metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊18D14, C14.1242+ (1+4), (C4×D7)⋊4D4, C4.32(D4×D7), (C2×Q8)⋊18D14, C4.4D4⋊8D7, C28.61(C2×D4), C28⋊D4⋊24C2, C4⋊D28⋊14C2, (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×C14).218C24, (C2×C28).186C23, 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
Subgroups: 2092 in 334 conjugacy classes, 103 normal (29 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×8], C22, C22 [×30], C7, C2×C4, C2×C4 [×4], C2×C4 [×11], D4 [×22], Q8 [×2], C23 [×2], C23 [×13], D7 [×6], C14, C14 [×2], C14 [×2], C42, C42, C22⋊C4 [×4], C22⋊C4 [×6], C4⋊C4 [×2], C22×C4 [×3], C2×D4, C2×D4 [×18], C2×Q8, C4○D4 [×4], C24 [×2], Dic7 [×2], Dic7 [×2], C28 [×2], C28 [×4], D14 [×2], D14 [×22], C2×C14, C2×C14 [×6], C42⋊C2, C22≀C2 [×4], C4⋊D4 [×4], C4.4D4, C4.4D4, C4⋊1D4 [×2], C22×D4, C2×C4○D4, C4×D7 [×4], C4×D7 [×4], D28 [×12], C2×Dic7, C2×Dic7 [×2], C7⋊D4 [×8], C2×C28, C2×C28 [×4], C7×D4 [×2], C7×Q8 [×2], C22×D7, C22×D7 [×4], C22×D7 [×8], C22×C14 [×2], C22.29C24, C4×Dic7, Dic7⋊C4 [×2], D14⋊C4 [×6], C4×C28, C7×C22⋊C4 [×4], C2×C4×D7, C2×C4×D7 [×2], C2×D28 [×2], C2×D28 [×6], D4×D7 [×4], Q8⋊2D7 [×4], C2×C7⋊D4 [×6], D4×C14, Q8×C14, C23×D7 [×2], C42⋊D7, C4⋊D28, C22⋊D28 [×4], D14⋊D4 [×4], C28⋊D4, C28.23D4, C7×C4.4D4, C2×D4×D7, C2×Q8⋊2D7, C42⋊18D14
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C22×D4, 2+ (1+4) [×2], C22×D7 [×7], C22.29C24, D4×D7 [×2], C23×D7, C2×D4×D7, D4⋊8D14 [×2], C42⋊18D14
Generators and relations
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 >
►On 112 points
(1 100 17 107)(2 108 18 101)(3 102 19 109)(4 110 20 103)(5 104 21 111)(6 112 15 105)(7 106 16 99)(8 83 27 76)(9 77 28 84)(10 71 22 78)(11 79 23 72)(12 73 24 80)(13 81 25 74)(14 75 26 82)(29 63 56 91)(30 92 43 64)(31 65 44 93)(32 94 45 66)(33 67 46 95)(34 96 47 68)(35 69 48 97)(36 98 49 70)(37 57 50 85)(38 86 51 58)(39 59 52 87)(40 88 53 60)(41 61 54 89)(42 90 55 62)
(1 56 23 36)(2 50 24 30)(3 44 25 38)(4 52 26 32)(5 46 27 40)(6 54 28 34)(7 48 22 42)(8 53 21 33)(9 47 15 41)(10 55 16 35)(11 49 17 29)(12 43 18 37)(13 51 19 31)(14 45 20 39)(57 73 64 101)(58 109 65 81)(59 75 66 103)(60 111 67 83)(61 77 68 105)(62 99 69 71)(63 79 70 107)(72 98 100 91)(74 86 102 93)(76 88 104 95)(78 90 106 97)(80 92 108 85)(82 94 110 87)(84 96 112 89)
(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 2)(3 7)(4 6)(9 14)(10 13)(11 12)(15 20)(16 19)(17 18)(22 25)(23 24)(26 28)(29 43)(30 56)(31 55)(32 54)(33 53)(34 52)(35 51)(36 50)(37 49)(38 48)(39 47)(40 46)(41 45)(42 44)(57 98)(58 97)(59 96)(60 95)(61 94)(62 93)(63 92)(64 91)(65 90)(66 89)(67 88)(68 87)(69 86)(70 85)(71 74)(72 73)(75 84)(76 83)(77 82)(78 81)(79 80)(99 102)(100 101)(103 112)(104 111)(105 110)(106 109)(107 108)
G:=sub<Sym(112)| (1,100,17,107)(2,108,18,101)(3,102,19,109)(4,110,20,103)(5,104,21,111)(6,112,15,105)(7,106,16,99)(8,83,27,76)(9,77,28,84)(10,71,22,78)(11,79,23,72)(12,73,24,80)(13,81,25,74)(14,75,26,82)(29,63,56,91)(30,92,43,64)(31,65,44,93)(32,94,45,66)(33,67,46,95)(34,96,47,68)(35,69,48,97)(36,98,49,70)(37,57,50,85)(38,86,51,58)(39,59,52,87)(40,88,53,60)(41,61,54,89)(42,90,55,62), (1,56,23,36)(2,50,24,30)(3,44,25,38)(4,52,26,32)(5,46,27,40)(6,54,28,34)(7,48,22,42)(8,53,21,33)(9,47,15,41)(10,55,16,35)(11,49,17,29)(12,43,18,37)(13,51,19,31)(14,45,20,39)(57,73,64,101)(58,109,65,81)(59,75,66,103)(60,111,67,83)(61,77,68,105)(62,99,69,71