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## G = C42⋊D14order 448 = 26·7

### 1st semidirect product of C42 and D14 acting via D14/C7=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C42⋊D14
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×C4×D7 — D7×C4○D4 — C42⋊D14
 Lower central C7 — C14 — C28 — C42⋊D14
 Upper central C1 — C4 — C2×C4 — C4≀C2

Generators and relations for C42⋊D14
G = < a,b,c,d | a4=b4=c14=d2=1, cac-1=ab=ba, dad=ab-1, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 828 in 154 conjugacy classes, 51 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D7, C14, C14, C42, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C4≀C2, C4≀C2, C42⋊C2, C2×M4(2), C2×C4○D4, C7⋊C8, C56, Dic14, Dic14, C4×D7, C4×D7, D28, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×D7, C22×D7, C42⋊C22, C8×D7, C8⋊D7, C4.Dic7, C4×Dic7, Dic7⋊C4, D14⋊C4, C4×C28, C7×M4(2), C2×C4×D7, C2×C4×D7, C4○D28, C4○D28, D4×D7, D4×D7, D42D7, D42D7, Q8×D7, Q82D7, C7×C4○D4, Dic14⋊C4, D284C4, D42Dic7, C7×C4≀C2, C42⋊D7, D7×M4(2), D7×C4○D4, C42⋊D14
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, D14, C2×C22⋊C4, C4×D7, C22×D7, C42⋊C22, C2×C4×D7, D4×D7, D7×C22⋊C4, C42⋊D14

Smallest permutation representation of C42⋊D14
On 112 points
Generators in S112
(1 83 108 85)(2 109)(3 71 110 87)(4 111)(5 73 112 89)(6 99)(7 75 100 91)(8 101)(9 77 102 93)(10 103)(11 79 104 95)(12 105)(13 81 106 97)(14 107)(16 35 43 69)(18 37 45 57)(20 39 47 59)(22 41 49 61)(24 29 51 63)(26 31 53 65)(28 33 55 67)(72 88)(74 90)(76 92)(78 94)(80 96)(82 98)(84 86)
(1 83 108 85)(2 86 109 84)(3 71 110 87)(4 88 111 72)(5 73 112 89)(6 90 99 74)(7 75 100 91)(8 92 101 76)(9 77 102 93)(10 94 103 78)(11 79 104 95)(12 96 105 80)(13 81 106 97)(14 98 107 82)(15 34 56 68)(16 69 43 35)(17 36 44 70)(18 57 45 37)(19 38 46 58)(20 59 47 39)(21 40 48 60)(22 61 49 41)(23 42 50 62)(24 63 51 29)(25 30 52 64)(26 65 53 31)(27 32 54 66)(28 67 55 33)
(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 60)(2 59)(3 58)(4 57)(5 70)(6 69)(7 68)(8 67)(9 66)(10 65)(11 64)(12 63)(13 62)(14 61)(15 91)(16 90)(17 89)(18 88)(19 87)(20 86)(21 85)(22 98)(23 97)(24 96)(25 95)(26 94)(27 93)(28 92)(29 105)(30 104)(31 103)(32 102)(33 101)(34 100)(35 99)(36 112)(37 111)(38 110)(39 109)(40 108)(41 107)(42 106)(43 74)(44 73)(45 72)(46 71)(47 84)(48 83)(49 82)(50 81)(51 80)(52 79)(53 78)(54 77)(55 76)(56 75)

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

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

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

64 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 7A 7B 7C 8A 8B 8C 8D 14A 14B 14C 14D 14E 14F 14G 14H 14I 28A ··· 28F 28G ··· 28U 28V 28W 28X 56A ··· 56F order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 7 7 7 8 8 8 8 14 14 14 14 14 14 14 14 14 28 ··· 28 28 ··· 28 28 28 28 56 ··· 56 size 1 1 2 4 14 14 28 1 1 2 4 4 4 14 14 28 28 28 2 2 2 4 4 28 28 2 2 2 4 4 4 8 8 8 2 ··· 2 4 ··· 4 8 8 8 8 ··· 8

64 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 D4 D4 D4 D7 D14 D14 D14 C4×D7 C4×D7 C42⋊C22 D4×D7 D4×D7 C42⋊D14 kernel C42⋊D14 Dic14⋊C4 D28⋊4C4 D4⋊2Dic7 C7×C4≀C2 C42⋊D7 D7×M4(2) D7×C4○D4 D4×D7 D4⋊2D7 Q8×D7 Q8⋊2D7 C4×D7 C2×Dic7 C22×D7 C4≀C2 C42 M4(2) C4○D4 D4 Q8 C7 C4 C22 C1 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 1 1 3 3 3 3 6 6 2 3 3 12

Matrix representation of C42⋊D14 in GL4(𝔽113) generated by

 98 0 0 0 19 112 0 0 72 16 15 0 32 43 58 1
,
 98 0 0 0 73 15 0 0 40 83 98 0 36 21 21 15
,
 99 67 0 0 38 14 0 0 27 8 8 76 26 102 34 105
,
 92 41 34 97 63 47 0 54 22 17 40 100 45 18 111 47
G:=sub<GL(4,GF(113))| [98,19,72,32,0,112,16,43,0,0,15,58,0,0,0,1],[98,73,40,36,0,15,83,21,0,0,98,21,0,0,0,15],[99,38,27,26,67,14,8,102,0,0,8,34,0,0,76,105],[92,63,22,45,41,47,17,18,34,0,40,111,97,54,100,47] >;

C42⋊D14 in GAP, Magma, Sage, TeX

C_4^2\rtimes D_{14}
% in TeX

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

G:=SmallGroup(448,355);
// by ID

G=gap.SmallGroup(448,355);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,219,58,136,851,438,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^14=d^2=1,c*a*c^-1=a*b=b*a,d*a*d=a*b^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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