metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊24D14, C14.1382+ 1+4, C4⋊C4⋊34D14, (C4×D28)⋊14C2, (C4×C28)⋊8C22, C42⋊2C2⋊3D7, C42⋊D7⋊6C2, D28⋊C4⋊40C2, D14⋊C4⋊63C22, D14⋊2Q8⋊40C2, D14⋊Q8⋊41C2, C22⋊D28.3C2, C4⋊Dic7⋊62C22, C22⋊C4.77D14, D14.26(C4○D4), D14.5D4⋊39C2, D14.D4⋊49C2, (C2×C14).249C24, (C2×C28).603C23, Dic7⋊C4⋊68C22, (C4×Dic7)⋊58C22, C2.63(D4⋊8D14), C23.55(C22×D7), Dic7.D4⋊45C2, C7⋊9(C22.45C24), (C2×Dic14)⋊33C22, (C2×D28).226C22, C23.D14⋊45C2, (C22×C14).63C23, (C23×D7).69C22, C22.270(C23×D7), C23.D7.65C22, (C2×Dic7).129C23, (C22×D7).223C23, C2.96(D7×C4○D4), (C2×C4×D7)⋊53C22, C4⋊C4⋊7D7⋊39C2, (C7×C4⋊C4)⋊33C22, (D7×C22⋊C4)⋊21C2, (C7×C42⋊2C2)⋊4C2, C14.207(C2×C4○D4), (C2×C4).86(C22×D7), (C2×C7⋊D4).69C22, (C7×C22⋊C4).74C22, SmallGroup(448,1158)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊24D14
G = < a,b,c,d | a4=b4=c14=d2=1, ab=ba, cac-1=dad=a-1b2, cbc-1=a2b, dbd=a2b-1, dcd=c-1 >
Subgroups: 1324 in 248 conjugacy classes, 95 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C22⋊C4, C42⋊C2, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, C42⋊2C2, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C22×D7, C22×D7, C22×C14, C22.45C24, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C2×C4×D7, C2×D28, C2×C7⋊D4, C23×D7, C42⋊D7, C4×D28, C23.D14, D7×C22⋊C4, C22⋊D28, D14.D4, Dic7.D4, C4⋊C4⋊7D7, D28⋊C4, D14.5D4, D14⋊Q8, D14⋊2Q8, C7×C42⋊2C2, C42⋊24D14
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, 2+ 1+4, C22×D7, C22.45C24, C23×D7, D7×C4○D4, D4⋊8D14, C42⋊24D14
►On 112 points
(1 75 12 61)(2 83 13 69)(3 77 14 63)(4 71 8 57)(5 79 9 65)(6 73 10 59)(7 81 11 67)(15 78 28 64)(16 72 22 58)(17 80 23 66)(18 74 24 60)(19 82 25 68)(20 76 26 62)(21 84 27 70)(29 86 36 111)(30 105 37 94)(31 88 38 99)(32 107 39 96)(33 90 40 101)(34 109 41 98)(35 92 42 103)(43 91 50 102)(44 110 51 85)(45 93 52 104)(46 112 53 87)(47 95 54 106)(48 100 55 89)(49 97 56 108)
(1 51 25 35)(2 45 26 29)(3 53 27 37)(4 47 28 31)(5 55 22 39)(6 49 23 33)(7 43 24 41)(8 54 15 38)(9 48 16 32)(10 56 17 40)(11 50 18 34)(12 44 19 42)(13 52 20 36)(14 46 21 30)(57 106 78 99)(58 96 79 89)(59 108 80 101)(60 98 81 91)(61 110 82 103)(62 86 83 93)(63 112 84 105)(64 88 71 95)(65 100 72 107)(66 90 73 97)(67 102 74 109)(68 92 75 85)(69 104 76 111)(70 94 77 87)
(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)(9 14)(10 13)(11 12)(16 21)(17 20)(18 19)(22 27)(23 26)(24 25)(29 56)(30 55)(31 54)(32 53)(33 52)(34 51)(35 50)(36 49)(37 48)(38 47)(39 46)(40 45)(41 44)(42 43)(57 64)(58 63)(59 62)(60 61)(65 70)(66 69)(67 68)(71 78)(72 77)(73 76)(74 75)(79 84)(80 83)(81 82)(85 91)(86 90)(87 89)(92 98)(93 97)(94 96)(100 112)(101 111)(102 110)(103 109)(104 108)(105 107)
