metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊18D14, C14.1242+ 1+4, (C4×D7)⋊4D4, C4.32(D4×D7), (C2×Q8)⋊18D14, C28.61(C2×D4), C4.4D4⋊8D7, C28⋊4D4⋊14C2, C28⋊D4⋊24C2, (C4×C28)⋊22C22, C22⋊C4⋊20D14, D14.45(C2×D4), (C2×D28)⋊9C22, C22⋊D28⋊23C2, D14⋊D4⋊39C2, D14⋊C4⋊23C22, (C2×D4).171D14, C42⋊D7⋊19C2, Dic7.50(C2×D4), (Q8×C14)⋊12C22, C14.88(C22×D4), C28.23D4⋊21C2, (C2×C28).186C23, (C2×C14).218C24, Dic7⋊C4⋊55C22, C7⋊4(C22.29C24), (C4×Dic7)⋊35C22, C2.48(D4⋊8D14), C23.40(C22×D7), (D4×C14).153C22, (C22×C14).48C23, (C23×D7).63C22, C22.239(C23×D7), (C2×Dic7).113C23, (C22×D7).213C23, (C2×D4×D7)⋊16C2, C2.61(C2×D4×D7), (C2×C4×D7)⋊25C22, (C2×Q8⋊2D7)⋊10C2, (C7×C4.4D4)⋊10C2, (C2×C7⋊D4)⋊22C22, (C7×C22⋊C4)⋊28C22, (C2×C4).193(C22×D7), SmallGroup(448,1127)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊18D14
G = < a,b,c,d | a4=b4=c14=d2=1, ab=ba, cac-1=dad=a-1, cbc-1=a2b-1, dbd=b-1, dcd=c-1 >
Subgroups: 2092 in 334 conjugacy classes, 103 normal (29 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C14, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C42⋊C2, C22≀C2, C4⋊D4, C4.4D4, C4.4D4, C4⋊1D4, C22×D4, C2×C4○D4, C4×D7, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×Q8, C22×D7, C22×D7, C22×D7, C22×C14, C22.29C24, C4×Dic7, Dic7⋊C4, D14⋊C4, C4×C28, C7×C22⋊C4, C2×C4×D7, C2×C4×D7, C2×D28, C2×D28, D4×D7, Q8⋊2D7, C2×C7⋊D4, D4×C14, Q8×C14, C23×D7, C42⋊D7, C28⋊4D4, C22⋊D28, D14⋊D4, C28⋊D4, C28.23D4, C7×C4.4D4, C2×D4×D7, C2×Q8⋊2D7, C42⋊18D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, 2+ 1+4, C22×D7, C22.29C24, D4×D7, C23×D7, C2×D4×D7, D4⋊8D14, C42⋊18D14
►On 112 points
Generators in S112
(1 101 19 108)(2 109 20 102)(3 103 21 110)(4 111 15 104)(5 105 16 112)(6 99 17 106)(7 107 18 100)(8 77 24 84)(9 71 25 78)(10 79 26 72)(11 73 27 80)(12 81 28 74)(13 75 22 82)(14 83 23 76)(29 56 90 57)(30 58 91 43)(31 44 92 59)(32 60 93 45)(33 46 94 61)(34 62 95 47)(35 48 96 63)(36 64 97 49)(37 50 98 65)(38 66 85 51)(39 52 86 67)(40 68 87 53)(41 54 88 69)(42 70 89 55)
(1 97 26 29)(2 91 27 37)(3 85 28 31)(4 93 22 39)(5 87 23 33)(6 95 24 41)(7 89 25 35)(8 88 17 34)(9 96 18 42)(10 90 19 36)(11 98 20 30)(12 92 21 38)(13 86 15 32)(14 94 16 40)(43 80 50 109)(44 103 51 74)(45 82 52 111)(46 105 53 76)(47 84 54 99)(48 107 55 78)(49 72 56 101)(57 108 64 79)(58 73 65 102)(59 110 66 81)(60 75 67 104)(61 112 68 83)(62 77 69 106)(63 100 70 71)
(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 7)(2 6)(3 5)(8 11)(9 10)(12 14)(16 21)(17 20)(18 19)(23 28)(24 27)(25 26)(29 89)(30 88)(31 87)(32 86)(33 85)(34 98)(35 97)(36 96)(37 95)(38 94)(39 93)(40 92)(41 91)(42 90)(43 69)(44 68)(45 67)(46 66)(47 65)(48 64)(49 63)(50 62)(51 61)(52 60)(53 59)(54 58)(55 57)(56 70)(71 72)(73 84)(74 83)(75 82)(76 81)(77 80)(78 79)(99 102)(100 101)(103 112)(104 111)(105 110)(106 109)(107 108)
G:=sub<Sym(112)| (1,101,19,108)(2,109,20,102)(3,103,21,110)(4,111,15,104)(5,105,16,112)(6,99,17,106)(7,107,18,100)(8,77,24,84)(9,71,25,78)(10,79,26,72)(11,73,27,80)(12,81,28,74)(13,75,22,82)(14,83,23,76)(29,56,90,57)(30,58,91,43)(31,44,92,59)(32,60,93,45)(33,46,94,61)(34,62,95,47)(35,48,96,63)(36,64,97,49)(37,50,98,65)(38,66,85,51)(39,52,86,67)(40,68,87,53)(41,54,88,69)(42,70,89,55), (1,97,26,29)(2,91,27,37)(3,85,28,31)(4,93,22,39)(5,87,23,33)(6,95,24,41)(7,89,25,35)(8,88,17,34)(9,96,18,42)(10,90,19,36)(11,98,20,30)(12,92,21,38)(13,86,15,32)(14,94,16,40)(43,80,50,109)(44,103,51,74)(45,82,52,111)(46,105,53,76)(47,84,54,99)(48,107,55,78)(49,72,56,101)(57,108,64,79)(58,73,65,102)(59,110,66,81)(60,75,67,104)(61,112,68,83)(62,77,69,106)(63,100,70,71), (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), (1,7)(2,6)(3,5)(8,11)(9,10)(12,14)(16,21)(17,20)(18,19)(23,28)(24,27)(25,26)(29,89)(30,88)(31,87)(32,86)(33,85)(34,98)(35,97)(36,96)(37,95)(38,94)(39,93)(40,92)(41,91)(42,90)(43,69)(44,68)(45,67)(46,66)(47,65)(48,64)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(56,70)(71,72)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)(99,102)(100,101)(103,112)(104,111)(105,110)(106,109)(107,108)>;
