metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊25D14, C14.1422+ (1+4), C4⋊C4⋊17D14, C4⋊D28⋊5C2, (C4×C28)⋊2C22, C28⋊1D4⋊38C2, C42⋊2C2⋊8D7, C42⋊2D7⋊1C2, C22⋊D28⋊28C2, D14⋊D4⋊46C2, D14⋊C4⋊24C22, (C2×D28)⋊10C22, Dic7⋊C4⋊5C22, C22⋊C4.41D14, D14.5D4⋊44C2, (C2×C14).255C24, (C2×C28).195C23, 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
Subgroups: 1644 in 252 conjugacy classes, 91 normal (16 characteristic)
C1, C2 [×3], C2 [×6], C4 [×9], C22, C22 [×22], C7, C2×C4 [×6], C2×C4 [×6], D4 [×12], C23, C23 [×8], D7 [×5], C14 [×3], C14, C42, C22⋊C4 [×3], C22⋊C4 [×9], C4⋊C4 [×3], C4⋊C4 [×3], C22×C4 [×3], C2×D4 [×12], C24, Dic7 [×3], C28 [×6], D14 [×19], C2×C14, C2×C14 [×3], C22≀C2 [×3], C4⋊D4 [×6], C22.D4 [×3], C42⋊2C2, C42⋊2C2, C4⋊1D4, C4×D7 [×3], D28 [×9], C2×Dic7 [×3], C7⋊D4 [×3], C2×C28 [×6], C22×D7 [×2], C22×D7 [×3], C22×D7 [×3], C22×C14, C22.54C24, Dic7⋊C4 [×3], D14⋊C4 [×9], C4×C28, C7×C22⋊C4 [×3], C7×C4⋊C4 [×3], C2×C4×D7 [×3], C2×D28 [×9], C2×C7⋊D4 [×3], C23×D7, C4⋊D28, C42⋊2D7, C22⋊D28 [×3], D14⋊D4 [×3], D14.5D4 [×3], C28⋊1D4 [×3], C7×C42⋊2C2, C42⋊25D14
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C24, D14 [×7], 2+ (1+4) [×3], C22×D7 [×7], C22.54C24, C23×D7, D4⋊8D14 [×3], C42⋊25D14
Generators and relations
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 >
►On 112 points
Generators in S112
(1 66 9 110)(2 60 10 104)(3 68 11 112)(4 62 12 106)(5 70 13 100)(6 64 14 108)(7 58 8 102)(15 63 23 107)(16 57 24 101)(17 65 25 109)(18 59 26 103)(19 67 27 111)(20 61 28 105)(21 69 22 99)(29 83 94 76)(30 55 95 48)(31 71 96 78)(32 43 97 50)(33 73 98 80)(34 45 85 52)(35 75 86 82)(36 47 87 54)(37 77 88 84)(38 49 89 56)(39 79 90 72)(40 51 91 44)(41 81 92 74)(42 53 93 46)
(1 50 26 72)(2 44 27 80)(3 52 28 74)(4 46 22 82)(5 54 23 76)(6 48 24 84)(7 56 25 78)(8 49 17 71)(9 43 18 79)(10 51 19 73)(11 45 20 81)(12 53 21 75)(13 47 15 83)(14 55 16 77)(29 70 36 107)(30 101 37 64)(31 58 38 109)(32 103 39 66)(33 60 40 111)(34 105 41 68)(35 62 42 99)(57 88 108 95)(59 90 110 97)(61 92 112 85)(63 94 100 87)(65 96 102 89)(67 98 104 91)(69 86 106 93)
(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 28)(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(13 16)(14 15)(29 95)(30 94)(31 93)(32 92)(33 91)(34 90)(35 89)(36 88)(37 87)(38 86)(39 85)(40 98)(41 97)(42 96)(43 52)(44 51)(45 50)(46 49)(47 48)(53 56)(54 55)(57 63)(58 62)(59 61)(64 70)(65 69)(66 68)(71 82)(72 81)(73 80)(74 79)(75 78)(76 77)(83 84)(99 109)(100 108)(101 107)(102 106)(103 105)(110 112)
G:=sub<Sym(112)| (1,66,9,110)(2,60,10,104)(3,68,11,112)(4,62,12,106)(5,70,13,100)(6,64,14,108)(7,58,8,102)(15,63,23,107)(16,57,24,101)(17,65,25,109)(18,59,26,103)(19,67,27,111)(20,61,28,105)(21,69,22,99)(29,83,94,76)(30,55,95,48)(31,71,96,78)(32,43,97,50)(33,73,98,80)(34,45,85,52)(35,75,86,82)(36,47,87,54)(37,77,88,84)(38,49,89,56)(39,79,90,72)(40,51,91,44)(41,81,92,74)(42,53,93,46), (1,50,26,72)(2,44,27,80)(3,52,28,74)(4,46,22,82)(5,54,23,76)(6,48,24,84)(7,56,25,78)(8,49,17,71)(9,43,18,79)(10,51,19,73)(11,45,20,81)(12,53,21,75)(13,47,15,83)(14,55,16,77)(29,70,36,107)(30,101,37,64)(31,58,38,109)(32,103,39,66)(33,60,40,111)(34,105,41,68)(35,62,42,99)(57,88,108,95)(59,90,110,97)(61,92,112,85)(63,94,100,87)(65,96,102,89)(67,98,104,91)(69,86,106,93), (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), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,95)(30,94)(31,93)(32,92)(33,91)(34,90)(35,89)(36,88)(37,87)(38,86)(39,85)(40,98)(41,97)(42,96)(43,52)(44,51)(45,50)(46,49)(47,48)(53,56)(54,55)(57,63)(58,62)(59,61)(64,70)(65,69)(66,68)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77)(83,84)(99,109)(100,108)(101,107)(102,106)(103,105)(110,112)>;
