metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊28D14, C14.802+ (1+4), C4⋊1D4⋊9D7, (C2×D4)⋊13D14, (C4×C28)⋊37C22, C23⋊D14⋊28C2, (D4×C14)⋊34C22, C42⋊2D7⋊19C2, Dic7⋊D4⋊39C2, (C2×C14).264C24, (C2×C28).638C23, Dic7⋊C4⋊37C22, D14⋊C4.75C22, C2.84(D4⋊6D14), C23.D7⋊38C22, C23.70(C22×D7), C7⋊5(C22.54C24), (C22×C14).78C23, (C23×D7).73C22, C22.285(C23×D7), C23.18D14⋊28C2, (C2×Dic7).138C23, (C22×Dic7)⋊30C22, (C22×D7).118C23, (C7×C4⋊1D4)⋊15C2, (C2×C4).216(C22×D7), (C2×C7⋊D4).80C22, SmallGroup(448,1173)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1356 in 252 conjugacy classes, 91 normal (12 characteristic)
C1, C2 [×3], C2 [×6], C4 [×9], C22, C22 [×22], C7, C2×C4 [×3], C2×C4 [×9], D4 [×12], C23, C23 [×3], C23 [×5], D7 [×2], C14 [×3], C14 [×4], C42, C22⋊C4 [×12], C4⋊C4 [×6], C22×C4 [×3], C2×D4 [×6], C2×D4 [×6], C24, Dic7 [×6], C28 [×3], D14 [×10], C2×C14, C2×C14 [×12], C22≀C2 [×3], C4⋊D4 [×6], C22.D4 [×3], C42⋊2C2 [×2], C4⋊1D4, C2×Dic7 [×6], C2×Dic7 [×3], C7⋊D4 [×6], C2×C28 [×3], C7×D4 [×6], C22×D7 [×2], C22×D7 [×3], C22×C14, C22×C14 [×3], C22.54C24, Dic7⋊C4 [×6], D14⋊C4 [×6], C23.D7 [×6], C4×C28, C22×Dic7 [×3], C2×C7⋊D4 [×6], D4×C14 [×6], C23×D7, C42⋊2D7 [×2], C23.18D14 [×3], C23⋊D14 [×3], Dic7⋊D4 [×6], C7×C4⋊1D4, C42⋊28D14
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C24, D14 [×7], 2+ (1+4) [×3], C22×D7 [×7], C22.54C24, C23×D7, D4⋊6D14 [×3], C42⋊28D14
Generators and relations
G = < a,b,c,d | a4=b4=c14=d2=1, ab=ba, cac-1=a-1, dad=a-1b2, cbc-1=b-1, dbd=a2b, dcd=c-1 >
►On 112 points
Generators in S112
(1 100 19 107)(2 108 20 101)(3 102 21 109)(4 110 15 103)(5 104 16 111)(6 112 17 105)(7 106 18 99)(8 60 24 67)(9 68 25 61)(10 62 26 69)(11 70 27 63)(12 64 28 57)(13 58 22 65)(14 66 23 59)(29 55 88 74)(30 75 89 56)(31 43 90 76)(32 77 91 44)(33 45 92 78)(34 79 93 46)(35 47 94 80)(36 81 95 48)(37 49 96 82)(38 83 97 50)(39 51 98 84)(40 71 85 52)(41 53 86 72)(42 73 87 54)
(1 84 10 77)(2 78 11 71)(3 72 12 79)(4 80 13 73)(5 74 14 81)(6 82 8 75)(7 76 9 83)(15 47 22 54)(16 55 23 48)(17 49 24 56)(18 43 25 50)(19 51 26 44)(20 45 27 52)(21 53 28 46)(29 66 95 104)(30 105 96 67)(31 68 97 106)(32 107 98 69)(33 70 85 108)(34 109 86 57)(35 58 87 110)(36 111 88 59)(37 60 89 112)(38 99 90 61)(39 62 91 100)(40 101 92 63)(41 64 93 102)(42 103 94 65)
(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 20)(9 19)(10 18)(11 17)(12 16)(13 15)(14 21)(29 86)(30 85)(31 98)(32 97)(33 96)(34 95)(35 94)(36 93)(37 92)(38 91)(39 90)(40 89)(41 88)(42 87)(43 44)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(57 59)(60 70)(61 69)(62 68)(63 67)(64 66)(71 82)(72 81)(73 80)(74 79)(75 78)(76 77)(83 84)(99 107)(100 106)(101 105)(102 104)(108 112)(109 111)
