metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊17D14, C14.212+ (1+4), C4⋊C4⋊50D14, (C4×D4)⋊22D7, (D4×C28)⋊24C2, C22⋊D28⋊7C2, C28⋊7D4⋊11C2, (C4×C28)⋊28C22, C22⋊C4⋊49D14, (C22×C4)⋊14D14, C23⋊D14⋊21C2, D14⋊D4⋊10C2, D14⋊C4⋊31C22, D14.5D4⋊8C2, (C2×D4).221D14, C4.D28⋊28C2, C42⋊2D7⋊10C2, C4⋊Dic7⋊10C22, Dic7⋊D4⋊27C2, (C2×C14).104C24, (C2×C28).162C23, Dic7⋊C4⋊33C22, (C22×C28)⋊11C22, C22⋊Dic14⋊9C2, Dic7.D4⋊9C2, C7⋊2(C22.32C24), (C2×Dic14)⋊7C22, (C4×Dic7)⋊53C22, (C2×D28).27C22, C22.6(C4○D28), C2.22(D4⋊6D14), C2.17(D4⋊8D14), C23.D7⋊10C22, (D4×C14).308C22, C23.23D14⋊2C2, (C2×Dic7).45C23, (C22×D7).38C23, (C23×D7).42C22, C23.101(C22×D7), C22.129(C23×D7), (C22×C14).174C23, (C22×Dic7).99C22, C4⋊C4⋊D7⋊8C2, (C4×C7⋊D4)⋊46C2, (C2×C4×D7)⋊49C22, (C2×D14⋊C4)⋊35C2, (C7×C4⋊C4)⋊62C22, C2.53(C2×C4○D28), C14.46(C2×C4○D4), (C2×C7⋊D4)⋊5C22, (C2×C14).17(C4○D4), (C7×C22⋊C4)⋊58C22, (C2×C4).162(C22×D7), SmallGroup(448,1013)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1364 in 250 conjugacy classes, 95 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C4 [×10], C22, C22 [×2], C22 [×18], C7, C2×C4 [×5], C2×C4 [×9], D4 [×9], Q8, C23 [×2], C23 [×7], D7 [×3], C14 [×3], C14 [×3], C42, C42, C22⋊C4 [×2], C22⋊C4 [×12], C4⋊C4, C4⋊C4 [×5], C22×C4 [×2], C22×C4 [×2], C2×D4, C2×D4 [×6], C2×Q8, C24, Dic7 [×5], C28 [×5], D14 [×13], C2×C14, C2×C14 [×2], C2×C14 [×5], C2×C22⋊C4, C4×D4, C4×D4, C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4 [×2], C42⋊2C2 [×2], Dic14, C4×D7, D28 [×2], C2×Dic7 [×5], C2×Dic7, C7⋊D4 [×5], C2×C28 [×5], C2×C28 [×2], C7×D4 [×2], C22×D7 [×3], C22×D7 [×4], C22×C14 [×2], C22.32C24, C4×Dic7, Dic7⋊C4 [×4], C4⋊Dic7, D14⋊C4 [×10], C23.D7 [×2], C4×C28, C7×C22⋊C4 [×2], C7×C4⋊C4, C2×Dic14, C2×C4×D7, C2×D28 [×2], C22×Dic7, C2×C7⋊D4 [×4], C22×C28 [×2], D4×C14, C23×D7, C4.D28, C42⋊2D7, C22⋊Dic14, C22⋊D28, D14⋊D4, Dic7.D4, D14.5D4, C4⋊C4⋊D7, C2×D14⋊C4, C4×C7⋊D4, C23.23D14, C28⋊7D4, C23⋊D14, Dic7⋊D4, D4×C28, C42⋊17D14
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C4○D4 [×2], C24, D14 [×7], C2×C4○D4, 2+ (1+4) [×2], C22×D7 [×7], C22.32C24, C4○D28 [×2], C23×D7, C2×C4○D28, D4⋊6D14, D4⋊8D14, C42⋊17D14
Generators and relations
G = < a,b,c,d | a4=b4=c14=d2=1, ab=ba, cac-1=a-1, dad=a-1b2, bc=cb, dbd=a2b, dcd=c-1 >
►On 112 points
(1 16 44 32)(2 33 45 17)(3 18 46 34)(4 35 47 19)(5 20 48 36)(6 37 49 21)(7 22 50 38)(8 39 51 23)(9 24 52 40)(10 41 53 25)(11 26 54 42)(12 29 55 27)(13 28 56 30)(14 31 43 15)(57 98 102 72)(58 73 103 85)(59 86 104 74)(60 75 105 87)(61 88 106 76)(62 77 107 89)(63 90 108 78)(64 79 109 91)(65 92 110 80)(66 81 111 93)(67 94 112 82)(68 83 99 95)(69 96 100 84)(70 71 101 97)
(1 108 8 101)(2 109 9 102)(3 110 10 103)(4 111 11 104)(5 112 12 105)(6 99 13 106)(7 100 14 107)(15 77 38 96)(16 78 39 97)(17 79 40 98)(18 80 41 85)(19 81 42 86)(20 82 29 87)(21 83 30 88)(22 84 31 89)(23 71 32 90)(24 72 33 91)(25 73 34 92)(26 74 35 93)(27 75 36 94)(28 76 37 95)(43 62 50 69)(44 63 51 70)(45 64 52 57)(46 65 53 58)(47 66 54 59)(48 67 55 60)(49 68 56 61)
