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## G = (D4×C14).16C4order 448 = 26·7

### 10th non-split extension by D4×C14 of C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — (D4×C14).16C4
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4.Dic7 — C2×C4.Dic7 — (D4×C14).16C4
 Lower central C7 — C14 — C2×C14 — (D4×C14).16C4
 Upper central C1 — C4 — C22×C4 — C2×C4○D4

Generators and relations for (D4×C14).16C4
G = < a,b,c,d | a14=b4=c2=1, d4=b2, ab=ba, ac=ca, dad-1=a-1b2, cbc=b-1, dbd-1=a7b, dcd-1=b2c >

Subgroups: 404 in 150 conjugacy classes, 67 normal (25 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C14, C14, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C28, C28, C28, C2×C14, C2×C14, C2×C14, C4.D4, C4.10D4, C2×M4(2), C2×C4○D4, C7⋊C8, C2×C28, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C22×C14, M4(2).8C22, C2×C7⋊C8, C4.Dic7, C4.Dic7, C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C7×C4○D4, C28.D4, C28.10D4, C2×C4.Dic7, C14×C4○D4, (D4×C14).16C4
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, M4(2).8C22, C23.D7, C22×Dic7, C2×C7⋊D4, C2×C23.D7, (D4×C14).16C4

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

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

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

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

82 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 7A 7B 7C 8A ··· 8H 14A ··· 14I 14J ··· 14AA 28A ··· 28L 28M ··· 28AD order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 7 7 7 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 2 2 2 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 4 4 type + + + + + + + - + - + + image C1 C2 C2 C2 C2 C4 C4 D4 D7 Dic7 D14 Dic7 D14 D14 C7⋊D4 M4(2).8C22 (D4×C14).16C4 kernel (D4×C14).16C4 C28.D4 C28.10D4 C2×C4.Dic7 C14×C4○D4 C22×C28 D4×C14 C2×C28 C2×C4○D4 C22×C4 C22×C4 C2×D4 C2×D4 C2×Q8 C2×C4 C7 C1 # reps 1 2 2 2 1 4 4 4 3 6 3 6 3 3 24 2 12

Matrix representation of (D4×C14).16C4 in GL4(𝔽113) generated by

 85 0 0 0 0 85 0 0 0 0 109 0 24 89 0 109
,
 0 98 0 0 98 0 0 0 1 112 98 1 0 0 0 15
,
 0 98 0 0 15 0 0 0 112 1 15 112 0 30 111 98
,
 0 0 1 0 15 98 1 15 0 1 0 0 30 0 0 15
G:=sub<GL(4,GF(113))| [85,0,0,24,0,85,0,89,0,0,109,0,0,0,0,109],[0,98,1,0,98,0,112,0,0,0,98,0,0,0,1,15],[0,15,112,0,98,0,1,30,0,0,15,111,0,0,112,98],[0,15,0,30,0,98,1,0,1,1,0,0,0,15,0,15] >;

(D4×C14).16C4 in GAP, Magma, Sage, TeX

(D_4\times C_{14})._{16}C_4
% in TeX

G:=Group("(D4xC14).16C4");
// GroupNames label

G:=SmallGroup(448,771);
// by ID

G=gap.SmallGroup(448,771);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,422,387,297,136,1684,18822]);
// Polycyclic

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

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