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G = C4⋊D28order 224 = 25·7

The semidirect product of C4 and D28 acting via D28/D14=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42D28, C281D4, D142D4, C4⋊C43D7, D14⋊C48C2, (C2×D28)⋊4C2, C72(C4⋊D4), C2.9(C2×D28), C2.13(D4×D7), C14.7(C2×D4), (C2×C4).12D14, (C2×C28).5C22, C14.34(C4○D4), (C2×C14).36C23, C2.6(Q82D7), (C22×D7).7C22, C22.50(C22×D7), (C2×Dic7).31C22, (C2×C4×D7)⋊1C2, (C7×C4⋊C4)⋊6C2, SmallGroup(224,90)

Series: Derived Chief Lower central Upper central

C1C2×C14 — C4⋊D28
C1C7C14C2×C14C22×D7C2×C4×D7 — C4⋊D28
C7C2×C14 — C4⋊D28
C1C22C4⋊C4

Generators and relations for C4⋊D28
 G = < a,b,c | a28=b4=c2=1, bab-1=a15, cac=a-1, cbc=b-1 >

Subgroups: 502 in 94 conjugacy classes, 35 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, C23, D7, C14, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic7, C28, C28, D14, D14, C2×C14, C4⋊D4, C4×D7, D28, C2×Dic7, C2×C28, C2×C28, C22×D7, C22×D7, D14⋊C4, C7×C4⋊C4, C2×C4×D7, C2×D28, C2×D28, C4⋊D28
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, D28, C22×D7, C2×D28, D4×D7, Q82D7, C4⋊D28

Smallest permutation representation of C4⋊D28
On 112 points
Generators in S112
(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 62 94 51)(2 77 95 38)(3 64 96 53)(4 79 97 40)(5 66 98 55)(6 81 99 42)(7 68 100 29)(8 83 101 44)(9 70 102 31)(10 57 103 46)(11 72 104 33)(12 59 105 48)(13 74 106 35)(14 61 107 50)(15 76 108 37)(16 63 109 52)(17 78 110 39)(18 65 111 54)(19 80 112 41)(20 67 85 56)(21 82 86 43)(22 69 87 30)(23 84 88 45)(24 71 89 32)(25 58 90 47)(26 73 91 34)(27 60 92 49)(28 75 93 36)
(1 94)(2 93)(3 92)(4 91)(5 90)(6 89)(7 88)(8 87)(9 86)(10 85)(11 112)(12 111)(13 110)(14 109)(15 108)(16 107)(17 106)(18 105)(19 104)(20 103)(21 102)(22 101)(23 100)(24 99)(25 98)(26 97)(27 96)(28 95)(29 45)(30 44)(31 43)(32 42)(33 41)(34 40)(35 39)(36 38)(46 56)(47 55)(48 54)(49 53)(50 52)(57 67)(58 66)(59 65)(60 64)(61 63)(68 84)(69 83)(70 82)(71 81)(72 80)(73 79)(74 78)(75 77)

G:=sub<Sym(112)| (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), (1,62,94,51)(2,77,95,38)(3,64,96,53)(4,79,97,40)(5,66,98,55)(6,81,99,42)(7,68,100,29)(8,83,101,44)(9,70,102,31)(10,57,103,46)(11,72,104,33)(12,59,105,48)(13,74,106,35)(14,61,107,50)(15,76,108,37)(16,63,109,52)(17,78,110,39)(18,65,111,54)(19,80,112,41)(20,67,85,56)(21,82,86,43)(22,69,87,30)(23,84,88,45)(24,71,89,32)(25,58,90,47)(26,73,91,34)(27,60,92,49)(28,75,93,36), (1,94)(2,93)(3,92)(4,91)(5,90)(6,89)(7,88)(8,87)(9,86)(10,85)(11,112)(12,111)(13,110)(14,109)(15,108)(16,107)(17,106)(18,105)(19,104)(20,103)(21,102)(22,101)(23,100)(24,99)(25,98)(26,97)(27,96)(28,95)(29,45)(30,44)(31,43)(32,42)(33,41)(34,40)(35,39)(36,38)(46,56)(47,55)(48,54)(49,53)(50,52)(57,67)(58,66)(59,65)(60,64)(61,63)(68,84)(69,83)(70,82)(71,81)(72,80)(73,79)(74,78)(75,77)>;

G:=Group( (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), (1,62,94,51)(2,77,95,38)(3,64,96,53)(4,79,97,40)(5,66,98,55)(6,81,99,42)(7,68,100,29)(8,83,101,44)(9,70,102,31)(10,57,103,46)(11,72,104,33)(12,59,105,48)(13,74,106,35)(14,61,107,50)(15,76,108,37)(16,63,109,52)(17,78,110,39)(18,65,111,54)(19,80,112,41)(20,67,85,56)(21,82,86,43)(22,69,87,30)(23,84,88,45)(24,71,89,32)(25,58,90,47)(26,73,91,34)(27,60,92,49)(28,75,93,36), (1,94)(2,93)(3,92)(4,91)(5,90)(6,89)(7,88)(8,87)(9,86)(10,85)(11,112)(12,111)(13,110)(14,109)(15,108)(16,107)(17,106)(18,105)(19,104)(20,103)(21,102)(22,101)(23,100)(24,99)(25,98)(26,97)(27,96)(28,95)(29,45)(30,44)(31,43)(32,42)(33,41)(34,40)(35,39)(36,38)(46,56)(47,55)(48,54)(49,53)(50,52)(57,67)(58,66)(59,65)(60,64)(61,63)(68,84)(69,83)(70,82)(71,81)(72,80)(73,79)(74,78)(75,77) );