(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 14)(2 13)(3 12)(4 11)(5 10)(6 9)(7 8)(15 39)(16 38)(17 37)(18 36)(19 35)(20 34)(21 33)(22 32)(23 31)(24 30)(25 29)(26 42)(27 41)(28 40)(43 44)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(57 99)(58 112)(59 111)(60 110)(61 109)(62 108)(63 107)(64 106)(65 105)(66 104)(67 103)(68 102)(69 101)(70 100)(71 77)(72 76)(73 75)(78 84)(79 83)(80 82)(85 87)(88 98)(89 97)(90 96)(91 95)(92 94)
G:=sub<Sym(112)| (1,16,44,32)(2,33,45,17)(3,18,46,34)(4,35,47,19)(5,20,48,36)(6,37,49,21)(7,22,50,38)(8,39,51,23)(9,24,52,40)(10,41,53,25)(11,26,54,42)(12,29,55,27)(13,28,56,30)(14,31,43,15)(57,98,102,72)(58,73,103,85)(59,86,104,74)(60,75,105,87)(61,88,106,76)(62,77,107,89)(63,90,108,78)(64,79,109,91)(65,92,110,80)(66,81,111,93)(67,94,112,82)(68,83,99,95)(69,96,100,84)(70,71,101,97), (1,108,8,101)(2,109,9,102)(3,110,10,103)(4,111,11,104)(5,112,12,105)(6,99,13,106)(7,100,14,107)(15,77,38,96)(16,78,39,97)(17,79,40,98)(18,80,41,85)(19,81,42,86)(20,82,29,87)(21,83,30,88)(22,84,31,89)(23,71,32,90)(24,72,33,91)(25,73,34,92)(26,74,35,93)(27,75,36,94)(28,76,37,95)(43,62,50,69)(44,63,51,70)(45,64,52,57)(46,65,53,58)(47,66,54,59)(48,67,55,60)(49,68,56,61), (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), (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(15,39)(16,38)(17,37)(18,36)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30)(25,29)(26,42)(27,41)(28,40)(43,44)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(57,99)(58,112)(59,111)(60,110)(61,109)(62,108)(63,107)(64,106)(65,105)(66,104)(67,103)(68,102)(69,101)(70,100)(71,77)(72,76)(73,75)(78,84)(79,83)(80,82)(85,87)(88,98)(89,97)(90,96)(91,95)(92,94)>;
G:=Group( (1,16,44,32)(2,33,45,17)(3,18,46,34)(4,35,47,19)(5,20,48,36)(6,37,49,21)(7,22,50,38)(8,39,51,23)(9,24,52,40)(10,41,53,25)(11,26,54,42)(12,29,55,27)(13,28,56,30)(14,31,43,15)(57,98,102,72)(58,73,103,85)(59,86,104,74)(60,75,105,87)(61,88,106,76)(62,77,107,89)(63,90,108,78)(64,79,109,91)(65,92,110,80)(66,81,111,93)(67,94,112,82)(68,83,99,95)(69,96,100,84)(70,71,101,97), (1,108,8,101)(2,109,9,102)(3,110,10,103)(4,111,11,104)(5,112,12,105)(6,99,13,106)(7,100,14,107)(15,77,38,96)(16,78,39,97)(17,79,40,98)(18,80,41,85)(19,81,42,86)(20,82,29,87)(21,83,30,88)(22,84,31,89)(23,71,32,90)(24,72,33,91)(25,73,34,92)(26,74,35,93)(27,75,36,94)(28,76,37,95)(43,62,50,69)(44,63,51,70)(45,64,52,57)(46,65,53,58)(47,66,54,59)(48,67,55,60)(49,68,56,61), (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), (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(15,39)(16,38)(17,37)(18,36)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30)(25,29)(26,42)(27,41)(28,40)(43,44)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(57,99)(58,112)(59,111)(60,110)(61,109)(62,108)(63,107)(64,106)(65,105)(66,104)(67,103)(68,102)(69,101)(70,100)(71,77)(72,76)(73,75)(78,84)(79,83)(80,82)(85,87)(88,98)(89,97)(90,96)(91,95)(92,94) );
