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## G = C2×C28.46D4order 448 = 26·7

### Direct product of C2 and C28.46D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C2×C28.46D4
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×D28 — C22×D28 — C2×C28.46D4
 Lower central C7 — C14 — C2×C14 — C2×C28.46D4
 Upper central C1 — C22 — C22×C4 — C2×M4(2)

Generators and relations for C2×C28.46D4
G = < a,b,c,d | a2=b28=d2=1, c4=b14, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=b7c3 >

Subgroups: 1252 in 186 conjugacy classes, 63 normal (39 characteristic)
C1, C2, C2, C2, C4, C22, C22, C7, C8, C2×C4, D4, C23, C23, D7, C14, C14, C14, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C24, C28, D14, C2×C14, C2×C14, C4.D4, C2×M4(2), C2×M4(2), C22×D4, C7⋊C8, C56, D28, C2×C28, C22×D7, C22×D7, C22×C14, C2×C4.D4, C2×C7⋊C8, C4.Dic7, C4.Dic7, C2×C56, C7×M4(2), C7×M4(2), C2×D28, C2×D28, C22×C28, C23×D7, C28.46D4, C2×C4.Dic7, C14×M4(2), C22×D28, C2×C28.46D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, D14, C4.D4, C2×C22⋊C4, C4×D7, D28, C7⋊D4, C22×D7, C2×C4.D4, D14⋊C4, C2×C4×D7, C2×D28, C2×C7⋊D4, C28.46D4, C2×D14⋊C4, C2×C28.46D4

Smallest permutation representation of C2×C28.46D4
On 112 points
Generators in S112
(1 32)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(17 48)(18 49)(19 50)(20 51)(21 52)(22 53)(23 54)(24 55)(25 56)(26 29)(27 30)(28 31)(57 110)(58 111)(59 112)(60 85)(61 86)(62 87)(63 88)(64 89)(65 90)(66 91)(67 92)(68 93)(69 94)(70 95)(71 96)(72 97)(73 98)(74 99)(75 100)(76 101)(77 102)(78 103)(79 104)(80 105)(81 106)(82 107)(83 108)(84 109)
(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 39 76 15 94 53 62)(2 107 40 75 16 93 54 61)(3 106 41 74 17 92 55 60)(4 105 42 73 18 91 56 59)(5 104 43 72 19 90 29 58)(6 103 44 71 20 89 30 57)(7 102 45 70 21 88 31 84)(8 101 46 69 22 87 32 83)(9 100 47 68 23 86 33 82)(10 99 48 67 24 85 34 81)(11 98 49 66 25 112 35 80)(12 97 50 65 26 111 36 79)(13 96 51 64 27 110 37 78)(14 95 52 63 28 109 38 77)
(1 32)(2 31)(3 30)(4 29)(5 56)(6 55)(7 54)(8 53)(9 52)(10 51)(11 50)(12 49)(13 48)(14 47)(15 46)(16 45)(17 44)(18 43)(19 42)(20 41)(21 40)(22 39)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(57 67)(58 66)(59 65)(60 64)(61 63)(68 84)(69 83)(70 82)(71 81)(72 80)(73 79)(74 78)(75 77)(85 89)(86 88)(90 112)(91 111)(92 110)(93 109)(94 108)(95 107)(96 106)(97 105)(98 104)(99 103)(100 102)

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

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

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

82 conjugacy classes

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

82 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 D4 D7 D14 D14 C4×D7 D28 C7⋊D4 C4×D7 C4.D4 C28.46D4 kernel C2×C28.46D4 C28.46D4 C2×C4.Dic7 C14×M4(2) C22×D28 C2×D28 C23×D7 C2×C28 C2×M4(2) M4(2) C22×C4 C2×C4 C2×C4 C2×C4 C23 C14 C2 # reps 1 4 1 1 1 4 4 4 3 6 3 6 12 12 6 2 12

Matrix representation of C2×C28.46D4 in GL6(𝔽113)

 112 0 0 0 0 0 0 112 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 24 10 0 0 0 0 79 0 0 0 0 0 0 0 96 36 0 0 0 0 77 23 0 0 0 0 59 46 23 36 0 0 46 28 77 96
,
 0 10 0 0 0 0 34 0 0 0 0 0 0 0 44 27 2 89 0 0 97 60 89 9 0 0 11 101 20 105 0 0 77 41 16 102
,
 0 10 0 0 0 0 34 0 0 0 0 0 0 0 103 89 0 0 0 0 103 10 0 0 0 0 34 105 1 0 0 0 98 19 24 112

G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[24,79,0,0,0,0,10,0,0,0,0,0,0,0,96,77,59,46,0,0,36,23,46,28,0,0,0,0,23,77,0,0,0,0,36,96],[0,34,0,0,0,0,10,0,0,0,0,0,0,0,44,97,11,77,0,0,27,60,101,41,0,0,2,89,20,16,0,0,89,9,105,102],[0,34,0,0,0,0,10,0,0,0,0,0,0,0,103,103,34,98,0,0,89,10,105,19,0,0,0,0,1,24,0,0,0,0,0,112] >;

C2×C28.46D4 in GAP, Magma, Sage, TeX

C_2\times C_{28}._{46}D_4
% in TeX

G:=Group("C2xC28.46D4");
// GroupNames label

G:=SmallGroup(448,664);
// by ID

G=gap.SmallGroup(448,664);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,422,58,1123,136,438,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^28=d^2=1,c^4=b^14,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=b^7*c^3>;
// generators/relations

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