G:=Group( (1,101,19,108)(2,109,20,102)(3,103,21,110)(4,111,15,104)(5,105,16,112)(6,99,17,106)(7,107,18,100)(8,77,24,84)(9,71,25,78)(10,79,26,72)(11,73,27,80)(12,81,28,74)(13,75,22,82)(14,83,23,76)(29,56,90,57)(30,58,91,43)(31,44,92,59)(32,60,93,45)(33,46,94,61)(34,62,95,47)(35,48,96,63)(36,64,97,49)(37,50,98,65)(38,66,85,51)(39,52,86,67)(40,68,87,53)(41,54,88,69)(42,70,89,55), (1,97,26,29)(2,91,27,37)(3,85,28,31)(4,93,22,39)(5,87,23,33)(6,95,24,41)(7,89,25,35)(8,88,17,34)(9,96,18,42)(10,90,19,36)(11,98,20,30)(12,92,21,38)(13,86,15,32)(14,94,16,40)(43,80,50,109)(44,103,51,74)(45,82,52,111)(46,105,53,76)(47,84,54,99)(48,107,55,78)(49,72,56,101)(57,108,64,79)(58,73,65,102)(59,110,66,81)(60,75,67,104)(61,112,68,83)(62,77,69,106)(63,100,70,71), (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), (1,7)(2,6)(3,5)(8,11)(9,10)(12,14)(16,21)(17,20)(18,19)(23,28)(24,27)(25,26)(29,89)(30,88)(31,87)(32,86)(33,85)(34,98)(35,97)(36,96)(37,95)(38,94)(39,93)(40,92)(41,91)(42,90)(43,69)(44,68)(45,67)(46,66)(47,65)(48,64)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(56,70)(71,72)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)(99,102)(100,101)(103,112)(104,111)(105,110)(106,109)(107,108) );
G=PermutationGroup([[(1,101,19,108),(2,109,20,102),(3,103,21,110),(4,111,15,104),(5,105,16,112),(6,99,17,106),(7,107,18,100),(8,77,24,84),(9,71,25,78),(10,79,26,72),(11,73,27,80),(12,81,28,74),(13,75,22,82),(14,83,23,76),(29,56,90,57),(30,58,91,43),(31,44,92,59),(32,60,93,45),(33,46,94,61),(34,62,95,47),(35,48,96,63),(36,64,97,49),(37,50,98,65),(38,66,85,51),(39,52,86,67),(40,68,87,53),(41,54,88,69),(42,70,89,55)], [(1,97,26,29),(2,91,27,37),(3,85,28,31),(4,93,22,39),(5,87,23,33),(6,95,24,41),(7,89,25,35),(8,88,17,34),(9,96,18,42),(10,90,19,36),(11,98,20,30),(12,92,21,38),(13,86,15,32),(14,94,16,40),(43,80,50,109),(44,103,51,74),(45,82,52,111),(46,105,53,76),(47,84,54,99),(48,107,55,78),(49,72,56,101),(57,108,64,79),(58,73,65,102),(59,110,66,81),(60,75,67,104),(61,112,68,83),(62,77,69,106),(63,100,70,71)], [(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)], [(1,7),(2,6),(3,5),(8,11),(9,10),(12,14),(16,21),(17,20),(18,19),(23,28),(24,27),(25,26),(29,89),(30,88),(31,87),(32,86),(33,85),(34,98),(35,97),(36,96),(37,95),(38,94),(39,93),(40,92),(41,91),(42,90),(43,69),(44,68),(45,67),(46,66),(47,65),(48,64),(49,63),(50,62),(51,61),(52,60),(53,59),(54,58),(55,57),(56,70),(71,72),(73,84),(74,83),(75,82),(76,81),(77,80),(78,79),(99,102),(100,101),(103,112),(104,111),(105,110),(106,109),(107,108)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28R | 28S | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 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 | 4 | 4 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | 14 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | D14 | D14 | D14 | D14 | 2+ 1+4 | D4×D7 | D4⋊8D14 |
| kernel | C42⋊18D14 | C42⋊D7 | C28⋊4D4 | C22⋊D28 | D14⋊D4 | C28⋊D4 | C28.23D4 | C7×C4.4D4 | C2×D4×D7 | C2×Q8⋊2D7 | C4×D7 | C4.4D4 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C14 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 4 | 3 | 3 | 12 | 3 | 3 | 2 | 6 | 12 |
Matrix representation of C42⋊18D14 ►in GL8(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 | 0 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 28 | 0 |
| 18 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 25 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 28 | 0 | 0 | 0 | 0 | 0 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(8,GF(29))| [28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0],[18,25,0,0,0,0,0,0,4,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1] >;
C42⋊18D14 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{18}D_{14} % in TeX
G:=Group("C4^2:18D14"); // GroupNames label
G:=SmallGroup(448,1127);
// by ID
G=gap.SmallGroup(448,1127);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,675,570,297,192,18822]);
// 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,c*b*c^-1=a^2*b^-1,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations