metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊25D14, C14.1422+ 1+4, C4⋊C4⋊17D14, C28⋊4D4⋊5C2, (C4×C28)⋊2C22, C4⋊D28⋊38C2, C42⋊2C2⋊8D7, C42⋊2D7⋊1C2, D14⋊D4⋊46C2, C22⋊D28⋊28C2, D14⋊C4⋊24C22, (C2×D28)⋊10C22, Dic7⋊C4⋊5C22, C22⋊C4.41D14, D14.5D4⋊44C2, (C2×C28).195C23, (C2×C14).255C24, C2.67(D4⋊8D14), C23.61(C22×D7), C7⋊4(C22.54C24), (C22×C14).69C23, (C23×D7).70C22, C22.276(C23×D7), (C2×Dic7).131C23, (C22×D7).114C23, (C2×C4×D7)⋊28C22, (C7×C4⋊C4)⋊34C22, (C7×C42⋊2C2)⋊10C2, (C2×C4).211(C22×D7), (C2×C7⋊D4).75C22, (C7×C22⋊C4).80C22, SmallGroup(448,1164)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊25D14
G = < a,b,c,d | a4=b4=c14=d2=1, ab=ba, cac-1=a-1b2, dad=ab2, cbc-1=a2b, dbd=a2b-1, dcd=c-1 >
Subgroups: 1644 in 252 conjugacy classes, 91 normal (16 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C24, Dic7, C28, D14, C2×C14, C2×C14, C22≀C2, C4⋊D4, C22.D4, C42⋊2C2, C42⋊2C2, C4⋊1D4, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C22×D7, C22×D7, C22×D7, C22×C14, C22.54C24, Dic7⋊C4, D14⋊C4, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×C4×D7, C2×D28, C2×C7⋊D4, C23×D7, C28⋊4D4, C42⋊2D7, C22⋊D28, D14⋊D4, D14.5D4, C4⋊D28, C7×C42⋊2C2, C42⋊25D14
Quotients: C1, C2, C22, C23, D7, C24, D14, 2+ 1+4, C22×D7, C22.54C24, C23×D7, D4⋊8D14, C42⋊25D14
►On 112 points
Generators in S112
(1 75 11 90)(2 83 12 98)(3 77 13 92)(4 71 14 86)(5 79 8 94)(6 73 9 88)(7 81 10 96)(15 74 28 89)(16 82 22 97)(17 76 23 91)(18 84 24 85)(19 78 25 93)(20 72 26 87)(21 80 27 95)(29 100 45 107)(30 68 46 61)(31 102 47 109)(32 70 48 63)(33 104 49 111)(34 58 50 65)(35 106 51 99)(36 60 52 67)(37 108 53 101)(38 62 54 69)(39 110 55 103)(40 64 56 57)(41 112 43 105)(42 66 44 59)
(1 102 22 62)(2 110 23 70)(3 104 24 64)(4 112 25 58)(5 106 26 66)(6 100 27 60)(7 108 28 68)(8 99 20 59)(9 107 21 67)(10 101 15 61)(11 109 16 69)(12 103 17 63)(13 111 18 57)(14 105 19 65)(29 80 36 88)(30 96 37 74)(31 82 38 90)(32 98 39 76)(33 84 40 92)(34 86 41 78)(35 72 42 94)(43 93 50 71)(44 79 51 87)(45 95 52 73)(46 81 53 89)(47 97 54 75)(48 83 55 91)(49 85 56 77)
(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 25)(2 24)(3 23)(4 22)(5 28)(6 27)(7 26)(8 15)(9 21)(10 20)(11 19)(12 18)(13 17)(14 16)(29 52)(30 51)(31 50)(32 49)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 56)(40 55)(41 54)(42 53)(57 70)(58 69)(59 68)(60 67)(61 66)(62 65)(63 64)(71 75)(72 74)(76 84)(77 83)(78 82)(79 81)(85 91)(86 90)(87 89)(92 98)(93 97)(94 96)(99 108)(100 107)(101 106)(102 105)(103 104)(109 112)(110 111)
G:=sub<Sym(112)| (1,75,11,90)(2,83,12,98)(3,77,13,92)(4,71,14,86)(5,79,8,94)(6,73,9,88)(7,81,10,96)(15,74,28,89)(16,82,22,97)(17,76,23,91)(18,84,24,85)(19,78,25,93)(20,72,26,87)(21,80,27,95)(29,100,45,107)(30,68,46,61)(31,102,47,109)(32,70,48,63)(33,104,49,111)(34,58,50,65)(35,106,51,99)(36,60,52,67)(37,108,53,101)(38,62,54,69)(39,110,55,103)(40,64,56,57)(41,112,43,105)(42,66,44,59), (1,102,22,62)(2,110,23,70)(3,104,24,64)(4,112,25,58)(5,106,26,66)(6,