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G = C7xC4oD8order 224 = 25·7

Direct product of C7 and C4oD8

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C7xC4oD8, D8:3C14, Q16:3C14, C28.69D4, SD16:3C14, C28.47C23, C56.28C22, (C2xC8):4C14, (C7xD8):7C2, (C2xC56):12C2, C4oD4:1C14, C8.6(C2xC14), (C7xQ16):7C2, C4.20(C7xD4), (C7xSD16):7C2, D4.2(C2xC14), C2.14(D4xC14), C14.77(C2xD4), (C2xC14).11D4, Q8.2(C2xC14), C22.1(C7xD4), C4.4(C22xC14), (C7xD4).12C22, (C7xQ8).13C22, (C2xC28).132C22, (C7xC4oD4):6C2, (C2xC4).28(C2xC14), SmallGroup(224,170)

Series: Derived Chief Lower central Upper central

C1C4 — C7xC4oD8
C1C2C4C28C7xD4C7xD8 — C7xC4oD8
C1C2C4 — C7xC4oD8
C1C28C2xC28 — C7xC4oD8

Generators and relations for C7xC4oD8
 G = < a,b,c,d | a7=b4=d2=1, c4=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c3 >

Subgroups: 92 in 62 conjugacy classes, 40 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2xC4, C2xC4, D4, D4, Q8, C14, C14, C2xC8, D8, SD16, Q16, C4oD4, C28, C28, C2xC14, C2xC14, C4oD8, C56, C2xC28, C2xC28, C7xD4, C7xD4, C7xQ8, C2xC56, C7xD8, C7xSD16, C7xQ16, C7xC4oD4, C7xC4oD8
Quotients: C1, C2, C22, C7, D4, C23, C14, C2xD4, C2xC14, C4oD8, C7xD4, C22xC14, D4xC14, C7xC4oD8

Smallest permutation representation of C7xC4oD8
On 112 points
Generators in S112
(1 73 43 23 93 65 35)(2 74 44 24 94 66 36)(3 75 45 17 95 67 37)(4 76 46 18 96 68 38)(5 77 47 19 89 69 39)(6 78 48 20 90 70 40)(7 79 41 21 91 71 33)(8 80 42 22 92 72 34)(9 112 84 54 26 104 57)(10 105 85 55 27 97 58)(11 106 86 56 28 98 59)(12 107 87 49 29 99 60)(13 108 88 50 30 100 61)(14 109 81 51 31 101 62)(15 110 82 52 32 102 63)(16 111 83 53 25 103 64)
(1 97 5 101)(2 98 6 102)(3 99 7 103)(4 100 8 104)(9 46 13 42)(10 47 14 43)(11 48 15 44)(12 41 16 45)(17 107 21 111)(18 108 22 112)(19 109 23 105)(20 110 24 106)(25 37 29 33)(26 38 30 34)(27 39 31 35)(28 40 32 36)(49 71 53 67)(50 72 54 68)(51 65 55 69)(52 66 56 70)(57 76 61 80)(58 77 62 73)(59 78 63 74)(60 79 64 75)(81 93 85 89)(82 94 86 90)(83 95 87 91)(84 96 88 92)
(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)
(2 8)(3 7)(4 6)(9 11)(12 16)(13 15)(17 21)(18 20)(22 24)(25 29)(26 28)(30 32)(33 37)(34 36)(38 40)(41 45)(42 44)(46 48)(49 53)(50 52)(54 56)(57 59)(60 64)(61 63)(66 72)(67 71)(68 70)(74 80)(75 79)(76 78)(82 88)(83 87)(84 86)(90 96)(91 95)(92 94)(98 104)(99 103)(100 102)(106 112)(107 111)(108 110)

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

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

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

C7xC4oD8 is a maximal subgroup of
D8:2Dic7  C28.58D8  Q16:D14  C56.30C23  C56.31C23  D8:5Dic7  D8:4Dic7  D8:10D14  D8:15D14  D8:11D14  D8.10D14
C7xC4oD8 is a maximal quotient of
D8xC28  SD16xC28  Q16xC28

98 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E7A···7F8A8B8C8D14A···14F14G···14L14M···14X28A···28L28M···28R28S···28AD56A···56X
order12222444447···7888814···1414···1414···1428···2828···2828···2856···56
size11244112441···122221···12···24···41···12···24···42···2

98 irreducible representations

dim111111111111222222
type++++++++
imageC1C2C2C2C2C2C7C14C14C14C14C14D4D4C4oD8C7xD4C7xD4C7xC4oD8
kernelC7xC4oD8C2xC56C7xD8C7xSD16C7xQ16C7xC4oD4C4oD8C2xC8D8SD16Q16C4oD4C28C2xC14C7C4C22C1
# reps111212666126121146624

Matrix representation of C7xC4oD8 in GL2(F113) generated by

160
016
,
980
098
,
5182
620
,
11
0112
G:=sub<GL(2,GF(113))| [16,0,0,16],[98,0,0,98],[51,62,82,0],[1,0,1,112] >;

C7xC4oD8 in GAP, Magma, Sage, TeX

C_7\times C_4\circ D_8
% in TeX

G:=Group("C7xC4oD8");
// GroupNames label

G:=SmallGroup(224,170);
// by ID

G=gap.SmallGroup(224,170);
# by ID

G:=PCGroup([6,-2,-2,-2,-7,-2,-2,697,518,5044,2530,88]);
// Polycyclic

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

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