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## G = C2×C4○D28order 224 = 25·7

### Direct product of C2 and C4○D28

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C2×C4○D28
 Chief series C1 — C7 — C14 — D14 — C22×D7 — C2×C4×D7 — C2×C4○D28
 Lower central C7 — C14 — C2×C4○D28
 Upper central C1 — C2×C4 — C22×C4

Generators and relations for C2×C4○D28
G = < a,b,c,d | a2=b4=d2=1, c14=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c13 >

Subgroups: 590 in 164 conjugacy classes, 89 normal (17 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C14, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C4○D4, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, C2×Dic14, C2×C4×D7, C2×D28, C4○D28, C2×C7⋊D4, C22×C28, C2×C4○D28
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, C22×D7, C4○D28, C23×D7, C2×C4○D28

Smallest permutation representation of C2×C4○D28
On 112 points
Generators in S112
(1 47)(2 48)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 55)(10 56)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(25 43)(26 44)(27 45)(28 46)(57 93)(58 94)(59 95)(60 96)(61 97)(62 98)(63 99)(64 100)(65 101)(66 102)(67 103)(68 104)(69 105)(70 106)(71 107)(72 108)(73 109)(74 110)(75 111)(76 112)(77 85)(78 86)(79 87)(80 88)(81 89)(82 90)(83 91)(84 92)
(1 97 15 111)(2 98 16 112)(3 99 17 85)(4 100 18 86)(5 101 19 87)(6 102 20 88)(7 103 21 89)(8 104 22 90)(9 105 23 91)(10 106 24 92)(11 107 25 93)(12 108 26 94)(13 109 27 95)(14 110 28 96)(29 71 43 57)(30 72 44 58)(31 73 45 59)(32 74 46 60)(33 75 47 61)(34 76 48 62)(35 77 49 63)(36 78 50 64)(37 79 51 65)(38 80 52 66)(39 81 53 67)(40 82 54 68)(41 83 55 69)(42 84 56 70)
(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 46)(2 45)(3 44)(4 43)(5 42)(6 41)(7 40)(8 39)(9 38)(10 37)(11 36)(12 35)(13 34)(14 33)(15 32)(16 31)(17 30)(18 29)(19 56)(20 55)(21 54)(22 53)(23 52)(24 51)(25 50)(26 49)(27 48)(28 47)(57 100)(58 99)(59 98)(60 97)(61 96)(62 95)(63 94)(64 93)(65 92)(66 91)(67 90)(68 89)(69 88)(70 87)(71 86)(72 85)(73 112)(74 111)(75 110)(76 109)(77 108)(78 107)(79 106)(80 105)(81 104)(82 103)(83 102)(84 101)

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

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

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

68 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 7A 7B 7C 14A ··· 14U 28A ··· 28X order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 7 7 7 14 ··· 14 28 ··· 28 size 1 1 1 1 2 2 14 14 14 14 1 1 1 1 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 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D7 C4○D4 D14 D14 C4○D28 kernel C2×C4○D28 C2×Dic14 C2×C4×D7 C2×D28 C4○D28 C2×C7⋊D4 C22×C28 C22×C4 C14 C2×C4 C23 C2 # reps 1 1 2 1 8 2 1 3 4 18 3 24

Matrix representation of C2×C4○D28 in GL3(𝔽29) generated by

 28 0 0 0 28 0 0 0 28
,
 1 0 0 0 12 0 0 0 12
,
 1 0 0 0 25 5 0 12 21
,
 1 0 0 0 27 16 0 27 2
G:=sub<GL(3,GF(29))| [28,0,0,0,28,0,0,0,28],[1,0,0,0,12,0,0,0,12],[1,0,0,0,25,12,0,5,21],[1,0,0,0,27,27,0,16,2] >;

C2×C4○D28 in GAP, Magma, Sage, TeX

C_2\times C_4\circ D_{28}
% in TeX

G:=Group("C2xC4oD28");
// GroupNames label

G:=SmallGroup(224,177);
// by ID

G=gap.SmallGroup(224,177);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,86,579,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=d^2=1,c^14=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^13>;
// generators/relations

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