100,27,60)(7,108,28,68)(8,99,20,59)(9,107,21,67)(10,101,15,61)(11,109,16,69)(12,103,17,63)(13,111,18,57)(14,105,19,65)(29,80,36,88)(30,96,37,74)(31,82,38,90)(32,98,39,76)(33,84,40,92)(34,86,41,78)(35,72,42,94)(43,93,50,71)(44,79,51,87)(45,95,52,73)(46,81,53,89)(47,97,54,75)(48,83,55,91)(49,85,56,77), (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), (1,25)(2,24)(3,23)(4,22)(5,28)(6,27)(7,26)(8,15)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)(29,52)(30,51)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,56)(40,55)(41,54)(42,53)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)(71,75)(72,74)(76,84)(77,83)(78,82)(79,81)(85,91)(86,90)(87,89)(92,98)(93,97)(94,96)(99,108)(100,107)(101,106)(102,105)(103,104)(109,112)(110,111)>;
G:=Group( (1,75,11,90)(2,83,12,98)(3,77,13,92)(4,71,14,86)(5,79,8,94)(6,73,9,88)(7,81,10,96)(15,74,28,89)(16,82,22,97)(17,76,23,91)(18,84,24,85)(19,78,25,93)(20,72,26,87)(21,80,27,95)(29,100,45,107)(30,68,46,61)(31,102,47,109)(32,70,48,63)(33,104,49,111)(34,58,50,65)(35,106,51,99)(36,60,52,67)(37,108,53,101)(38,62,54,69)(39,110,55,103)(40,64,56,57)(41,112,43,105)(42,66,44,59), (1,102,22,62)(2,110,23,70)(3,104,24,64)(4,112,25,58)(5,106,26,66)(6,100,27,60)(7,108,28,68)(8,99,20,59)(9,107,21,67)(10,101,15,61)(11,109,16,69)(12,103,17,63)(13,111,18,57)(14,105,19,65)(29,80,36,88)(30,96,37,74)(31,82,38,90)(32,98,39,76)(33,84,40,92)(34,86,41,78)(35,72,42,94)(43,93,50,71)(44,79,51,87)(45,95,52,73)(46,81,53,89)(47,97,54,75)(48,83,55,91)(49,85,56,77), (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), (1,25)(2,24)(3,23)(4,22)(5,28)(6,27)(7,26)(8,15)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)(29,52)(30,51)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,56)(40,55)(41,54)(42,53)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)(71,75)(72,74)(76,84)(77,83)(78,82)(79,81)(85,91)(86,90)(87,89)(92,98)(93,97)(94,96)(99,108)(100,107)(101,106)(102,105)(103,104)(109,112)(110,111) );
G=PermutationGroup([[(1,75,11,90),(2,83,12,98),(3,77,13,92),(4,71,14,86),(5,79,8,94),(6,73,9,88),(7,81,10,96),(15,74,28,89),(16,82,22,97),(17,76,23,91),(18,84,24,85),(19,78,25,93),(20,72,26,87),(21,80,27,95),(29,100,45,107),(30,68,46,61),(31,102,47,109),(32,70,48,63),(33,104,49,111),(34,58,50,65),(35,106,51,99),(36,60,52,67),(37,108,53,101),(38,62,54,69),(39,110,55,103),(40,64,56,57),(41,112,43,105),(42,66,44,59)], [(1,102,22,62),(2,110,23,70),(3,104,24,64),(4,112,25,58),(5,106,26,66),(6,100,27,60),(7,108,28,68),(8,99,20,59),(9,107,21,67),(10,101,15,61),(11,109,16,69),(12,103,17,63),(13,111,18,57),(14,105,19,65),(29,80,36,88),(30,96,37,74),(31,82,38,90),(32,98,39,76),(33,84,40,92),(34,86,41,78),(35,72,42,94),(43,93,50,71),(44,79,51,87),(45,95,52,73),(46,81,53,89),(47,97,54,75),(48,83,55,91),(49,85,56,77)], [(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)], [(1,25),(2,24),(3,23),(4,22),(5,28),(6,27),(7,26),(8,15),(9,21),(10,20),(11,19),(12,18),(13,17),(14,16),(29,52),(30,51),(31,50),(32,49),(33,48),(34,47),(35,46),(36,45),(37,44),(38,43),(39,56),(40,55),(41,54),(42,53),(57,70),(58,69),(59,68),(60,67),(61,66),(62,65),(63,64),(71,75),(72,74),(76,84),(77,83),(78,82),(79,81),(85,91),(86,90),(87,89),(92,98),(93,97),(94,96),(99,108),(100,107),(101,106),(102,105),(103,104),(109,112),(110,111)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | ··· | 