metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28⋊4D4, C4⋊1D28, C42⋊5D7, (C4×C28)⋊4C2, (C2×D28)⋊1C2, C7⋊1(C4⋊1D4), C2.5(C2×D28), C14.3(C2×D4), (C2×C4).76D14, (C2×C14).15C23, (C2×C28).87C22, (C22×D7).1C22, C22.36(C22×D7), SmallGroup(224,69)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28⋊4D4
G = < a,b,c | a4=b28=c2=1, ab=ba, cac=a-1, cbc=b-1 >
Subgroups: 630 in 108 conjugacy classes, 41 normal (7 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, D4, C23, D7, C14, C42, C2×D4, C28, D14, C2×C14, C4⋊1D4, D28, C2×C28, C22×D7, C4×C28, C2×D28, C28⋊4D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C4⋊1D4, D28, C22×D7, C2×D28, C28⋊4D4
►On 112 points
Generators in S112
(1 89 71 51)(2 90 72 52)(3 91 73 53)(4 92 74 54)(5 93 75 55)(6 94 76 56)(7 95 77 29)(8 96 78 30)(9 97 79 31)(10 98 80 32)(11 99 81 33)(12 100 82 34)(13 101 83 35)(14 102 84 36)(15 103 57 37)(16 104 58 38)(17 105 59 39)(18 106 60 40)(19 107 61 41)(20 108 62 42)(21 109 63 43)(22 110 64 44)(23 111 65 45)(24 112 66 46)(25 85 67 47)(26 86 68 48)(27 87 69 49)(28 88 70 50)
(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)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)(29 89)(30 88)(31 87)(32 86)(33 85)(34 112)(35 111)(36 110)(37 109)(38 108)(39 107)(40 106)(41 105)(42 104)(43 103)(44 102)(45 101)(46 100)(47 99)(48 98)(49 97)(50 96)(51 95)(52 94)(53 93)(54 92)(55 91)(56 90)(57 63)(58 62)(59 61)(64 84)(65 83)(66 82)(67 81)(68 80)(69 79)(70 78)(71 77)(72 76)(73 75)
G:=sub<Sym(112)| (1,89,71,51)(2,90,72,52)(3,91,73,53)(4,92,74,54)(5,93,75,55)(6,94,76,56)(7,95,77,29)(8,96,78,30)(9,97,79,31)(10,98,80,32)(11,99,81,33)(12,100,82,34)(13,101,83,35)(14,102,84,36)(15,103,57,37)(16,104,58,38)(17,105,59,39)(18,106,60,40)(19,107,61,41)(20,108,62,42)(21,109,63,43)(22,110,64,44)(23,111,65,45)(24,112,66,46)(25,85,67,47)(26,86,68,48)(27,87,69,49)(28,88,70,50), (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)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,89)(30,88)(31,87)(32,86)(33,85)(34,112)(35,111)(36,110)(37,109)(38,108)(39,107)(40,106)(41,105)(42,104)(43,103)(44,102)(45,101)(46,100)(47,99)(48,98)(49,97)(50,96)(51,95)(52,94)(53,93)(54,92)(55,91)(56,90)(57,63)(58,62)(59,61)(64,84)(65,83)(66,82)(67,81)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75)>;
G:=Group( (1,89,71,51)(2,90,72,52)(3,91,73,53)(4,92,74,54)(5,93,75,55)(6,94,76,56)(7,95,77,29)(8,96,78,30)(9,97,79,31)(10,98,80,32)(11,99,81,33)(12,100,82,34)(13,101,83,35)(14,102,84,36)(15,103,57,37)(16,104,58,38)(17,105,59,39)(18,106,60,40)(19,107,61,41)(20,108,62,42)(21,109,63,43)(22,110,64,44)(23,111,65,45)(24,112,66,46)(25,85,67,47)(26,86,68,48)(27,87,69,49)(28,88,70,50), (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)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,89)(30,88)(31,87)(32,86)(33,85)(34,112)(35,111)(36,110)(37,109)(38,108)(39,107)(40,106)(41,105)(42,104)(43,103)(44,102)(45,101)(46,100)(47,99)(48,98)(49,97)(50,96)(51,95)(52,94)(53,93)(54,92)(55,91)(56,90)(57,63)(58,62)(59,61)(64,84)(65,83)(66,82)(67,81)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75) );
