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## G = C4×D28order 224 = 25·7

### Direct product of C4 and D28

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C4×D28
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C2×D28 — C4×D28
 Lower central C7 — C14 — C4×D28
 Upper central C1 — C2×C4 — C42

Generators and relations for C4×D28
G = < a,b,c | a4=b28=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 406 in 94 conjugacy classes, 45 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, D4, C23, D7, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic7, C28, C28, D14, D14, C2×C14, C4×D4, C4×D7, D28, C2×Dic7, C2×C28, C22×D7, C4⋊Dic7, D14⋊C4, C4×C28, C2×C4×D7, C2×D28, C4×D28
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22×C4, C2×D4, C4○D4, D14, C4×D4, C4×D7, D28, C22×D7, C2×C4×D7, C2×D28, C4○D28, C4×D28

Smallest permutation representation of C4×D28
On 112 points
Generators in S112
(1 66 51 95)(2 67 52 96)(3 68 53 97)(4 69 54 98)(5 70 55 99)(6 71 56 100)(7 72 29 101)(8 73 30 102)(9 74 31 103)(10 75 32 104)(11 76 33 105)(12 77 34 106)(13 78 35 107)(14 79 36 108)(15 80 37 109)(16 81 38 110)(17 82 39 111)(18 83 40 112)(19 84 41 85)(20 57 42 86)(21 58 43 87)(22 59 44 88)(23 60 45 89)(24 61 46 90)(25 62 47 91)(26 63 48 92)(27 64 49 93)(28 65 50 94)
(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 7)(2 6)(3 5)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)(29 51)(30 50)(31 49)(32 48)(33 47)(34 46)(35 45)(36 44)(37 43)(38 42)(39 41)(52 56)(53 55)(57 81)(58 80)(59 79)(60 78)(61 77)(62 76)(63 75)(64 74)(65 73)(66 72)(67 71)(68 70)(82 84)(85 111)(86 110)(87 109)(88 108)(89 107)(90 106)(91 105)(92 104)(93 103)(94 102)(95 101)(96 100)(97 99)

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

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

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

68 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 7A 7B 7C 14A ··· 14I 28A ··· 28AJ order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 7 7 7 14 ··· 14 28 ··· 28 size 1 1 1 1 14 14 14 14 1 1 1 1 2 2 2 2 14 14 14 14 2 2 2 2 ··· 2 2 ··· 2

68 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 D4 D7 C4○D4 D14 C4×D7 D28 C4○D28 kernel C4×D28 C4⋊Dic7 D14⋊C4 C4×C28 C2×C4×D7 C2×D28 D28 C28 C42 C14 C2×C4 C4 C4 C2 # reps 1 1 2 1 2 1 8 2 3 2 9 12 12 12

Matrix representation of C4×D28 in GL3(𝔽29) generated by

 12 0 0 0 12 0 0 0 12
,
 28 0 0 0 2 24 0 5 17
,
 28 0 0 0 21 8 0 3 8
G:=sub<GL(3,GF(29))| [12,0,0,0,12,0,0,0,12],[28,0,0,0,2,5,0,24,17],[28,0,0,0,21,3,0,8,8] >;

C4×D28 in GAP, Magma, Sage, TeX

C_4\times D_{28}
% in TeX

G:=Group("C4xD28");
// GroupNames label

G:=SmallGroup(224,68);
// by ID

G=gap.SmallGroup(224,68);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,217,103,50,6917]);
// Polycyclic

G:=Group<a,b,c|a^4=b^28=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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