G:=sub<Sym(112)| (1,75,12,61)(2,83,13,69)(3,77,14,63)(4,71,8,57)(5,79,9,65)(6,73,10,59)(7,81,11,67)(15,78,28,64)(16,72,22,58)(17,80,23,66)(18,74,24,60)(19,82,25,68)(20,76,26,62)(21,84,27,70)(29,86,36,111)(30,105,37,94)(31,88,38,99)(32,107,39,96)(33,90,40,101)(34,109,41,98)(35,92,42,103)(43,91,50,102)(44,110,51,85)(45,93,52,104)(46,112,53,87)(47,95,54,106)(48,100,55,89)(49,97,56,108), (1,51,25,35)(2,45,26,29)(3,53,27,37)(4,47,28,31)(5,55,22,39)(6,49,23,33)(7,43,24,41)(8,54,15,38)(9,48,16,32)(10,56,17,40)(11,50,18,34)(12,44,19,42)(13,52,20,36)(14,46,21,30)(57,106,78,99)(58,96,79,89)(59,108,80,101)(60,98,81,91)(61,110,82,103)(62,86,83,93)(63,112,84,105)(64,88,71,95)(65,100,72,107)(66,90,73,97)(67,102,74,109)(68,92,75,85)(69,104,76,111)(70,94,77,87), (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)(9,14)(10,13)(11,12)(16,21)(17,20)(18,19)(22,27)(23,26)(24,25)(29,56)(30,55)(31,54)(32,53)(33,52)(34,51)(35,50)(36,49)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43)(57,64)(58,63)(59,62)(60,61)(65,70)(66,69)(67,68)(71,78)(72,77)(73,76)(74,75)(79,84)(80,83)(81,82)(85,91)(86,90)(87,89)(92,98)(93,97)(94,96)(100,112)(101,111)(102,110)(103,109)(104,108)(105,107)>;
G:=Group( (1,75,12,61)(2,83,13,69)(3,77,14,63)(4,71,8,57)(5,79,9,65)(6,73,10,59)(7,81,11,67)(15,78,28,64)(16,72,22,58)(17,80,23,66)(18,74,24,60)(19,82,25,68)(20,76,26,62)(21,84,27,70)(29,86,36,111)(30,105,37,94)(31,88,38,99)(32,107,39,96)(33,90,40,101)(34,109,41,98)(35,92,42,103)(43,91,50,102)(44,110,51,85)(45,93,52,104)(46,112,53,87)(47,95,54,106)(48,100,55,89)(49,97,56,108), (1,51,25,35)(2,45,26,29)(3,53,27,37)(4,47,28,31)(5,55,22,39)(6,49,23,33)(7,43,24,41)(8,54,15,38)(9,48,16,32)(10,56,17,40)(11,50,18,34)(12,44,19,42)(13,52,20,36)(14,46,21,30)(57,106,78,99)(58,96,79,89)(59,108,80,101)(60,98,81,91)(61,110,82,103)(62,86,83,93)(63,112,84,105)(64,88,71,95)(65,100,72,107)(66,90,73,97)(67,102,74,109)(68,92,75,85)(69,104,76,111)(70,94,77,87), (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)(9,14)(10,13)(11,12)(16,21)(17,20)(18,19)(22,27)(23,26)(24,25)(29,56)(30,55)(31,54)(32,53)(33,52)(34,51)(35,50)(36,49)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43)(57,64)(58,63)(59,62)(60,61)(65,70)(66,69)(67,68)(71,78)(72,77)(73,76)(74,75)(79,84)(80,83)(81,82)(85,91)(86,90)(87,89)(92,98)(93,97)(94,96)(100,112)(101,111)(102,110)(103,109)(104,108)(105,107) );
