Copied to
clipboard

## G = C14×C4○D4order 224 = 25·7

### Direct product of C14 and C4○D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C14×C4○D4
 Chief series C1 — C2 — C14 — C2×C14 — C7×D4 — C7×C4○D4 — C14×C4○D4
 Lower central C1 — C2 — C14×C4○D4
 Upper central C1 — C2×C28 — C14×C4○D4

Generators and relations for C14×C4○D4
G = < a,b,c,d | a14=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >

Subgroups: 188 in 164 conjugacy classes, 140 normal (12 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C14, C14, C14, C22×C4, C2×D4, C2×Q8, C4○D4, C28, C2×C14, C2×C14, C2×C14, C2×C4○D4, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C22×C28, D4×C14, Q8×C14, C7×C4○D4, C14×C4○D4
Quotients: C1, C2, C22, C7, C23, C14, C4○D4, C24, C2×C14, C2×C4○D4, C22×C14, C7×C4○D4, C23×C14, C14×C4○D4

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

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

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

140 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 4A 4B 4C 4D 4E ··· 4J 7A ··· 7F 14A ··· 14R 14S ··· 14BB 28A ··· 28X 28Y ··· 28BH order 1 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 7 ··· 7 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

140 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 type + + + + + image C1 C2 C2 C2 C2 C7 C14 C14 C14 C14 C4○D4 C7×C4○D4 kernel C14×C4○D4 C22×C28 D4×C14 Q8×C14 C7×C4○D4 C2×C4○D4 C22×C4 C2×D4 C2×Q8 C4○D4 C14 C2 # reps 1 3 3 1 8 6 18 18 6 48 4 24

Matrix representation of C14×C4○D4 in GL3(𝔽29) generated by

 28 0 0 0 20 0 0 0 20
,
 1 0 0 0 17 0 0 0 17
,
 28 0 0 0 27 1 0 24 2
,
 28 0 0 0 2 28 0 3 27
G:=sub<GL(3,GF(29))| [28,0,0,0,20,0,0,0,20],[1,0,0,0,17,0,0,0,17],[28,0,0,0,27,24,0,1,2],[28,0,0,0,2,3,0,28,27] >;

C14×C4○D4 in GAP, Magma, Sage, TeX

C_{14}\times C_4\circ D_4
% in TeX

G:=Group("C14xC4oD4");
// GroupNames label

G:=SmallGroup(224,192);
// by ID

G=gap.SmallGroup(224,192);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-7,-2,1369,518]);
// Polycyclic

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

׿
×
𝔽