2I | 4A | ··· | 4F | 4G | 4H | 4I | 7A | 7B | 7C | 14A | ··· | 14I | 14J | 14K | 14L | 28A | ··· | 28R | 28S | ··· | 28AA |
| order | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 28 | ··· | 28 | 4 | ··· | 4 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | D14 | D14 | D14 | 2+ 1+4 | D4⋊8D14 |
| kernel | C42⋊25D14 | C28⋊4D4 | C42⋊2D7 | C22⋊D28 | D14⋊D4 | D14.5D4 | C4⋊D28 | C7×C42⋊2C2 | C42⋊2C2 | C42 | C22⋊C4 | C4⋊C4 | C14 | C2 |
| # reps | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 1 | 3 | 3 | 9 | 9 | 3 | 18 |
Matrix representation of C42⋊25D14 ►in GL8(𝔽29)
| 8 | 24 | 13 | 10 | 0 | 0 | 0 | 0 |
| 13 | 21 | 3 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 5 | 16 | 16 |
| 0 | 0 | 0 | 0 | 28 | 2 | 3 | 19 |
| 0 | 0 | 0 | 0 | 4 | 18 | 8 | 24 |
| 0 | 0 | 0 | 0 | 7 | 11 | 13 | 21 |
| 1 | 0 | 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 24 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 8 | 24 |
| 0 | 0 | 0 | 0 | 0 | 0 | 13 | 21 |
| 7 | 25 | 0 | 0 | 0 | 0 | 0 | 0 |
| 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 25 | 22 | 4 | 0 | 0 | 0 | 0 |
| 22 | 0 | 7 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 10 | 22 | 4 |
| 0 | 0 | 0 | 0 | 21 | 15 | 7 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 9 | 28 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 28 | 0 | 0 | 0 | 0 | 0 |
| 9 | 28 | 20 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 28 |
G:=sub<GL(8,GF(29))| [8,13,0,0,0,0,0,0,24,21,0,0,0,0,0,0,13,3,21,16,0,0,0,0,10,16,5,8,0,0,0,0,0,0,0,0,27,28,4,7,0,0,0,0,5,2,18,11,0,0,0,0,16,3,8,13,0,0,0,0,16,19,24,21],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,27,0,28,0,0,0,0,0,0,27,0,28,0,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,24,27,0,0,0,0,0,0,0,0,8,13,0,0,0,0,0,0,24,21],[7,22,7,22,0,0,0,0,25,0,25,0,0,0,0,0,0,0,22,7,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,8,24,24,21,0,0,0,0,25,28,10,15,0,0,0,0,0,0,22,7,0,0,0,0,0,0,4,0],[1,9,1,9,0,0,0,0,0,28,0,28,0,0,0,0,0,0,28,20,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,26,23,0,0,0,0,0,0,11,3,0,0,0,0,0,0,0,0,1,9,0,0,0,0,0,0,0,28] >;
C42⋊25D14 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{25}D_{14} % in TeX
G:=Group("C4^2:25D14"); // GroupNames label
G:=SmallGroup(448,1164);
// by ID
G=gap.SmallGroup(448,1164);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,219,184,1571,570,192,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^14=d^2=1,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d=a*b^2,c*b*c^-1=a^2*b,d*b*d=a^2*b^-1,d*c*d=c^-1>;
// generators/relations