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## G = (D4×C14)⋊9C4order 448 = 26·7

### 5th semidirect product of D4×C14 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — (D4×C14)⋊9C4
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4.Dic7 — C2×C4.Dic7 — (D4×C14)⋊9C4
 Lower central C7 — C14 — C28 — (D4×C14)⋊9C4
 Upper central C1 — C4 — C22×C4 — C2×C4○D4

Generators and relations for (D4×C14)⋊9C4
G = < a,b,c,d | a14=b4=c2=d4=1, ab=ba, ac=ca, dad-1=a-1b2, cbc=b-1, bd=db, dcd-1=bc >

Subgroups: 468 in 154 conjugacy classes, 67 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic7, C28, C28, C2×C14, C2×C14, C4≀C2, C42⋊C2, C2×M4(2), C2×C4○D4, C7⋊C8, C2×Dic7, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C22×C14, C22×C14, C42⋊C22, C2×C7⋊C8, C4.Dic7, C4.Dic7, C4×Dic7, C4⋊Dic7, C23.D7, C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C7×C4○D4, C7×C4○D4, D42Dic7, C2×C4.Dic7, C23.21D14, C14×C4○D4, (D4×C14)⋊9C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, Dic7, D14, C2×C22⋊C4, C2×Dic7, C7⋊D4, C22×D7, C42⋊C22, C23.D7, C22×Dic7, C2×C7⋊D4, C2×C23.D7, (D4×C14)⋊9C4

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

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

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

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

82 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 ··· 14I 14J ··· 14AA 28A ··· 28L 28M ··· 28AD 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 28 ··· 28 28 ··· 28 size 1 1 2 2 2 4 4 1 1 2 2 2 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 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + - - - + image C1 C2 C2 C2 C2 C4 C4 C4 D4 D4 D7 D14 Dic7 Dic7 Dic7 D14 C7⋊D4 C7⋊D4 C42⋊C22 (D4×C14)⋊9C4 kernel (D4×C14)⋊9C4 D4⋊2Dic7 C2×C4.Dic7 C23.21D14 C14×C4○D4 D4×C14 Q8×C14 C7×C4○D4 C2×C28 C22×C14 C2×C4○D4 C22×C4 C2×D4 C2×Q8 C4○D4 C4○D4 C2×C4 C23 C7 C1 # reps 1 4 1 1 1 2 2 4 3 1 3 3 3 3 6 6 18 6 2 12

Matrix representation of (D4×C14)⋊9C4 in GL6(𝔽113)

 104 80 0 0 0 0 33 33 0 0 0 0 0 0 1 0 106 0 0 0 108 0 74 112 0 0 0 0 112 0 0 0 108 112 74 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 15 0 0 0 0 0 38 98 0 0 0 0 0 0 15 0 0 0 38 0 0 98
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 93 0 0 105 0 0 36 0 98 20 0 0 75 15 0 15 0 0 64 0 0 20
,
 98 0 0 0 0 0 91 15 0 0 0 0 0 0 1 0 0 0 0 0 35 15 0 0 0 0 81 0 112 0 0 0 73 0 0 98

G:=sub<GL(6,GF(113))| [104,33,0,0,0,0,80,33,0,0,0,0,0,0,1,108,0,108,0,0,0,0,0,112,0,0,106,74,112,74,0,0,0,112,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,38,0,38,0,0,0,98,0,0,0,0,0,0,15,0,0,0,0,0,0,98],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,93,36,75,64,0,0,0,0,15,0,0,0,0,98,0,0,0,0,105,20,15,20],[98,91,0,0,0,0,0,15,0,0,0,0,0,0,1,35,81,73,0,0,0,15,0,0,0,0,0,0,112,0,0,0,0,0,0,98] >;

(D4×C14)⋊9C4 in GAP, Magma, Sage, TeX

(D_4\times C_{14})\rtimes_9C_4
% in TeX

G:=Group("(D4xC14):9C4");
// GroupNames label

G:=SmallGroup(448,770);
// by ID

G=gap.SmallGroup(448,770);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,422,387,136,1684,438,102,18822]);
// 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*b^2,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b*c>;
// generators/relations

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