G=PermutationGroup([[(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)], [(1,62,94,51),(2,77,95,38),(3,64,96,53),(4,79,97,40),(5,66,98,55),(6,81,99,42),(7,68,100,29),(8,83,101,44),(9,70,102,31),(10,57,103,46),(11,72,104,33),(12,59,105,48),(13,74,106,35),(14,61,107,50),(15,76,108,37),(16,63,109,52),(17,78,110,39),(18,65,111,54),(19,80,112,41),(20,67,85,56),(21,82,86,43),(22,69,87,30),(23,84,88,45),(24,71,89,32),(25,58,90,47),(26,73,91,34),(27,60,92,49),(28,75,93,36)], [(1,94),(2,93),(3,92),(4,91),(5,90),(6,89),(7,88),(8,87),(9,86),(10,85),(11,112),(12,111),(13,110),(14,109),(15,108),(16,107),(17,106),(18,105),(19,104),(20,103),(21,102),(22,101),(23,100),(24,99),(25,98),(26,97),(27,96),(28,95),(29,45),(30,44),(31,43),(32,42),(33,41),(34,40),(35,39),(36,38),(46,56),(47,55),(48,54),(49,53),(50,52),(57,67),(58,66),(59,65),(60,64),(61,63),(68,84),(69,83),(70,82),(71,81),(72,80),(73,79),(74,78),(75,77)]])

C4⋊D28 is a maximal subgroup of
D4⋊D28  D14⋊D8  D43D28  C7⋊C8⋊D4  Q82D28  D142SD16  D284D4  C7⋊(C8⋊D4)  D14.4SD16  C88D28  C567D4  C4.Q8⋊D7  D14.5D8  C87D28  C2.D8⋊D7  C83D28  C14.2- 1+4  C14.2+ 1+4  C14.112+ 1+4  C428D14  C429D14  C42.95D14  C42.97D14  C42.228D14  D4×D28  D45D28  C42.116D14  Q85D28  Q86D28  C42.131D14  C42.133D14  Dic1420D4  D7×C4⋊D4  C14.382+ 1+4  D2819D4  C4⋊C426D14  C14.172- 1+4  D2821D4  Dic1422D4  C14.562+ 1+4  C14.262- 1+4  C14.1202+ 1+4  C14.1212+ 1+4  C14.662+ 1+4  C14.682+ 1+4  C42.237D14  C42.150D14  C42.153D14  C42.156D14  C42.158D14  C4223D14  C42.163D14  C4225D14  C42.240D14  D2812D4  C42.178D14  C42.179D14
C4⋊D28 is a maximal quotient of
C14.(C4⋊Q8)  (C2×C4)⋊9D28  C2.(C4×D28)  (C2×C28)⋊5D4  (C2×Dic7)⋊3D4  (C2×C4).21D28  C28⋊SD16  C4⋊D56  D28.19D4  C42.36D14  Dic148D4  C4⋊Dic28  C88D28  C567D4  C8.2D28  C87D28  C83D28  D142Q16  C8.20D28  C8.21D28  C8.24D28  C4⋊(C4⋊Dic7)  C4⋊(D14⋊C4)  (C2×D28)⋊10C4  (C2×C4)⋊3D28  (C2×C4).45D28

44 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F7A7B7C14A···14I28A···28R
order1222222244444477714···1428···28
size111114142828224414142222···24···4

44 irreducible representations

dim1111122222244
type++++++++++++
imageC1C2C2C2C2D4D4D7C4○D4D14D28D4×D7Q82D7
kernelC4⋊D28D14⋊C4C7×C4⋊C4C2×C4×D7C2×D28C28D14C4⋊C4C14C2×C4C4C2C2
# reps12113223291233

Matrix representation of C4⋊D28 in GL4(𝔽29) generated by

19700
222800
002113
00248
,
21600
23800
00280
0011
,
19700
191000
0010
002828
G:=sub<GL(4,GF(29))| [19,22,0,0,7,28,0,0,0,0,21,24,0,0,13,8],[21,23,0,0,6,8,0,0,0,0,28,1,0,0,0,1],[19,19,0,0,7,10,0,0,0,0,1,28,0,0,0,28] >;

C4⋊D28 in GAP, Magma, Sage, TeX

C_4\rtimes D_{28}
% in TeX

G:=Group("C4:D28");
// GroupNames label

G:=SmallGroup(224,90);
// by ID

G=gap.SmallGroup(224,90);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,103,218,188,50,6917]);
// Polycyclic

G:=Group<a,b,c|a^28=b^4=c^2=1,b*a*b^-1=a^15,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

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