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## G = (D4×C14)⋊C4order 448 = 26·7

### 1st semidirect product of D4×C14 and C4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — (D4×C14)⋊C4
 Chief series C1 — C7 — C14 — C2×C14 — C22×C14 — C22×C28 — C14.C42 — (D4×C14)⋊C4
 Lower central C7 — C2×C14 — C2×C28 — (D4×C14)⋊C4
 Upper central C1 — C22 — C22×C4 — C4⋊D4

Generators and relations for (D4×C14)⋊C4
G = < a,b,c,d | a14=b4=c2=d4=1, ab=ba, ac=ca, dad-1=a-1, cbc=b-1, dbd-1=a7b, dcd-1=bc >

Subgroups: 396 in 90 conjugacy classes, 31 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, C23, C14, C14, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, Dic7, C28, C2×C14, C2×C14, C2.C42, C22⋊C8, C4⋊D4, C7⋊C8, C2×Dic7, C2×C28, C2×C28, C7×D4, C22×C14, C22×C14, C22.SD16, C2×C7⋊C8, C7×C22⋊C4, C7×C4⋊C4, C22×Dic7, C22×C28, D4×C14, D4×C14, C28.55D4, C14.C42, C7×C4⋊D4, (D4×C14)⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, D8, SD16, Dic7, D14, C23⋊C4, D4⋊C4, C4≀C2, C2×Dic7, C7⋊D4, C22.SD16, D4⋊D7, D4.D7, C23.D7, D4⋊Dic7, C23⋊Dic7, D42Dic7, (D4×C14)⋊C4

Smallest permutation representation of (D4×C14)⋊C4
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 46 23 62)(2 47 24 63)(3 48 25 64)(4 49 26 65)(5 50 27 66)(6 51 28 67)(7 52 15 68)(8 53 16 69)(9 54 17 70)(10 55 18 57)(11 56 19 58)(12 43 20 59)(13 44 21 60)(14 45 22 61)(29 99 87 81)(30 100 88 82)(31 101 89 83)(32 102 90 84)(33 103 91 71)(34 104 92 72)(35 105 93 73)(36 106 94 74)(37 107 95 75)(38 108 96 76)(39 109 97 77)(40 110 98 78)(41 111 85 79)(42 112 86 80)
(1 40)(2 41)(3 42)(4 29)(5 30)(6 31)(7 32)(8 33)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 90)(16 91)(17 92)(18 93)(19 94)(20 95)(21 96)(22 97)(23 98)(24 85)(25 86)(26 87)(27 88)(28 89)(43 75)(44 76)(45 77)(46 78)(47 79)(48 80)(49 81)(50 82)(51 83)(52 84)(53 71)(54 72)(55 73)(56 74)(57 105)(58 106)(59 107)(60 108)(61 109)(62 110)(63 111)(64 112)(65 99)(66 100)(67 101)(68 102)(69 103)(70 104)
(1 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 16)(9 15)(10 28)(11 27)(12 26)(13 25)(14 24)(29 82 94 107)(30 81 95 106)(31 80 96 105)(32 79 97 104)(33 78 98 103)(34 77 85 102)(35 76 86 101)(36 75 87 100)(37 74 88 99)(38 73 89 112)(39 72 90 111)(40 71 91 110)(41 84 92 109)(42 83 93 108)(43 58)(44 57)(45 70)(46 69)(47 68)(48 67)(49 66)(50 65)(51 64)(52 63)(53 62)(54 61)(55 60)(56 59)

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

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

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

61 conjugacy classes

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

61 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + - + - + + - image C1 C2 C2 C2 C4 C4 D4 D4 D7 D8 SD16 Dic7 D14 Dic7 C4≀C2 C7⋊D4 C7⋊D4 C23⋊C4 D4⋊D7 D4.D7 C23⋊Dic7 D4⋊2Dic7 kernel (D4×C14)⋊C4 C28.55D4 C14.C42 C7×C4⋊D4 C7×C4⋊C4 D4×C14 C2×C28 C22×C14 C4⋊D4 C2×C14 C2×C14 C4⋊C4 C22×C4 C2×D4 C14 C2×C4 C23 C14 C22 C22 C2 C2 # reps 1 1 1 1 2 2 1 1 3 2 2 3 3 3 4 6 6 1 3 3 6 6

Matrix representation of (D4×C14)⋊C4 in GL6(𝔽113)

 112 0 0 0 0 0 0 112 0 0 0 0 0 0 89 10 0 0 0 0 103 103 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 15 2 0 0 0 0 0 98 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 98 0 0 0 0 0 31 15
,
 69 51 0 0 0 0 95 44 0 0 0 0 0 0 112 0 0 0 0 0 0 112 0 0 0 0 0 0 69 85 0 0 0 0 57 44
,
 15 2 0 0 0 0 1 98 0 0 0 0 0 0 15 0 0 0 0 0 21 98 0 0 0 0 0 0 112 0 0 0 0 0 91 98

G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,89,103,0,0,0,0,10,103,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[15,0,0,0,0,0,2,98,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,98,31,0,0,0,0,0,15],[69,95,0,0,0,0,51,44,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,69,57,0,0,0,0,85,44],[15,1,0,0,0,0,2,98,0,0,0,0,0,0,15,21,0,0,0,0,0,98,0,0,0,0,0,0,112,91,0,0,0,0,0,98] >;

(D4×C14)⋊C4 in GAP, Magma, Sage, TeX

(D_4\times C_{14})\rtimes C_4
% in TeX

G:=Group("(D4xC14):C4");
// GroupNames label

G:=SmallGroup(448,94);
// by ID

G=gap.SmallGroup(448,94);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,219,1571,570,136,18822]);
// Polycyclic

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

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