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G = C7×C8○D8order 448 = 26·7

Direct product of C7 and C8○D8

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

Aliases: C7×C8○D8, D85C28, Q165C28, C56.77D4, SD163C28, C4≀C27C14, (C4×C56)⋊26C2, (C4×C8)⋊10C14, C8○D46C14, (C7×D8)⋊11C4, C8.30(C7×D4), C8.11(C2×C28), C56.67(C2×C4), (C7×Q16)⋊11C4, C4○D8.5C14, (C7×SD16)⋊7C4, D4.3(C2×C28), C2.18(D4×C28), C4.82(D4×C14), Q8.3(C2×C28), C8.C48C14, C14.120(C4×D4), C28.487(C2×D4), C42.73(C2×C14), C4.15(C22×C28), (C4×C28).358C22, C28.160(C22×C4), (C2×C56).433C22, (C2×C28).910C23, M4(2).11(C2×C14), (C7×M4(2)).45C22, (C7×C4≀C2)⋊15C2, (C7×C8○D4)⋊15C2, (C7×C4○D8).10C2, C4○D4.8(C2×C14), (C7×D4).20(C2×C4), (C7×Q8).21(C2×C4), (C7×C8.C4)⋊17C2, C22.1(C7×C4○D4), (C2×C8).101(C2×C14), (C2×C14).49(C4○D4), (C2×C4).85(C22×C14), (C7×C4○D4).53C22, SmallGroup(448,851)

Series: Derived Chief Lower central Upper central

C1C4 — C7×C8○D8
C1C2C4C2×C4C2×C28C7×M4(2)C7×C4≀C2 — C7×C8○D8
C1C2C4 — C7×C8○D8
C1C56C2×C56 — C7×C8○D8

Generators and relations for C7×C8○D8
 G = < a,b,c,d | a7=b8=d2=1, c4=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c3 >

Subgroups: 154 in 106 conjugacy classes, 66 normal (38 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×4], C22, C22 [×2], C7, C8 [×4], C8 [×2], C2×C4, C2×C4 [×3], D4 [×2], D4 [×2], Q8 [×2], C14, C14 [×3], C42, C2×C8 [×2], C2×C8 [×2], M4(2) [×2], M4(2) [×2], D8, SD16 [×2], Q16, C4○D4 [×2], C28 [×2], C28 [×4], C2×C14, C2×C14 [×2], C4×C8, C4≀C2 [×2], C8.C4, C8○D4 [×2], C4○D8, C56 [×4], C56 [×2], C2×C28, C2×C28 [×3], C7×D4 [×2], C7×D4 [×2], C7×Q8 [×2], C8○D8, C4×C28, C2×C56 [×2], C2×C56 [×2], C7×M4(2) [×2], C7×M4(2) [×2], C7×D8, C7×SD16 [×2], C7×Q16, C7×C4○D4 [×2], C4×C56, C7×C4≀C2 [×2], C7×C8.C4, C7×C8○D4 [×2], C7×C4○D8, C7×C8○D8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C7, C2×C4 [×6], D4 [×2], C23, C14 [×7], C22×C4, C2×D4, C4○D4, C28 [×4], C2×C14 [×7], C4×D4, C2×C28 [×6], C7×D4 [×2], C22×C14, C8○D8, C22×C28, D4×C14, C7×C4○D4, D4×C28, C7×C8○D8

Smallest permutation representation of C7×C8○D8
On 112 points
Generators in S112
(1 59 19 55 10 47 39)(2 60 20 56 11 48 40)(3 61 21 49 12 41 33)(4 62 22 50 13 42 34)(5 63 23 51 14 43 35)(6 64 24 52 15 44 36)(7 57 17 53 16 45 37)(8 58 18 54 9 46 38)(25 76 105 96 85 68 97)(26 77 106 89 86 69 98)(27 78 107 90 87 70 99)(28 79 108 91 88 71 100)(29 80 109 92 81 72 101)(30 73 110 93 82 65 102)(31 74 111 94 83 66 103)(32 75 112 95 84 67 104)
(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 6 3 8 5 2 7 4)(9 14 11 16 13 10 15 12)(17 22 19 24 21 18 23 20)(25 28 31 26 29 32 27 30)(33 38 35 40 37 34 39 36)(41 46 43 48 45 42 47 44)(49 54 51 56 53 50 55 52)(57 62 59 64 61 58 63 60)(65 68 71 66 69 72 67 70)(73 76 79 74 77 80 75 78)(81 84 87 82 85 88 83 86)(89 92 95 90 93 96 91 94)(97 100 103 98 101 104 99 102)(105 108 111 106 109 112 107 110)
(1 103)(2 104)(3 97)(4 98)(5 99)(6 100)(7 101)(8 102)(9 93)(10 94)(11 95)(12 96)(13 89)(14 90)(15 91)(16 92)(17 80)(18 73)(19 74)(20 75)(21 76)(22 77)(23 78)(24 79)(25 61)(26 62)(27 63)(28 64)(29 57)(30 58)(31 59)(32 60)(33 68)(34 69)(35 70)(36 71)(37 72)(38 65)(39 66)(40 67)(41 85)(42 86)(43 87)(44 88)(45 81)(46 82)(47 83)(48 84)(49 105)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)

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

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

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

196 conjugacy classes

class 1 2A2B2C2D4A4B4C···4G4H4I7A···7F8A8B8C8D8E···8J8K8L8M8N14A···14F14G···14L14M···14X28A···28L28M···28AP28AQ···28BB56A···56X56Y···56BH56BI···56CF
order12222444···4447···788888···8888814···1414···1414···1428···2828···2828···2856···5656···5656···56
size11244112···2441···111112···244441···12···24···41···12···24···41···12···24···4

196 irreducible representations

dim111111111111111111222222
type+++++++
imageC1C2C2C2C2C2C4C4C4C7C14C14C14C14C14C28C28C28D4C4○D4C7×D4C8○D8C7×C4○D4C7×C8○D8
kernelC7×C8○D8C4×C56C7×C4≀C2C7×C8.C4C7×C8○D4C7×C4○D8C7×D8C7×SD16C7×Q16C8○D8C4×C8C4≀C2C8.C4C8○D4C4○D8D8SD16Q16C56C2×C14C8C7C22C1
# reps11212124266126126122412221281248

Matrix representation of C7×C8○D8 in GL3(𝔽113) generated by

3000
010
001
,
100
0180
0018
,
11200
0950
01369
,
11200
01111
00112
G:=sub<GL(3,GF(113))| [30,0,0,0,1,0,0,0,1],[1,0,0,0,18,0,0,0,18],[112,0,0,0,95,13,0,0,69],[112,0,0,0,1,0,0,111,112] >;

C7×C8○D8 in GAP, Magma, Sage, TeX

C_7\times C_8\circ D_8
% in TeX

G:=Group("C7xC8oD8");
// GroupNames label

G:=SmallGroup(448,851);
// by ID

G=gap.SmallGroup(448,851);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,604,9804,4911,172,124]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^8=d^2=1,c^4=b^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^4*c^3>;
// generators/relations

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