G:=Group( (1,66,9,110)(2,60,10,104)(3,68,11,112)(4,62,12,106)(5,70,13,100)(6,64,14,108)(7,58,8,102)(15,63,23,107)(16,57,24,101)(17,65,25,109)(18,59,26,103)(19,67,27,111)(20,61,28,105)(21,69,22,99)(29,83,94,76)(30,55,95,48)(31,71,96,78)(32,43,97,50)(33,73,98,80)(34,45,85,52)(35,75,86,82)(36,47,87,54)(37,77,88,84)(38,49,89,56)(39,79,90,72)(40,51,91,44)(41,81,92,74)(42,53,93,46), (1,50,26,72)(2,44,27,80)(3,52,28,74)(4,46,22,82)(5,54,23,76)(6,48,24,84)(7,56,25,78)(8,49,17,71)(9,43,18,79)(10,51,19,73)(11,45,20,81)(12,53,21,75)(13,47,15,83)(14,55,16,77)(29,70,36,107)(30,101,37,64)(31,58,38,109)(32,103,39,66)(33,60,40,111)(34,105,41,68)(35,62,42,99)(57,88,108,95)(59,90,110,97)(61,92,112,85)(63,94,100,87)(65,96,102,89)(67,98,104,91)(69,86,106,93), (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), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,95)(30,94)(31,93)(32,92)(33,91)(34,90)(35,89)(36,88)(37,87)(38,86)(39,85)(40,98)(41,97)(42,96)(43,52)(44,51)(45,50)(46,49)(47,48)(53,56)(54,55)(57,63)(58,62)(59,61)(64,70)(65,69)(66,68)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77)(83,84)(99,109)(100,108)(101,107)(102,106)(103,105)(110,112) );
G=PermutationGroup([(1,66,9,110),(2,60,10,104),(3,68,11,112),(4,62,12,106),(5,70,13,100),(6,64,14,108),(7,58,8,102),(15,63,23,107),(16,57,24,101),(17,65,25,109),(18,59,26,103),(19,67,27,111),(20,61,28,105),(21,69,22,99),(29,83,94,76),(30,55,95,48),(31,71,96,78),(32,43,97,50),(33,73,98,80),(34,45,85,52),(35,75,86,82),(36,47,87,54),(37,77,88,84),(38,49,89,56),(39,79,90,72),(40,51,91,44),(41,81,92,74),(42,53,93,46)], [(1,50,26,72),(2,44,27,80),(3,52,28,74),(4,46,22,82),(5,54,23,76),(6,48,24,84),(7,56,25,78),(8,49,17,71),(9,43,18,79),(10,51,19,73),(11,45,20,81),(12,53,21,75),(13,47,15,83),(14,55,16,77),(29,70,36,107),(30,101,37,64),(31,58,38,109),(32,103,39,66),(33,60,40,111),(34,105,41,68),(35,62,42,99),(57,88,108,95),(59,90,110,97),(61,92,112,85),(63,94,100,87),(65,96,102,89),(67,98,104,91),(69,86,106,93)], [(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)], [(1,28),(2,27),(3,26),(4,25),(5,24),(6,23),(7,22),(8,21),(9,20),(10,19),(11,18),(12,17),(13,16),(14,15),(29,95),(30,94),(31,93),(32,92),(33,91),(34,90),(35,89),(36,88),(37,87),(38,86),(39,85),(40,98),(41,97),(42,96),(43,52),(44,51),(45,50),(46,49),(47,48),(53,56),(54,55),(57,63),(58,62),(59,61),(64,70),(65,69),(66,68),(71,82),(72,81),(73,80),(74,79),(75,78),(76,77),(83,84),(99,109),(100,108),(101,107),(102,106),(103,105),(110,112)])
Matrix representation ►G ⊆ 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] >;
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 | C4⋊D28 | C42⋊2D7 | C22⋊D28 | D14⋊D4 | D14.5D4 | C28⋊1D4 | 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 |
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
×
⋊
ℤ
𝔽
○
≀