)(63,79,70,107)(72,98,100,91)(74,86,102,93)(76,88,104,95)(78,90,106,97)(80,92,108,85)(82,94,110,87)(84,96,112,89), (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), (1,2)(3,7)(4,6)(9,14)(10,13)(11,12)(15,20)(16,19)(17,18)(22,25)(23,24)(26,28)(29,43)(30,56)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(37,49)(38,48)(39,47)(40,46)(41,45)(42,44)(57,98)(58,97)(59,96)(60,95)(61,94)(62,93)(63,92)(64,91)(65,90)(66,89)(67,88)(68,87)(69,86)(70,85)(71,74)(72,73)(75,84)(76,83)(77,82)(78,81)(79,80)(99,102)(100,101)(103,112)(104,111)(105,110)(106,109)(107,108)>;
G:=Group( (1,100,17,107)(2,108,18,101)(3,102,19,109)(4,110,20,103)(5,104,21,111)(6,112,15,105)(7,106,16,99)(8,83,27,76)(9,77,28,84)(10,71,22,78)(11,79,23,72)(12,73,24,80)(13,81,25,74)(14,75,26,82)(29,63,56,91)(30,92,43,64)(31,65,44,93)(32,94,45,66)(33,67,46,95)(34,96,47,68)(35,69,48,97)(36,98,49,70)(37,57,50,85)(38,86,51,58)(39,59,52,87)(40,88,53,60)(41,61,54,89)(42,90,55,62), (1,56,23,36)(2,50,24,30)(3,44,25,38)(4,52,26,32)(5,46,27,40)(6,54,28,34)(7,48,22,42)(8,53,21,33)(9,47,15,41)(10,55,16,35)(11,49,17,29)(12,43,18,37)(13,51,19,31)(14,45,20,39)(57,73,64,101)(58,109,65,81)(59,75,66,103)(60,111,67,83)(61,77,68,105)(62,99,69,71)(63,79,70,107)(72,98,100,91)(74,86,102,93)(76,88,104,95)(78,90,106,97)(80,92,108,85)(82,94,110,87)(84,96,112,89), (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), (1,2)(3,7)(4,6)(9,14)(10,13)(11,12)(15,20)(16,19)(17,18)(22,25)(23,24)(26,28)(29,43)(30,56)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(37,49)(38,48)(39,47)(40,46)(41,45)(42,44)(57,98)(58,97)(59,96)(60,95)(61,94)(62,93)(63,92)(64,91)(65,90)(66,89)(67,88)(68,87)(69,86)(70,85)(71,74)(72,73)(75,84)(76,83)(77,82)(78,81)(79,80)(99,102)(100,101)(103,112)(104,111)(105,110)(106,109)(107,108) );
G=PermutationGroup([(1,100,17,107),(2,108,18,101),(3,102,19,109),(4,110,20,103),(5,104,21,111),(6,112,15,105),(7,106,16,99),(8,83,27,76),(9,77,28,84),(10,71,22,78),(11,79,23,72),(12,73,24,80),(13,81,25,74),(14,75,26,82),(29,63,56,91),(30,92,43,64),(31,65,44,93),(32,94,45,66),(33,67,46,95),(34,96,47,68),(35,69,48,97),(36,98,49,70),(37,57,50,85),(38,86,51,58),(39,59,52,87),(40,88,53,60),(41,61,54,89),(42,90,55,62)], [(1,56,23,36),(2,50,24,30),(3,44,25,38),(4,52,26,32),(5,46,27,40),(6,54,28,34),(7,48,22,42),(8,53,21,33),(9,47,15,41),(10,55,16,35),(11,49,17,29),(12,43,18,37),(13,51,19,31),(14,45,20,39),(57,73,64,101),(58,109,65,81),(59,75,66,103),(60,111,67,83),(61,77,68,105),(62,99,69,71),(63,79,70,107),(72,98,100,91),(74,86,102,93),(76,88,104,95),(78,90,106,97),(80,92,108,85),(82,94,110,87),(84,96,112,89)], [(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)], [(1,2),(3,7),(4,6),(9,14),(10,13),(11,12),(15,20),(16,19),(17,18),(22,25),(23,24),(26,28),(29,43),(30,56),(31,55),(32,54),(33,53),(34,52),(35,51),(36,50),(37,49),(38,48),(39,47),(40,46),(41,45),(42,44),(57,98),(58,97),(59,96),(60,95),(61,94),(62,93),(63,92),(64,91),(65,90),(66,89),(67,88),(68,87),(69,86),(70,85),(71,74),(72,73),(75,84),(76,83),(77,82),(78,81),(79,80),(99,102),(100,101),(103,112),(104,111),(105,110),(106,109),(107,108)])
Matrix representation ►G ⊆ 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] >;
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 | C4⋊D28 | 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 |
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