metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (D4×C14)⋊10C4, (C22×C28)⋊5C4, (Q8×C14)⋊10C4, (C2×Q8)⋊8Dic7, (C2×D4)⋊10Dic7, (C2×C28).200D4, (C22×C4)⋊4Dic7, (C2×D4).206D14, C23⋊Dic7⋊11C2, C23.4(C2×Dic7), (C22×C4).166D14, C4.35(C23.D7), C28.100(C22⋊C4), C23.77(C22×D7), (D4×C14).281C22, C7⋊3(C23.C23), C23.D7.79C22, C23.21D14⋊21C2, C22.9(C22×Dic7), (C22×C14).116C23, (C22×C28).212C22, (C2×C4○D4).8D7, (C14×C4○D4).8C2, (C2×C14).41(C2×D4), (C2×C4).6(C2×Dic7), (C2×C28).308(C2×C4), C14.87(C2×C22⋊C4), C22.13(C2×C7⋊D4), C2.23(C2×C23.D7), (C2×C4).201(C7⋊D4), (C22×C14).16(C2×C4), (C2×C14).202(C22×C4), SmallGroup(448,774)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (D4×C14)⋊10C4
G = < a,b,c,d | a14=b4=c2=d4=1, ab=ba, ac=ca, dad-1=a-1b2, cbc=b-1, bd=db, dcd-1=a7b2c >
Subgroups: 532 in 158 conjugacy classes, 67 normal (25 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic7, C28, C28, C28, C2×C14, C2×C14, C2×C14, C23⋊C4, C42⋊C2, C2×C4○D4, C2×Dic7, C2×C28, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C22×C14, C23.C23, C4×Dic7, C4⋊Dic7, C23.D7, C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C7×C4○D4, C23⋊Dic7, C23.21D14, C14×C4○D4, (D4×C14)⋊10C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, Dic7, D14, C2×C22⋊C4, C2×Dic7, C7⋊D4, C22×D7, C23.C23, C23.D7, C22×Dic7, C2×C7⋊D4, C2×C23.D7, (D4×C14)⋊10C4
►On 112 points
Generators in S112
(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 60 36 23)(2 61 37 24)(3 62 38 25)(4 63 39 26)(5 64 40 27)(6 65 41 28)(7 66 42 15)(8 67 29 16)(9 68 30 17)(10 69 31 18)(11 70 32 19)(12 57 33 20)(13 58 34 21)(14 59 35 22)(43 107 73 86)(44 108 74 87)(45 109 75 88)(46 110 76 89)(47 111 77 90)(48 112 78 91)(49 99 79 92)(50 100 80 93)(51 101 81 94)(52 102 82 95)(53 103 83 96)(54 104 84 97)(55 105 71 98)(56 106 72 85)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 43)(10 44)(11 45)(12 46)(13 47)(14 48)(15 105)(16 106)(17 107)(18 108)(19 109)(20 110)(21 111)(22 112)(23 99)(24 100)(25 101)(26 102)(27 103)(28 104)(29 72)(30 73)(31 74)(32 75)(33 76)(34 77)(35 78)(36 79)(37 80)(38 81)(39 82)(40 83)(41 84)(42 71)(57 89)(58 90)(59 91)(60 92)(61 93)(62 94)(63 95)(64 96)(65 97)(66 98)(67 85)(68 86)(69 87)(70 88)
(2 35)(3 13)(4 33)(5 11)(6 31)(7 9)(8 29)(10 41)(12 39)(14 37)(15 17)(16 67)(18 65)(19 27)(20 63)(21 25)(22 61)(24 59)(26 57)(28 69)(30 42)(32 40)(34 38)(43 48 73 78)(44 77 74 47)(45 46 75 76)(49 56 79 72)(50 71 80 55)(51 54 81 84)(52 83 82 53)(58 62)(64 70)(66 68)(85 99 106 92)(86 91 107 112)(87 111 108 90)(88 89 109 110)(93 105 100 98)(94 97 101 104)(95 103 102 96)
