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## G = C2×D28⋊4C4order 448 = 26·7

### Direct product of C2 and D28⋊4C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C2×D28⋊4C4
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4○D28 — C2×C4○D28 — C2×D28⋊4C4
 Lower central C7 — C14 — C28 — C2×D28⋊4C4
 Upper central C1 — C2×C4 — C22×C4 — C2×M4(2)

Generators and relations for C2×D284C4
G = < a,b,c,d | a2=b28=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b13, dcd-1=b19c >

Subgroups: 868 in 170 conjugacy classes, 63 normal (41 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C14, C42, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic7, C28, D14, C2×C14, C2×C14, C4≀C2, C2×C42, C2×M4(2), C2×C4○D4, C56, Dic14, Dic14, C4×D7, D28, D28, C2×Dic7, C7⋊D4, C2×C28, C22×D7, C22×C14, C2×C4≀C2, C4×Dic7, C4×Dic7, C2×C56, C7×M4(2), C7×M4(2), C2×Dic14, C2×C4×D7, C2×D28, C4○D28, C4○D28, C22×Dic7, C2×C7⋊D4, C22×C28, D284C4, C2×C4×Dic7, C14×M4(2), C2×C4○D28, C2×D284C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, D14, C4≀C2, C2×C22⋊C4, C4×D7, D28, C7⋊D4, C22×D7, C2×C4≀C2, D14⋊C4, C2×C4×D7, C2×D28, C2×C7⋊D4, D284C4, C2×D14⋊C4, C2×D284C4

Smallest permutation representation of C2×D284C4
On 112 points
Generators in S112
(1 62)(2 63)(3 64)(4 65)(5 66)(6 67)(7 68)(8 69)(9 70)(10 71)(11 72)(12 73)(13 74)(14 75)(15 76)(16 77)(17 78)(18 79)(19 80)(20 81)(21 82)(22 83)(23 84)(24 57)(25 58)(26 59)(27 60)(28 61)(29 104)(30 105)(31 106)(32 107)(33 108)(34 109)(35 110)(36 111)(37 112)(38 85)(39 86)(40 87)(41 88)(42 89)(43 90)(44 91)(45 92)(46 93)(47 94)(48 95)(49 96)(50 97)(51 98)(52 99)(53 100)(54 101)(55 102)(56 103)
(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 55)(2 54)(3 53)(4 52)(5 51)(6 50)(7 49)(8 48)(9 47)(10 46)(11 45)(12 44)(13 43)(14 42)(15 41)(16 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(28 56)(57 107)(58 106)(59 105)(60 104)(61 103)(62 102)(63 101)(64 100)(65 99)(66 98)(67 97)(68 96)(69 95)(70 94)(71 93)(72 92)(73 91)(74 90)(75 89)(76 88)(77 87)(78 86)(79 85)(80 112)(81 111)(82 110)(83 109)(84 108)
(1 69 15 83)(2 82 16 68)(3 67 17 81)(4 80 18 66)(5 65 19 79)(6 78 20 64)(7 63 21 77)(8 76 22 62)(9 61 23 75)(10 74 24 60)(11 59 25 73)(12 72 26 58)(13 57 27 71)(14 70 28 84)(29 88)(30 101)(31 86)(32 99)(33 112)(34 97)(35 110)(36 95)(37 108)(38 93)(39 106)(40 91)(41 104)(42 89)(43 102)(44 87)(45 100)(46 85)(47 98)(48 111)(49 96)(50 109)(51 94)(52 107)(53 92)(54 105)(55 90)(56 103)

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

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

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

88 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G ··· 4N 4O 4P 7A 7B 7C 8A 8B 8C 8D 14A ··· 14I 14J ··· 14O 28A ··· 28L 28M ··· 28R 56A ··· 56X order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 ··· 4 4 4 7 7 7 8 8 8 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 1 1 2 2 28 28 1 1 1 1 2 2 14 ··· 14 28 28 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 4 ··· 4

88 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 D4 D4 D7 D14 D14 C4≀C2 C4×D7 D28 C7⋊D4 D28 D28⋊4C4 kernel C2×D28⋊4C4 D28⋊4C4 C2×C4×Dic7 C14×M4(2) C2×C4○D28 C2×Dic14 C2×D28 C4○D28 C2×C28 C22×C14 C2×M4(2) M4(2) C22×C4 C14 C2×C4 C2×C4 C2×C4 C23 C2 # reps 1 4 1 1 1 2 2 4 3 1 3 6 3 8 12 6 12 6 12

Matrix representation of C2×D284C4 in GL4(𝔽113) generated by

 112 0 0 0 0 112 0 0 0 0 112 0 0 0 0 112
,
 112 9 0 0 104 80 0 0 0 0 15 0 0 0 0 98
,
 1 104 0 0 0 112 0 0 0 0 0 1 0 0 1 0
,
 112 0 0 0 104 1 0 0 0 0 15 0 0 0 0 112
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,112,0,0,0,0,112],[112,104,0,0,9,80,0,0,0,0,15,0,0,0,0,98],[1,0,0,0,104,112,0,0,0,0,0,1,0,0,1,0],[112,104,0,0,0,1,0,0,0,0,15,0,0,0,0,112] >;

C2×D284C4 in GAP, Magma, Sage, TeX

C_2\times D_{28}\rtimes_4C_4
% in TeX

G:=Group("C2xD28:4C4");
// GroupNames label

G:=SmallGroup(448,672);
// by ID

G=gap.SmallGroup(448,672);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,422,58,136,1684,438,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^28=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^13,d*c*d^-1=b^19*c>;
// generators/relations

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