,98),(50,102),(51,86),(52,104),(53,88),(54,106),(55,90),(56,108),(57,73),(59,75),(61,77),(63,79),(65,81),(67,83),(69,71)], [(1,68),(2,69),(3,70),(4,57),(5,58),(6,59),(7,60),(8,61),(9,62),(10,63),(11,64),(12,65),(13,66),(14,67),(15,50),(16,51),(17,52),(18,53),(19,54),(20,55),(21,56),(22,43),(23,44),(24,45),(25,46),(26,47),(27,48),(28,49),(29,82),(30,83),(31,84),(32,71),(33,72),(34,73),(35,74),(36,75),(37,76),(38,77),(39,78),(40,79),(41,80),(42,81),(85,102),(86,103),(87,104),(88,105),(89,106),(90,107),(91,108),(92,109),(93,110),(94,111),(95,112),(96,99),(97,100),(98,101)], [(2,69),(4,57),(6,59),(8,61),(10,63),(12,65),(14,67),(15,102,50,85),(16,103,51,86),(17,104,52,87),(18,105,53,88),(19,106,54,89),(20,107,55,90),(21,108,56,91),(22,109,43,92),(23,110,44,93),(24,111,45,94),(25,112,46,95),(26,99,47,96),(27,100,48,97),(28,101,49,98),(29,82),(31,84),(33,72),(35,74),(37,76),(39,78),(41,80)], [(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),(85,86,87,88,89,90,91,92,93,94,95,96,97,98),(99,100,101,102,103,104,105,106,107,108,109,110,111,112)], [(1,108),(2,107),(3,106),(4,105),(5,104),(6,103),(7,102),(8,101),(9,100),(10,99),(11,112),(12,111),(13,110),(14,109),(15,37),(16,36),(17,35),(18,34),(19,33),(20,32),(21,31),(22,30),(23,29),(24,42),(25,41),(26,40),(27,39),(28,38),(43,83),(44,82),(45,81),(46,80),(47,79),(48,78),(49,77),(50,76),(51,75),(52,74),(53,73),(54,72),(55,71),(56,84),(57,88),(58,87),(59,86),(60,85),(61,98),(62,97),(63,96),(64,95),(65,94),(66,93),(67,92),(68,91),(69,90),(70,89)]])
58 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 14P | 14Q | 14R | 28A | ··· | 28L | 28M | ··· | 28U |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 28 | 28 | 4 | 4 | 4 | 8 | 28 | 28 | 56 | 56 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | C7⋊D4 | C7⋊D4 | C23.7D4 | D4×D7 | C22⋊C4⋊D14 |
| kernel | C22⋊C4⋊D14 | C23.1D14 | C23⋊Dic7 | C23.23D14 | C7×C22.D4 | D4⋊6D14 | C2×Dic7 | C2×C28 | C22×D7 | C22×C14 | C22.D4 | C22⋊C4 | C22×C4 | C2×D4 | C2×C4 | C23 | C7 | C22 | C1 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 1 | 3 | 3 | 3 | 3 | 6 | 6 | 2 | 6 | 12 |
Matrix representation of C22⋊C4⋊D14 ►in GL4(𝔽29) generated by
| 1 | 0 | 0 | 0 |
| 3 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 26 | 1 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 1 | 9 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 9 |
| 0 | 0 | 3 | 28 |
| 18 | 17 | 0 | 0 |
| 25 | 11 | 0 | 0 |
| 0 | 0 | 8 | 14 |
| 0 | 0 | 24 | 21 |
| 0 | 0 | 8 | 14 |
| 0 | 0 | 24 | 21 |
| 18 | 17 | 0 | 0 |
| 25 | 11 | 0 | 0 |
G:=sub<GL(4,GF(29))| [1,3,0,0,0,28,0,0,0,0,28,26,0,0,0,1],[28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[1,0,0,0,9,28,0,0,0,0,1,3,0,0,9,28],[18,25,0,0,17,11,0,0,0,0,8,24,0,0,14,21],[0,0,18,25,0,0,17,11,8,24,0,0,14,21,0,0] >;
C22⋊C4⋊D14 in GAP, Magma, Sage, TeX
C_2^2\rtimes C_4\rtimes D_{14} % in TeX
G:=Group("C2^2:C4:D14"); // GroupNames label
G:=SmallGroup(448,587);
// by ID
G=gap.SmallGroup(448,587);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,254,219,184,570,1684,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^4=d^14=e^2=1,c*a*c^-1=d*a*d^-1=a*b=b*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=b*c^-1,e*c*e=a*b*c,e*d*e=d^-1>;
// generators/relations