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## G = C42⋊17D14order 448 = 26·7

### 17th semidirect product of C42 and D14 acting via D14/C7=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C42⋊17D14
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C23×D7 — C2×D14⋊C4 — C42⋊17D14
 Lower central C7 — C2×C14 — C42⋊17D14
 Upper central C1 — C22 — C4×D4

Generators and relations for C4217D14
G = < a,b,c,d | a4=b4=c14=d2=1, ab=ba, cac-1=a-1, dad=a-1b2, bc=cb, dbd=a2b, dcd=c-1 >

Subgroups: 1364 in 250 conjugacy classes, 95 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, Dic7, C28, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C4×D4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C422C2, Dic14, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C22×D7, C22×D7, C22×C14, C22.32C24, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C2×C4×D7, C2×D28, C22×Dic7, C2×C7⋊D4, C22×C28, D4×C14, C23×D7, C4.D28, C422D7, C22⋊Dic14, C22⋊D28, D14⋊D4, Dic7.D4, D14.5D4, C4⋊C4⋊D7, C2×D14⋊C4, C4×C7⋊D4, C23.23D14, C287D4, C23⋊D14, Dic7⋊D4, D4×C28, C4217D14
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, 2+ 1+4, C22×D7, C22.32C24, C4○D28, C23×D7, C2×C4○D28, D46D14, D48D14, C4217D14

Smallest permutation representation of C4217D14
On 112 points
Generators in S112
(1 88 58 43)(2 44 59 89)(3 90 60 45)(4 46 61 91)(5 92 62 47)(6 48 63 93)(7 94 64 49)(8 50 65 95)(9 96 66 51)(10 52 67 97)(11 98 68 53)(12 54 69 85)(13 86 70 55)(14 56 57 87)(15 99 40 84)(16 71 41 100)(17 101 42 72)(18 73 29 102)(19 103 30 74)(20 75 31 104)(21 105 32 76)(22 77 33 106)(23 107 34 78)(24 79 35 108)(25 109 36 80)(26 81 37 110)(27 111 38 82)(28 83 39 112)
(1 24 8 17)(2 25 9 18)(3 26 10 19)(4 27 11 20)(5 28 12 21)(6 15 13 22)(7 16 14 23)(29 59 36 66)(30 60 37 67)(31 61 38 68)(32 62 39 69)(33 63 40 70)(34 64 41 57)(35 65 42 58)(43 108 95 72)(44 109 96 73)(45 110 97 74)(46 111 98 75)(47 112 85 76)(48 99 86 77)(49 100 87 78)(50 101 88 79)(51 102 89 80)(52 103 90 81)(53 104 91 82)(54 105 92 83)(55 106 93 84)(56 107 94 71)
(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 14)(2 13)(3 12)(4 11)(5 10)(6 9)(7 8)(15 29)(16 42)(17 41)(18 40)(19 39)(20 38)(21 37)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)(28 30)(43 94)(44 93)(45 92)(46 91)(47 90)(48 89)(49 88)(50 87)(51 86)(52 85)(53 98)(54 97)(55 96)(56 95)(57 58)(59 70)(60 69)(61 68)(62 67)(63 66)(64 65)(71 79)(72 78)(73 77)(74 76)(80 84)(81 83)(99 109)(100 108)(101 107)(102 106)(103 105)(110 112)

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

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

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

82 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H ··· 4L 7A 7B 7C 14A ··· 14I 14J ··· 14U 28A ··· 28L 28M ··· 28AJ order 1 2 2 2 2 2 2 2 2 2 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 2 2 4 28 28 28 2 2 2 2 4 4 4 28 ··· 28 2 2 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

82 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D7 C4○D4 D14 D14 D14 D14 D14 C4○D28 2+ 1+4 D4⋊6D14 D4⋊8D14 kernel C42⋊17D14 C4.D28 C42⋊2D7 C22⋊Dic14 C22⋊D28 D14⋊D4 Dic7.D4 D14.5D4 C4⋊C4⋊D7 C2×D14⋊C4 C4×C7⋊D4 C23.23D14 C28⋊7D4 C23⋊D14 Dic7⋊D4 D4×C28 C4×D4 C2×C14 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C22 C14 C2 C2 # reps 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 4 3 6 3 6 3 24 2 6 6

Matrix representation of C4217D14 in GL6(𝔽29)

 11 24 0 0 0 0 24 18 0 0 0 0 0 0 9 16 0 4 0 0 24 20 25 11 0 0 24 15 24 13 0 0 14 0 16 5
,
 17 0 0 0 0 0 0 17 0 0 0 0 0 0 19 6 0 0 0 0 17 10 0 0 0 0 20 0 8 6 0 0 26 9 23 21
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 20 19 0 0 0 0 20 6 0 0 0 0 13 26 10 10 0 0 6 12 19 22
,
 28 13 0 0 0 0 0 1 0 0 0 0 0 0 22 22 0 0 0 0 11 7 0 0 0 0 7 17 19 19 0 0 1 19 7 10

G:=sub<GL(6,GF(29))| [11,24,0,0,0,0,24,18,0,0,0,0,0,0,9,24,24,14,0,0,16,20,15,0,0,0,0,25,24,16,0,0,4,11,13,5],[17,0,0,0,0,0,0,17,0,0,0,0,0,0,19,17,20,26,0,0,6,10,0,9,0,0,0,0,8,23,0,0,0,0,6,21],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,20,20,13,6,0,0,19,6,26,12,0,0,0,0,10,19,0,0,0,0,10,22],[28,0,0,0,0,0,13,1,0,0,0,0,0,0,22,11,7,1,0,0,22,7,17,19,0,0,0,0,19,7,0,0,0,0,19,10] >;

C4217D14 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{17}D_{14}
% in TeX

G:=Group("C4^2:17D14");
// GroupNames label

G:=SmallGroup(448,1013);
// by ID

G=gap.SmallGroup(448,1013);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,100,675,570,18822]);
// 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,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations

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