G:=sub<Sym(112)| (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), (1,60,36,23)(2,61,37,24)(3,62,38,25)(4,63,39,26)(5,64,40,27)(6,65,41,28)(7,66,42,15)(8,67,29,16)(9,68,30,17)(10,69,31,18)(11,70,32,19)(12,57,33,20)(13,58,34,21)(14,59,35,22)(43,107,73,86)(44,108,74,87)(45,109,75,88)(46,110,76,89)(47,111,77,90)(48,112,78,91)(49,99,79,92)(50,100,80,93)(51,101,81,94)(52,102,82,95)(53,103,83,96)(54,104,84,97)(55,105,71,98)(56,106,72,85), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,43)(10,44)(11,45)(12,46)(13,47)(14,48)(15,105)(16,106)(17,107)(18,108)(19,109)(20,110)(21,111)(22,112)(23,99)(24,100)(25,101)(26,102)(27,103)(28,104)(29,72)(30,73)(31,74)(32,75)(33,76)(34,77)(35,78)(36,79)(37,80)(38,81)(39,82)(40,83)(41,84)(42,71)(57,89)(58,90)(59,91)(60,92)(61,93)(62,94)(63,95)(64,96)(65,97)(66,98)(67,85)(68,86)(69,87)(70,88), (2,35)(3,13)(4,33)(5,11)(6,31)(7,9)(8,29)(10,41)(12,39)(14,37)(15,17)(16,67)(18,65)(19,27)(20,63)(21,25)(22,61)(24,59)(26,57)(28,69)(30,42)(32,40)(34,38)(43,48,73,78)(44,77,74,47)(45,46,75,76)(49,56,79,72)(50,71,80,55)(51,54,81,84)(52,83,82,53)(58,62)(64,70)(66,68)(85,99,106,92)(86,91,107,112)(87,111,108,90)(88,89,109,110)(93,105,100,98)(94,97,101,104)(95,103,102,96)>;
G:=Group( (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), (1,60,36,23)(2,61,37,24)(3,62,38,25)(4,63,39,26)(5,64,40,27)(6,65,41,28)(7,66,42,15)(8,67,29,16)(9,68,30,17)(10,69,31,18)(11,70,32,19)(12,57,33,20)(13,58,34,21)(14,59,35,22)(43,107,73,86)(44,108,74,87)(45,109,75,88)(46,110,76,89)(47,111,77,90)(48,112,78,91)(49,99,79,92)(50,100,80,93)(51,101,81,94)(52,102,82,95)(53,103,83,96)(54,104,84,97)(55,105,71,98)(56,106,72,85), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,43)(10,44)(11,45)(12,46)(13,47)(14,48)(15,105)(16,106)(17,107)(18,108)(19,109)(20,110)(21,111)(22,112)(23,99)(24,100)(25,101)(26,102)(27,103)(28,104)(29,72)(30,73)(31,74)(32,75)(33,76)(34,77)(35,78)(36,79)(37,80)(38,81)(39,82)(40,83)(41,84)(42,71)(57,89)(58,90)(59,91)(60,92)(61,93)(62,94)(63,95)(64,96)(65,97)(66,98)(67,85)(68,86)(69,87)(70,88), (2,35)(3,13)(4,33)(5,11)(6,31)(7,9)(8,29)(10,41)(12,39)(14,37)(15,17)(16,67)(18,65)(19,27)(20,63)(21,25)(22,61)(24,59)(26,57)(28,69)(30,42)(32,40)(34,38)(43,48,73,78)(44,77,74,47)(45,46,75,76)(49,56,79,72)(50,71,80,55)(51,54,81,84)(52,83,82,53)(58,62)(64,70)(66,68)(85,99,106,92)(86,91,107,112)(87,111,108,90)(88,89,109,110)(93,105,100,98)(94,97,101,104)(95,103,102,96) );
G=PermutationGroup([[(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)], [(1,60,36,23),(2,61,37,24),(3,62,38,25),(4,63,39,26),(5,64,40,27),(6,65,41,28),(7,66,42,15),(8,67,29,16),(9,68,30,17),(10,69,31,18),(11,70,32,19),(12,57,33,20),(13,58,34,21),(14,59,35,22),(43,107,73,86),(44,108,74,87),(45,109,75,88),(46,110,76,89),(47,111,77,90),(48,112,78,91),(49,99,79,92),(50,100,80,93),(51,101,81,94),(52,102,82,95),(53,103,83,96),(54,104,84,97),(55,105,71,98),(56,106,72,85)], [(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,55),(8,56),(9,43),(10,44),(11,45),(12,46),(13,47),(14,48),(15,105),(16,106),(17,107),(18,108),(19,109),(20,110),(21,111),(22,112),(23,99),(24,100),(25,101),(26,102),(27,103),(28,104),(29,72),(30,73),(31,74),(32,75),(33,76),(34,77),(35,78),(36,79),(37,80),(38,81),(39,82),(40,83),(41,84),(42,71),(57,89),(58,90),(59,91),(60,92),(61,93),(62,94),(63,95),(64,96),(65,97),(66,98),(67,85),(68,86),(69,87),(70,88)], [(2,35),(3,13),(4,33),(5,11),(6,31),(7,9),(8,29),(10,41),(12,39),(14,37),(15,17),(16,67),(18,65),(19,27),(20,63),(21,25),(22,61),(24,59),(26,57),(28,69),(30,42),(32,40),(34,38),(43,48,73,78),(44,77,74,47),(45,46,75,76),(49,56,79,72),(50,71,80,55),(51,54,81,84),(52,83,82,53),(58,62),(64,70),(66,68),(85,99,106,92),(86,91,107,112),(87,111,108,90),(88,89,109,110),(93,105,100,98),(94,97,101,104),(95,103,102,96)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4O | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14AA | 28A | ··· | 28L | 28M | ··· | 28AD |
| order | 1 | 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 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D7 | Dic7 | D14 | Dic7 | D14 | Dic7 | C7⋊D4 | C23.C23 | (D4×C14)⋊10C4 |
| kernel | (D4×C14)⋊10C4 | C23⋊Dic7 | C23.21D14 | C14×C4○D4 | C22×C28 | D4×C14 | Q8×C14 | C2×C28 | C2×C4○D4 | C22×C4 | C22×C4 | C2×D4 | C2×D4 | C2×Q8 | C2×C4 | C7 | C1 |
| # reps | 1 | 4 | 2 | 1 | 4 | 2 | 2 | 4 | 3 | 6 | 3 | 3 | 6 | 3 | 24 | 2 | 12 |
Matrix representation of (D4×C14)⋊10C4 ►in GL4(𝔽29) generated by
| 8 | 21 | 0 | 0 |
| 8 | 10 | 0 | 0 |
| 0 | 0 | 8 | 21 |
| 0 | 0 | 8 | 10 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 17 |
| 0 | 0 | 21 | 23 |
| 0 | 0 | 6 | 8 |
| 8 | 6 | 0 | 0 |
| 23 | 21 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 7 | 28 | 0 | 0 |
| 0 | 0 | 9 | 14 |
| 0 | 0 | 19 | 20 |
G:=sub<GL(4,GF(29))| [8,8,0,0,21,10,0,0,0,0,8,8,0,0,21,10],[12,0,0,0,0,12,0,0,0,0,17,0,0,0,0,17],[0,0,8,23,0,0,6,21,21,6,0,0,23,8,0,0],[1,7,0,0,0,28,0,0,0,0,9,19,0,0,14,20] >;
(D4×C14)⋊10C4 in GAP, Magma, Sage, TeX
(D_4\times C_{14})\rtimes_{10}C_4 % in TeX
G:=Group("(D4xC14):10C4"); // GroupNames label
G:=SmallGroup(448,774);
// by ID
G=gap.SmallGroup(448,774);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,232,422,297,1684,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^14=b^4=c^2=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,c*b*c=b^-1,b*d=d*b,d*c*d^-1=a^7*b^2*c>;
// generators/relations