G=PermutationGroup([[(1,89,71,51),(2,90,72,52),(3,91,73,53),(4,92,74,54),(5,93,75,55),(6,94,76,56),(7,95,77,29),(8,96,78,30),(9,97,79,31),(10,98,80,32),(11,99,81,33),(12,100,82,34),(13,101,83,35),(14,102,84,36),(15,103,57,37),(16,104,58,38),(17,105,59,39),(18,106,60,40),(19,107,61,41),(20,108,62,42),(21,109,63,43),(22,110,64,44),(23,111,65,45),(24,112,66,46),(25,85,67,47),(26,86,68,48),(27,87,69,49),(28,88,70,50)], [(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),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,20),(17,19),(29,89),(30,88),(31,87),(32,86),(33,85),(34,112),(35,111),(36,110),(37,109),(38,108),(39,107),(40,106),(41,105),(42,104),(43,103),(44,102),(45,101),(46,100),(47,99),(48,98),(49,97),(50,96),(51,95),(52,94),(53,93),(54,92),(55,91),(56,90),(57,63),(58,62),(59,61),(64,84),(65,83),(66,82),(67,81),(68,80),(69,79),(70,78),(71,77),(72,76),(73,75)]])
C28⋊4D4 is a maximal subgroup of
C4.D56 C8⋊5D28 C4.5D56 C28⋊4D8 C8⋊D28 C42.19D14 D4⋊4D28 C4⋊D56 Dic14⋊8D4 C28⋊7D8 Q8⋊D28 C42.64D14 C42.70D14 C28⋊D8 C28⋊6SD16 C28.D8 C42.276D14 C42⋊9D14 C42.100D14 D4×D28 Dic14⋊24D4 Q8⋊6D28 C42.136D14 C42⋊18D14 C42.156D14 C42⋊25D14 D7×C4⋊1D4 C42.240D14
C28⋊4D4 is a maximal quotient of
(C2×C28)⋊5D4 (C2×C28).33D4 C8⋊5D28 C28⋊4D8 C8.8D28 C28⋊4Q16 C8⋊D28 C8.D28 C42⋊8Dic7 (C2×C4)⋊6D28
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
62 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | D4 | D7 | D14 | D28 |
| kernel | C28⋊4D4 | C4×C28 | C2×D28 | C28 | C42 | C2×C4 | C4 |
| # reps | 1 | 1 | 6 | 6 | 3 | 9 | 36 |
Matrix representation of C28⋊4D4 ►in GL4(𝔽29) generated by
| 27 | 23 | 0 | 0 |
| 25 | 2 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 26 | 22 | 0 | 0 |
| 5 | 21 | 0 | 0 |
| 0 | 0 | 9 | 4 |
| 0 | 0 | 25 | 8 |
| 0 | 19 | 0 | 0 |
| 26 | 0 | 0 | 0 |
| 0 | 0 | 10 | 7 |
| 0 | 0 | 19 | 19 |
G:=sub<GL(4,GF(29))| [27,25,0,0,23,2,0,0,0,0,28,0,0,0,0,28],[26,5,0,0,22,21,0,0,0,0,9,25,0,0,4,8],[0,26,0,0,19,0,0,0,0,0,10,19,0,0,7,19] >;
C28⋊4D4 in GAP, Magma, Sage, TeX
C_{28}\rtimes_4D_4 % in TeX
G:=Group("C28:4D4"); // GroupNames label
G:=SmallGroup(224,69);
// by ID
G=gap.SmallGroup(224,69);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,217,103,218,50,6917]);
// Polycyclic
G:=Group<a,b,c|a^4=b^28=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations