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## G = C14×SD16order 224 = 25·7

### Direct product of C14 and SD16

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C14×SD16
 Chief series C1 — C2 — C4 — C28 — C7×Q8 — C7×SD16 — C14×SD16
 Lower central C1 — C2 — C4 — C14×SD16
 Upper central C1 — C2×C14 — C2×C28 — C14×SD16

Generators and relations for C14×SD16
G = < a,b,c | a14=b8=c2=1, ab=ba, ac=ca, cbc=b3 >

Subgroups: 108 in 68 conjugacy classes, 44 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C14, C14, C14, C2×C8, SD16, C2×D4, C2×Q8, C28, C28, C2×C14, C2×C14, C2×SD16, C56, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C22×C14, C2×C56, C7×SD16, D4×C14, Q8×C14, C14×SD16
Quotients: C1, C2, C22, C7, D4, C23, C14, SD16, C2×D4, C2×C14, C2×SD16, C7×D4, C22×C14, C7×SD16, D4×C14, C14×SD16

Smallest permutation representation of C14×SD16
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)
(1 108 96 28 47 61 32 75)(2 109 97 15 48 62 33 76)(3 110 98 16 49 63 34 77)(4 111 85 17 50 64 35 78)(5 112 86 18 51 65 36 79)(6 99 87 19 52 66 37 80)(7 100 88 20 53 67 38 81)(8 101 89 21 54 68 39 82)(9 102 90 22 55 69 40 83)(10 103 91 23 56 70 41 84)(11 104 92 24 43 57 42 71)(12 105 93 25 44 58 29 72)(13 106 94 26 45 59 30 73)(14 107 95 27 46 60 31 74)
(15 109)(16 110)(17 111)(18 112)(19 99)(20 100)(21 101)(22 102)(23 103)(24 104)(25 105)(26 106)(27 107)(28 108)(29 93)(30 94)(31 95)(32 96)(33 97)(34 98)(35 85)(36 86)(37 87)(38 88)(39 89)(40 90)(41 91)(42 92)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)

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

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

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

98 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 7A ··· 7F 8A 8B 8C 8D 14A ··· 14R 14S ··· 14AD 28A ··· 28L 28M ··· 28X 56A ··· 56X order 1 2 2 2 2 2 4 4 4 4 7 ··· 7 8 8 8 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 1 1 4 4 2 2 4 4 1 ··· 1 2 2 2 2 1 ··· 1 4 ··· 4 2 ··· 2 4 ··· 4 2 ··· 2

98 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C2 C7 C14 C14 C14 C14 D4 D4 SD16 C7×D4 C7×D4 C7×SD16 kernel C14×SD16 C2×C56 C7×SD16 D4×C14 Q8×C14 C2×SD16 C2×C8 SD16 C2×D4 C2×Q8 C28 C2×C14 C14 C4 C22 C2 # reps 1 1 4 1 1 6 6 24 6 6 1 1 4 6 6 24

Matrix representation of C14×SD16 in GL4(𝔽113) generated by

 4 0 0 0 0 4 0 0 0 0 109 0 0 0 0 109
,
 0 112 0 0 1 0 0 0 0 0 87 87 0 0 13 0
,
 1 0 0 0 0 112 0 0 0 0 1 0 0 0 112 112
G:=sub<GL(4,GF(113))| [4,0,0,0,0,4,0,0,0,0,109,0,0,0,0,109],[0,1,0,0,112,0,0,0,0,0,87,13,0,0,87,0],[1,0,0,0,0,112,0,0,0,0,1,112,0,0,0,112] >;

C14×SD16 in GAP, Magma, Sage, TeX

C_{14}\times {\rm SD}_{16}
% in TeX

G:=Group("C14xSD16");
// GroupNames label

G:=SmallGroup(224,168);
// by ID

G=gap.SmallGroup(224,168);
# by ID

G:=PCGroup([6,-2,-2,-2,-7,-2,-2,672,697,5044,2530,88]);
// Polycyclic

G:=Group<a,b,c|a^14=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^3>;
// generators/relations

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