metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊16D14, C14.182+ (1+4), C4⋊C4⋊49D14, (C4×D4)⋊18D7, (D4×C28)⋊20C2, (C22×C4)⋊5D14, (C4×C28)⋊32C22, D14⋊Q8⋊8C2, C22⋊C4⋊48D14, C4⋊Dic7⋊9C22, D14.D4⋊7C2, C23⋊D14.5C2, (C2×D4).217D14, C42⋊2D7⋊16C2, C42⋊D7⋊32C2, D14.17(C4○D4), C28.48D4⋊11C2, C23.D7⋊9C22, (C2×C14).100C24, (C2×C28).699C23, Dic7⋊C4⋊42C22, D14⋊C4.85C22, (C22×C28)⋊37C22, Dic7.D4⋊7C2, (C2×Dic14)⋊6C22, (C4×Dic7)⋊52C22, C2.19(D4⋊6D14), C7⋊3(C22.45C24), (D4×C14).307C22, C22.12(C4○D28), (C23×D7).41C22, (C22×D7).35C23, C23.174(C22×D7), C22.125(C23×D7), C23.11D14⋊29C2, C23.18D14⋊18C2, C23.23D14⋊16C2, (C22×C14).170C23, (C2×Dic7).207C23, (C22×Dic7).98C22, C4⋊C4⋊D7⋊7C2, (C4×C7⋊D4)⋊43C2, C2.23(D7×C4○D4), (C2×D14⋊C4)⋊22C2, (C7×C4⋊C4)⋊61C22, (D7×C22⋊C4)⋊29C2, C2.49(C2×C4○D28), C14.140(C2×C4○D4), (C2×C4×D7).201C22, (C2×C14).16(C4○D4), (C7×C22⋊C4)⋊57C22, (C2×C4).284(C22×D7), (C2×C7⋊D4).16C22, SmallGroup(448,1009)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1204 in 248 conjugacy classes, 97 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C4 [×11], C22, C22 [×2], C22 [×16], C7, C2×C4 [×5], C2×C4 [×13], D4 [×5], Q8, C23 [×2], C23 [×7], D7 [×3], C14 [×3], C14 [×3], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×12], C4⋊C4, C4⋊C4 [×7], C22×C4 [×2], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, C24, Dic7 [×6], C28 [×5], D14 [×2], D14 [×9], C2×C14, C2×C14 [×2], C2×C14 [×5], C2×C22⋊C4 [×2], C42⋊C2 [×2], C4×D4, C4×D4, C22≀C2, C22⋊Q8 [×2], C22.D4 [×3], C4.4D4, C42⋊2C2 [×2], Dic14, C4×D7 [×3], C2×Dic7 [×6], C2×Dic7 [×2], C7⋊D4 [×3], C2×C28 [×5], C2×C28 [×2], C7×D4 [×2], C22×D7 [×2], C22×D7 [×5], C22×C14 [×2], C22.45C24, C4×Dic7 [×2], Dic7⋊C4 [×6], C4⋊Dic7, D14⋊C4 [×8], C23.D7 [×4], C4×C28, C7×C22⋊C4 [×2], C7×C4⋊C4, C2×Dic14, C2×C4×D7 [×2], C22×Dic7, C2×C7⋊D4 [×2], C22×C28 [×2], D4×C14, C23×D7, C42⋊D7, C42⋊2D7, C23.11D14, D7×C22⋊C4, D14.D4, Dic7.D4, D14⋊Q8, C4⋊C4⋊D7, C28.48D4, C2×D14⋊C4, C4×C7⋊D4, C23.23D14, C23.18D14, C23⋊D14, D4×C28, C42⋊16D14
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C4○D4 [×4], C24, D14 [×7], C2×C4○D4 [×2], 2+ (1+4), C22×D7 [×7], C22.45C24, C4○D28 [×2], C23×D7, C2×C4○D28, D4⋊6D14, D7×C4○D4, C42⋊16D14
Generators and relations
G = < a,b,c,d | a4=b4=c14=d2=1, ab=ba, cac-1=dad=a-1b2, bc=cb, dbd=a2b, dcd=c-1 >
►On 112 points
(1 68 27 79)(2 62 28 73)(3 70 22 81)(4 64 23 75)(5 58 24 83)(6 66 25 77)(7 60 26 71)(8 80 20 69)(9 74 21 63)(10 82 15 57)(11 76 16 65)(12 84 17 59)(13 78 18 67)(14 72 19 61)(29 112 53 94)(30 106 54 88)(31 100 55 96)(32 108 56 90)(33 102 50 98)(34 110 51 92)(35 104 52 86)(36 91 43 109)(37 85 44 103)(38 93 45 111)(39 87 46 105)(40 95 47 99)(41 89 48 107)(42 97 49 101)
(1 36 14 33)(2 37 8 34)(3 38 9 35)(4 39 10 29)(5 40 11 30)(6 41 12 31)(7 42 13 32)(15 53 23 46)(16 54 24 47)(17 55 25 48)(18 56 26 49)(19 50 27 43)(20 51 28 44)(21 52 22 45)(57 94 75 105)(58 95 76 106)(59 96 77 107)(60 97 78 108)(61 98 79 109)(62 85 80 110)(63 86 81 111)(64 87 82 112)(65 88 83 99)(66 89 84 100)(67 90 71 101)(68 91 72 102)(69 92 73 103)(70 93 74 104)