G=PermutationGroup([(1,16,44,32),(2,33,45,17),(3,18,46,34),(4,35,47,19),(5,20,48,36),(6,37,49,21),(7,22,50,38),(8,39,51,23),(9,24,52,40),(10,41,53,25),(11,26,54,42),(12,29,55,27),(13,28,56,30),(14,31,43,15),(57,98,102,72),(58,73,103,85),(59,86,104,74),(60,75,105,87),(61,88,106,76),(62,77,107,89),(63,90,108,78),(64,79,109,91),(65,92,110,80),(66,81,111,93),(67,94,112,82),(68,83,99,95),(69,96,100,84),(70,71,101,97)], [(1,108,8,101),(2,109,9,102),(3,110,10,103),(4,111,11,104),(5,112,12,105),(6,99,13,106),(7,100,14,107),(15,77,38,96),(16,78,39,97),(17,79,40,98),(18,80,41,85),(19,81,42,86),(20,82,29,87),(21,83,30,88),(22,84,31,89),(23,71,32,90),(24,72,33,91),(25,73,34,92),(26,74,35,93),(27,75,36,94),(28,76,37,95),(43,62,50,69),(44,63,51,70),(45,64,52,57),(46,65,53,58),(47,66,54,59),(48,67,55,60),(49,68,56,61)], [(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)], [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(15,39),(16,38),(17,37),(18,36),(19,35),(20,34),(21,33),(22,32),(23,31),(24,30),(25,29),(26,42),(27,41),(28,40),(43,44),(45,56),(46,55),(47,54),(48,53),(49,52),(50,51),(57,99),(58,112),(59,111),(60,110),(61,109),(62,108),(63,107),(64,106),(65,105),(66,104),(67,103),(68,102),(69,101),(70,100),(71,77),(72,76),(73,75),(78,84),(79,83),(80,82),(85,87),(88,98),(89,97),(90,96),(91,95),(92,94)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 11 | 24 | 0 | 0 | 0 | 0 |
| 24 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 16 | 0 | 4 |
| 0 | 0 | 24 | 20 | 25 | 11 |
| 0 | 0 | 24 | 15 | 24 | 13 |
| 0 | 0 | 14 | 0 | 16 | 5 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 6 | 0 | 0 |
| 0 | 0 | 17 | 10 | 0 | 0 |
| 0 | 0 | 20 | 0 | 8 | 6 |
| 0 | 0 | 26 | 9 | 23 | 21 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 19 | 0 | 0 |
| 0 | 0 | 20 | 6 | 0 | 0 |
| 0 | 0 | 13 | 26 | 10 | 10 |
| 0 | 0 | 6 | 12 | 19 | 22 |
| 28 | 13 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 22 | 0 | 0 |
| 0 | 0 | 11 | 7 | 0 | 0 |
| 0 | 0 | 7 | 17 | 19 | 19 |
| 0 | 0 | 1 | 19 | 7 | 10 |
G:=sub<GL(6,GF(29))| [11,24,0,0,0,0,24,18,0,0,0,0,0,0,9,24,24,14,0,0,16,20,15,0,0,0,0,25,24,16,0,0,4,11,13,5],[17,0,0,0,0,0,0,17,0,0,0,0,0,0,19,17,20,26,0,0,6,10,0,9,0,0,0,0,8,23,0,0,0,0,6,21],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,20,20,13,6,0,0,19,6,26,12,0,0,0,0,10,19,0,0,0,0,10,22],[28,0,0,0,0,0,13,1,0,0,0,0,0,0,22,11,7,1,0,0,22,7,17,19,0,0,0,0,19,7,0,0,0,0,19,10] >;
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4L | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | D14 | D14 | D14 | D14 | D14 | C4○D28 | 2+ (1+4) | D4⋊6D14 | D4⋊8D14 |
| kernel | C42⋊17D14 | C4.D28 | C42⋊2D7 | C22⋊Dic14 | C22⋊D28 | D14⋊D4 | Dic7.D4 | D14.5D4 | C4⋊C4⋊D7 | C2×D14⋊C4 | C4×C7⋊D4 | C23.23D14 | C28⋊7D4 | C23⋊D14 | Dic7⋊D4 | D4×C28 | C4×D4 | C2×C14 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C22 | C14 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 3 | 6 | 3 | 6 | 3 | 24 | 2 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_4^2\rtimes_{17}D_{14} % in TeX
G:=Group("C4^2:17D14"); // GroupNames label
G:=SmallGroup(448,1013);
// by ID
G=gap.SmallGroup(448,1013);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,100,675,570,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^14=d^2=1,a*b=b*a,c*a*c^-1=a^-1,d*a*d=a^-1*b^2,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations