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## G = C42⋊Dic7order 448 = 26·7

### 1st semidirect product of C42 and Dic7 acting via Dic7/C7=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C42⋊Dic7
 Chief series C1 — C7 — C14 — C2×C14 — C22×C14 — C22×C28 — C23.21D14 — C42⋊Dic7
 Lower central C7 — C28 — C42⋊Dic7
 Upper central C1 — C4 — C42⋊C2

Generators and relations for C42⋊Dic7
G = < a,b,c,d | a4=b4=c14=1, d2=c7, ab=ba, cac-1=ab2, dad-1=ab-1, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 340 in 94 conjugacy classes, 47 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C23, C14, C14, C42, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, Dic7, C28, C28, C2×C14, C2×C14, C42⋊C2, C42⋊C2, C2×M4(2), C7⋊C8, C2×Dic7, C2×C28, C2×C28, C22×C14, C4.9C42, C2×C7⋊C8, C4.Dic7, C4×Dic7, C4⋊Dic7, C23.D7, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C22×C28, C2×C4.Dic7, C23.21D14, C7×C42⋊C2, C42⋊Dic7
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D7, C42, C22⋊C4, C4⋊C4, Dic7, D14, C2.C42, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C4.9C42, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C14.C42, C42⋊Dic7

Smallest permutation representation of C42⋊Dic7
On 112 points
Generators in S112
(1 63 42 107)(2 57 36 101)(3 65 37 109)(4 59 38 103)(5 67 39 111)(6 61 40 105)(7 69 41 99)(8 75 54 94)(9 83 55 88)(10 77 56 96)(11 71 50 90)(12 79 51 98)(13 73 52 92)(14 81 53 86)(15 87 30 82)(16 95 31 76)(17 89 32 84)(18 97 33 78)(19 91 34 72)(20 85 35 80)(21 93 29 74)(22 60 45 104)(23 68 46 112)(24 62 47 106)(25 70 48 100)(26 64 49 108)(27 58 43 102)(28 66 44 110)
(1 9 25 31)(2 10 26 32)(3 11 27 33)(4 12 28 34)(5 13 22 35)(6 14 23 29)(7 8 24 30)(15 41 54 47)(16 42 55 48)(17 36 56 49)(18 37 50 43)(19 38 51 44)(20 39 52 45)(21 40 53 46)(57 77 64 84)(58 78 65 71)(59 79 66 72)(60 80 67 73)(61 81 68 74)(62 82 69 75)(63 83 70 76)(85 111 92 104)(86 112 93 105)(87 99 94 106)(88 100 95 107)(89 101 96 108)(90 102 97 109)(91 103 98 110)
(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 56)(2 55)(3 54)(4 53)(5 52)(6 51)(7 50)(8 43)(9 49)(10 48)(11 47)(12 46)(13 45)(14 44)(15 27)(16 26)(17 25)(18 24)(19 23)(20 22)(21 28)(29 38)(30 37)(31 36)(32 42)(33 41)(34 40)(35 39)(57 100 64 107)(58 99 65 106)(59 112 66 105)(60 111 67 104)(61 110 68 103)(62 109 69 102)(63 108 70 101)(71 87 78 94)(72 86 79 93)(73 85 80 92)(74 98 81 91)(75 97 82 90)(76 96 83 89)(77 95 84 88)

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

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

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

82 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 7A 7B 7C 8A 8B 8C 8D 14A ··· 14I 14J ··· 14O 28A ··· 28L 28M ··· 28AP order 1 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 4 7 7 7 8 8 8 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 2 2 2 1 1 2 2 2 4 4 4 4 28 28 28 28 2 2 2 28 28 28 28 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

82 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + - + + - + - + image C1 C2 C2 C2 C4 C4 C4 D4 Q8 D4 D7 Dic7 D14 Dic14 C4×D7 D28 C7⋊D4 C7⋊D4 C4.9C42 C42⋊Dic7 kernel C42⋊Dic7 C2×C4.Dic7 C23.21D14 C7×C42⋊C2 C2×C7⋊C8 C4×Dic7 C4×C28 C2×C28 C2×C28 C22×C14 C42⋊C2 C42 C22×C4 C2×C4 C2×C4 C2×C4 C2×C4 C23 C7 C1 # reps 1 1 1 1 4 4 4 2 1 1 3 6 3 6 12 6 6 6 2 12

Matrix representation of C42⋊Dic7 in GL6(𝔽113)

 96 46 0 0 0 0 33 17 0 0 0 0 0 0 0 0 1 0 0 0 8 103 0 36 0 0 112 69 0 0 0 0 65 82 50 10
,
 112 0 0 0 0 0 0 112 0 0 0 0 0 0 98 0 0 0 0 0 0 98 0 0 0 0 0 0 98 0 0 0 0 0 0 98
,
 10 112 0 0 0 0 100 24 0 0 0 0 0 0 112 0 0 0 0 0 0 112 0 0 0 0 0 0 1 0 0 0 13 12 0 1
,
 15 36 0 0 0 0 0 98 0 0 0 0 0 0 1 44 0 0 0 0 77 112 0 0 0 0 82 67 15 30 0 0 112 82 98 98

G:=sub<GL(6,GF(113))| [96,33,0,0,0,0,46,17,0,0,0,0,0,0,0,8,112,65,0,0,0,103,69,82,0,0,1,0,0,50,0,0,0,36,0,10],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,98,0,0,0,0,0,0,98,0,0,0,0,0,0,98,0,0,0,0,0,0,98],[10,100,0,0,0,0,112,24,0,0,0,0,0,0,112,0,0,13,0,0,0,112,0,12,0,0,0,0,1,0,0,0,0,0,0,1],[15,0,0,0,0,0,36,98,0,0,0,0,0,0,1,77,82,112,0,0,44,112,67,82,0,0,0,0,15,98,0,0,0,0,30,98] >;

C42⋊Dic7 in GAP, Magma, Sage, TeX

C_4^2\rtimes {\rm Dic}_7
% in TeX

G:=Group("C4^2:Dic7");
// GroupNames label

G:=SmallGroup(448,88);
// by ID

G=gap.SmallGroup(448,88);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,253,64,387,1123,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^14=1,d^2=c^7,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a*b^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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