(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 13)(2 12)(3 11)(4 10)(5 9)(6 8)(7 14)(15 23)(16 22)(17 28)(18 27)(19 26)(20 25)(21 24)(29 46)(30 45)(31 44)(32 43)(33 49)(34 48)(35 47)(36 56)(37 55)(38 54)(39 53)(40 52)(41 51)(42 50)(57 82)(58 81)(59 80)(60 79)(61 78)(62 77)(63 76)(64 75)(65 74)(66 73)(67 72)(68 71)(69 84)(70 83)(85 89)(86 88)(90 98)(91 97)(92 96)(93 95)(99 111)(100 110)(101 109)(102 108)(103 107)(104 106)
G:=sub<Sym(112)| (1,68,27,79)(2,62,28,73)(3,70,22,81)(4,64,23,75)(5,58,24,83)(6,66,25,77)(7,60,26,71)(8,80,20,69)(9,74,21,63)(10,82,15,57)(11,76,16,65)(12,84,17,59)(13,78,18,67)(14,72,19,61)(29,112,53,94)(30,106,54,88)(31,100,55,96)(32,108,56,90)(33,102,50,98)(34,110,51,92)(35,104,52,86)(36,91,43,109)(37,85,44,103)(38,93,45,111)(39,87,46,105)(40,95,47,99)(41,89,48,107)(42,97,49,101), (1,36,14,33)(2,37,8,34)(3,38,9,35)(4,39,10,29)(5,40,11,30)(6,41,12,31)(7,42,13,32)(15,53,23,46)(16,54,24,47)(17,55,25,48)(18,56,26,49)(19,50,27,43)(20,51,28,44)(21,52,22,45)(57,94,75,105)(58,95,76,106)(59,96,77,107)(60,97,78,108)(61,98,79,109)(62,85,80,110)(63,86,81,111)(64,87,82,112)(65,88,83,99)(66,89,84,100)(67,90,71,101)(68,91,72,102)(69,92,73,103)(70,93,74,104), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(7,14)(15,23)(16,22)(17,28)(18,27)(19,26)(20,25)(21,24)(29,46)(30,45)(31,44)(32,43)(33,49)(34,48)(35,47)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(57,82)(58,81)(59,80)(60,79)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,84)(70,83)(85,89)(86,88)(90,98)(91,97)(92,96)(93,95)(99,111)(100,110)(101,109)(102,108)(103,107)(104,106)>;
G:=Group( (1,68,27,79)(2,62,28,73)(3,70,22,81)(4,64,23,75)(5,58,24,83)(6,66,25,77)(7,60,26,71)(8,80,20,69)(9,74,21,63)(10,82,15,57)(11,76,16,65)(12,84,17,59)(13,78,18,67)(14,72,19,61)(29,112,53,94)(30,106,54,88)(31,100,55,96)(32,108,56,90)(33,102,50,98)(34,110,51,92)(35,104,52,86)(36,91,43,109)(37,85,44,103)(38,93,45,111)(39,87,46,105)(40,95,47,99)(41,89,48,107)(42,97,49,101), (1,36,14,33)(2,37,8,34)(3,38,9,35)(4,39,10,29)(5,40,11,30)(6,41,12,31)(7,42,13,32)(15,53,23,46)(16,54,24,47)(17,55,25,48)(18,56,26,49)(19,50,27,43)(20,51,28,44)(21,52,22,45)(57,94,75,105)(58,95,76,106)(59,96,77,107)(60,97,78,108)(61,98,79,109)(62,85,80,110)(63,86,81,111)(64,87,82,112)(65,88,83,99)(66,89,84,100)(67,90,71,101)(68,91,72,102)(69,92,73,103)(70,93,74,104), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(7,14)(15,23)(16,22)(17,28)(18,27)(19,26)(20,25)(21,24)(29,46)(30,45)(31,44)(32,43)(33,49)(34,48)(35,47)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(57,82)(58,81)(59,80)(60,79)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,84)(70,83)(85,89)(86,88)(90,98)(91,97)(92,96)(93,95)(99,111)(100,110)(101,109)(102,108)(103,107)(104,106) );