G=PermutationGroup([[(1,75,12,61),(2,83,13,69),(3,77,14,63),(4,71,8,57),(5,79,9,65),(6,73,10,59),(7,81,11,67),(15,78,28,64),(16,72,22,58),(17,80,23,66),(18,74,24,60),(19,82,25,68),(20,76,26,62),(21,84,27,70),(29,86,36,111),(30,105,37,94),(31,88,38,99),(32,107,39,96),(33,90,40,101),(34,109,41,98),(35,92,42,103),(43,91,50,102),(44,110,51,85),(45,93,52,104),(46,112,53,87),(47,95,54,106),(48,100,55,89),(49,97,56,108)], [(1,51,25,35),(2,45,26,29),(3,53,27,37),(4,47,28,31),(5,55,22,39),(6,49,23,33),(7,43,24,41),(8,54,15,38),(9,48,16,32),(10,56,17,40),(11,50,18,34),(12,44,19,42),(13,52,20,36),(14,46,21,30),(57,106,78,99),(58,96,79,89),(59,108,80,101),(60,98,81,91),(61,110,82,103),(62,86,83,93),(63,112,84,105),(64,88,71,95),(65,100,72,107),(66,90,73,97),(67,102,74,109),(68,92,75,85),(69,104,76,111),(70,94,77,87)], [(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),(9,14),(10,13),(11,12),(16,21),(17,20),(18,19),(22,27),(23,26),(24,25),(29,56),(30,55),(31,54),(32,53),(33,52),(34,51),(35,50),(36,49),(37,48),(38,47),(39,46),(40,45),(41,44),(42,43),(57,64),(58,63),(59,62),(60,61),(65,70),(66,69),(67,68),(71,78),(72,77),(73,76),(74,75),(79,84),(80,83),(81,82),(85,91),(86,90),(87,89),(92,98),(93,97),(94,96),(100,112),(101,111),(102,110),(103,109),(104,108),(105,107)]])
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 7A | 7B | 7C | 14A | ··· | 14I | 14J | 14K | 14L | 28A | ··· | 28R | 28S | ··· | 28AA |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 14 | 14 | 14 | 14 | 28 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
67 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | D14 | D14 | D14 | 2+ 1+4 | D7×C4○D4 | D4⋊8D14 |
| kernel | C42⋊24D14 | C42⋊D7 | C4×D28 | C23.D14 | D7×C22⋊C4 | C22⋊D28 | D14.D4 | Dic7.D4 | C4⋊C4⋊7D7 | D28⋊C4 | D14.5D4 | D14⋊Q8 | D14⋊2Q8 | C7×C42⋊2C2 | C42⋊2C2 | D14 | C42 | C22⋊C4 | C4⋊C4 | C14 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 3 | 8 | 3 | 9 | 9 | 1 | 12 | 6 |
Matrix representation of C42⋊24D14 ►in GL6(𝔽29)
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 4 | 0 | 0 |
| 0 | 0 | 25 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 4 | 0 | 0 |
| 0 | 0 | 18 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(29))| [0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,0,0,0,0,0,0,17],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,28,0,0,0,0,1,0],[1,0,0,0,0,0,0,28,0,0,0,0,0,0,4,25,0,0,0,0,4,18,0,0,0,0,0,0,1,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,28,0,0,0,0,0,0,4,18,0,0,0,0,4,25,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C42⋊24D14 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{24}D_{14} % in TeX
G:=Group("C4^2:24D14"); // GroupNames label
G:=SmallGroup(448,1158);
// by ID
G=gap.SmallGroup(448,1158);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,184,1571,570,192,18822]);
// 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*b^2,c*b*c^-1=a^2*b,d*b*d=a^2*b^-1,d*c*d=c^-1>;
// generators/relations