G:=sub<Sym(112)| (1,100,19,107)(2,108,20,101)(3,102,21,109)(4,110,15,103)(5,104,16,111)(6,112,17,105)(7,106,18,99)(8,60,24,67)(9,68,25,61)(10,62,26,69)(11,70,27,63)(12,64,28,57)(13,58,22,65)(14,66,23,59)(29,55,88,74)(30,75,89,56)(31,43,90,76)(32,77,91,44)(33,45,92,78)(34,79,93,46)(35,47,94,80)(36,81,95,48)(37,49,96,82)(38,83,97,50)(39,51,98,84)(40,71,85,52)(41,53,86,72)(42,73,87,54), (1,84,10,77)(2,78,11,71)(3,72,12,79)(4,80,13,73)(5,74,14,81)(6,82,8,75)(7,76,9,83)(15,47,22,54)(16,55,23,48)(17,49,24,56)(18,43,25,50)(19,51,26,44)(20,45,27,52)(21,53,28,46)(29,66,95,104)(30,105,96,67)(31,68,97,106)(32,107,98,69)(33,70,85,108)(34,109,86,57)(35,58,87,110)(36,111,88,59)(37,60,89,112)(38,99,90,61)(39,62,91,100)(40,101,92,63)(41,64,93,102)(42,103,94,65), (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)(8,20)(9,19)(10,18)(11,17)(12,16)(13,15)(14,21)(29,86)(30,85)(31,98)(32,97)(33,96)(34,95)(35,94)(36,93)(37,92)(38,91)(39,90)(40,89)(41,88)(42,87)(43,44)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(57,59)(60,70)(61,69)(62,68)(63,67)(64,66)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77)(83,84)(99,107)(100,106)(101,105)(102,104)(108,112)(109,111)>;
G:=Group( (1,100,19,107)(2,108,20,101)(3,102,21,109)(4,110,15,103)(5,104,16,111)(6,112,17,105)(7,106,18,99)(8,60,24,67)(9,68,25,61)(10,62,26,69)(11,70,27,63)(12,64,28,57)(13,58,22,65)(14,66,23,59)(29,55,88,74)(30,75,89,56)(31,43,90,76)(32,77,91,44)(33,45,92,78)(34,79,93,46)(35,47,94,80)(36,81,95,48)(37,49,96,82)(38,83,97,50)(39,51,98,84)(40,71,85,52)(41,53,86,72)(42,73,87,54), (1,84,10,77)(2,78,11,71)(3,72,12,79)(4,80,13,73)(5,74,14,81)(6,82,8,75)(7,76,9,83)(15,47,22,54)(16,55,23,48)(17,49,24,56)(18,43,25,50)(19,51,26,44)(20,45,27,52)(21,53,28,46)(29,66,95,104)(30,105,96,67)(31,68,97,106)(32,107,98,69)(33,70,85,108)(34,109,86,57)(35,58,87,110)(36,111,88,59)(37,60,89,112)(38,99,90,61)(39,62,91,100)(40,101,92,63)(41,64,93,102)(42,103,94,65), (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)(8,20)(9,19)(10,18)(11,17)(12,16)(13,15)(14,21)(29,86)(30,85)(31,98)(32,97)(33,96)(34,95)(35,94)(36,93)(37,92)(38,91)(39,90)(40,89)(41,88)(42,87)(43,44)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(57,59)(60,70)(61,69)(62,68)(63,67)(64,66)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77)(83,84)(99,107)(100,106)(101,105)(102,104)(108,112)(109,111) );
G=PermutationGroup([(1,100,19,107),(2,108,20,101),(3,102,21,109),(4,110,15,103),(5,104,16,111),(6,112,17,105),(7,106,18,99),(8,60,24,67),(9,68,25,61),(10,62,26,69),(11,70,27,63),(12,64,28,57),(13,58,22,65),(14,66,23,59),(29,55,88,74),(30,75,89,56),(31,43,90,76),(32,77,91,44),(33,45,92,78),(34,79,93,46),(35,47,94,80),(36,81,95,48),(37,49,96,82),(38,83,97,50),(39,51,98,84),(40,71,85,52),(41,53,86,72),(42,73,87,54)], [(1,84,10,77),(2,78,11,71),(3,72,12,79),(4,80,13,73),(5,74,14,81),(6,82,8,75),(7,76,9,83),(15,47,22,54),(16,55,23,48),(17,49,24,56),(18,43,25,50),(19,51,26,44),(20,45,27,52),(21,53,28,46),(29,66,95,104),(30,105,96,67),(31,68,97,106),(32,107,98,69),(33,70,85,108),(34,109,86,57),(35,58,87,110),(36,111,88,59),(37,60,89,112),(38,99,90,61),(39,62,91,100),(40,101,92,63),(41,64,93,102),(42,103,94,65)], [(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),(8,20),(9,19),(10,18),(11,17),(12,16),(13,15),(14,21),(29,86),(30,85),(31,98),(32,97),(33,96),(34,95),(35,94),(36,93),(37,92),(38,91),(39,90),(40,89),(41,88),(42,87),(43,44),(45,56),(46,55),(47,54),(48,53),(49,52),(50,51),(57,59),(60,70),(61,69),(62,68),(63,67),(64,66),(71,82),(72,81),(73,80),(74,79),(75,78),(76,77),(83,84),(99,107),(100,106),(101,105),(102,104),(108,112),(109,111)])
Matrix representation ►G ⊆ GL8(𝔽29)
| 5 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 13 | 24 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 24 | 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 |
| 1 | 0 | 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 27 | 0 | 0 | 0 | 0 |
| 1 | 0 | 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 20 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 14 |
| 0 | 0 | 0 | 0 | 0 | 0 | 15 | 20 |
| 21 | 21 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 26 | 0 | 0 | 0 | 0 | 0 | 0 |
| 21 | 21 | 8 | 8 | 0 | 0 | 0 | 0 |
| 8 | 26 | 21 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 19 | 19 |
| 0 | 0 | 0 | 0 | 0 | 0 | 10 | 7 |
| 8 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 21 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 19 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 10 | 10 |
| 0 | 0 | 0 | 0 | 0 | 0 | 22 | 19 |
G:=sub<GL(8,GF(29))| [5,13,0,0,0,0,0,0,16,24,0,0,0,0,0,0,0,0,5,13,0,0,0,0,0,0,16,24,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0],[1,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,27,0,28,0,0,0,0,0,0,27,0,28,0,0,0,0,0,0,0,0,9,15,0,0,0,0,0,0,14,20,0,0,0,0,0,0,0,0,9,15,0,0,0,0,0,0,14,20],[21,8,21,8,0,0,0,0,21,26,21,26,0,0,0,0,0,0,8,21,0,0,0,0,0,0,8,3,0,0,0,0,0,0,0,0,10,19,0,0,0,0,0,0,10,22,0,0,0,0,0,0,0,0,19,10,0,0,0,0,0,0,19,7],[8,3,0,0,0,0,0,0,8,21,0,0,0,0,0,0,0,0,8,3,0,0,0,0,0,0,8,21,0,0,0,0,0,0,0,0,19,7,0,0,0,0,0,0,19,10,0,0,0,0,0,0,0,0,10,22,0,0,0,0,0,0,10,19] >;
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | ··· | 4I | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 28 | 28 | 4 | 4 | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D7 | D14 | D14 | 2+ (1+4) | D4⋊6D14 |
| kernel | C42⋊28D14 | C42⋊2D7 | C23.18D14 | C23⋊D14 | Dic7⋊D4 | C7×C4⋊1D4 | C4⋊1D4 | C42 | C2×D4 | C14 | C2 |
| # reps | 1 | 2 | 3 | 3 | 6 | 1 | 3 | 3 | 18 | 3 | 18 |
In GAP, Magma, Sage, TeX
C_4^2\rtimes_{28}D_{14} % in TeX
G:=Group("C4^2:28D14"); // GroupNames label
G:=SmallGroup(448,1173);
// by ID
G=gap.SmallGroup(448,1173);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,219,1571,570,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^14=d^2=1,a*b=b*a,c*a*c^-1=a^-1,d*a*d=a^-1*b^2,c*b*c^-1=b^-1,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