G=PermutationGroup([(1,68,27,79),(2,62,28,73),(3,70,22,81),(4,64,23,75),(5,58,24,83),(6,66,25,77),(7,60,26,71),(8,80,20,69),(9,74,21,63),(10,82,15,57),(11,76,16,65),(12,84,17,59),(13,78,18,67),(14,72,19,61),(29,112,53,94),(30,106,54,88),(31,100,55,96),(32,108,56,90),(33,102,50,98),(34,110,51,92),(35,104,52,86),(36,91,43,109),(37,85,44,103),(38,93,45,111),(39,87,46,105),(40,95,47,99),(41,89,48,107),(42,97,49,101)], [(1,36,14,33),(2,37,8,34),(3,38,9,35),(4,39,10,29),(5,40,11,30),(6,41,12,31),(7,42,13,32),(15,53,23,46),(16,54,24,47),(17,55,25,48),(18,56,26,49),(19,50,27,43),(20,51,28,44),(21,52,22,45),(57,94,75,105),(58,95,76,106),(59,96,77,107),(60,97,78,108),(61,98,79,109),(62,85,80,110),(63,86,81,111),(64,87,82,112),(65,88,83,99),(66,89,84,100),(67,90,71,101),(68,91,72,102),(69,92,73,103),(70,93,74,104)], [(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)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,8),(7,14),(15,23),(16,22),(17,28),(18,27),(19,26),(20,25),(21,24),(29,46),(30,45),(31,44),(32,43),(33,49),(34,48),(35,47),(36,56),(37,55),(38,54),(39,53),(40,52),(41,51),(42,50),(57,82),(58,81),(59,80),(60,79),(61,78),(62,77),(63,76),(64,75),(65,74),(66,73),(67,72),(68,71),(69,84),(70,83),(85,89),(86,88),(90,98),(91,97),(92,96),(93,95),(99,111),(100,110),(101,109),(102,108),(103,107),(104,106)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 1 |
| 0 | 0 | 0 | 0 | 7 | 20 |
| 2 | 2 | 0 | 0 | 0 | 0 |
| 12 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 10 | 0 | 0 |
| 0 | 0 | 19 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 11 | 28 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 2 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 18 | 1 |
G:=sub<GL(6,GF(29))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,7,0,0,0,0,1,20],[2,12,0,0,0,0,2,27,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,17,0,0,0,0,0,0,17],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,22,19,0,0,0,0,10,10,0,0,0,0,0,0,1,11,0,0,0,0,0,28],[28,2,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,28,18,0,0,0,0,0,1] >;
85 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4O | 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 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 14 | 14 | 28 | 2 | ··· | 2 | 4 | 4 | 14 | 14 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
85 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 | 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 | C4○D4 | D14 | D14 | D14 | D14 | D14 | C4○D28 | 2+ (1+4) | D4⋊6D14 | D7×C4○D4 |
| kernel | C42⋊16D14 | C42⋊D7 | C42⋊2D7 | C23.11D14 | D7×C22⋊C4 | D14.D4 | Dic7.D4 | D14⋊Q8 | C4⋊C4⋊D7 | C28.48D4 | C2×D14⋊C4 | C4×C7⋊D4 | C23.23D14 | C23.18D14 | C23⋊D14 | D4×C28 | C4×D4 | D14 | 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 | 4 | 3 | 6 | 3 | 6 | 3 | 24 | 1 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_4^2\rtimes_{16}D_{14} % in TeX
G:=Group("C4^2:16D14"); // GroupNames label
G:=SmallGroup(448,1009);
// by ID
G=gap.SmallGroup(448,1009);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,184,1571,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^14=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b^2,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations