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G = C7×D4○SD16order 448 = 26·7

Direct product of C7 and D4○SD16

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

Aliases: C7×D4○SD16, C56.53C23, C28.86C24, 2+ 1+44C14, 2- 1+42C14, C4○D85C14, C8○D44C14, D85(C2×C14), C8⋊C225C14, Q165(C2×C14), (C7×D4).46D4, D4.12(C7×D4), C4.46(D4×C14), Q8.12(C7×D4), (C7×Q8).46D4, (C2×C56)⋊32C22, (C2×SD16)⋊6C14, SD166(C2×C14), C28.407(C2×D4), (C7×D8)⋊22C22, C8.C224C14, C4.9(C23×C14), C22.8(D4×C14), (C14×SD16)⋊17C2, M4(2)⋊7(C2×C14), C8.13(C22×C14), (C7×Q16)⋊19C22, (Q8×C14)⋊32C22, D4.6(C22×C14), (C7×D4).39C23, (C7×Q8).40C23, Q8.6(C22×C14), (C2×C28).688C23, (C7×SD16)⋊21C22, C14.207(C22×D4), (C7×2- 1+4)⋊7C2, (D4×C14).226C22, (C7×M4(2))⋊33C22, (C7×2+ 1+4)⋊10C2, (C2×C8)⋊5(C2×C14), C2.31(D4×C2×C14), (C7×C4○D8)⋊12C2, (C7×C8○D4)⋊13C2, C4○D42(C2×C14), (C2×Q8)⋊7(C2×C14), (C7×C8⋊C22)⋊12C2, (C2×D4).39(C2×C14), (C2×C14).185(C2×D4), (C7×C4○D4)⋊15C22, (C7×C8.C22)⋊11C2, (C2×C4).49(C22×C14), SmallGroup(448,1360)

Series: Derived Chief Lower central Upper central

C1C4 — C7×D4○SD16
C1C2C4C28C7×Q8C7×SD16C14×SD16 — C7×D4○SD16
C1C2C4 — C7×D4○SD16
C1C14C7×C4○D4 — C7×D4○SD16

Generators and relations for C7×D4○SD16
 G = < a,b,c,d,e | a7=b4=c2=e2=1, d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d3 >

Subgroups: 410 in 258 conjugacy classes, 158 normal (26 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, D4, Q8, Q8, Q8, C23, C14, C14, C2×C8, M4(2), D8, SD16, SD16, Q16, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C4○D4, C28, C28, C28, C2×C14, C2×C14, C8○D4, C2×SD16, C4○D8, C8⋊C22, C8.C22, 2+ 1+4, 2- 1+4, C56, C56, C2×C28, C2×C28, C7×D4, C7×D4, C7×D4, C7×Q8, C7×Q8, C7×Q8, C22×C14, D4○SD16, C2×C56, C7×M4(2), C7×D8, C7×SD16, C7×SD16, C7×Q16, D4×C14, D4×C14, Q8×C14, Q8×C14, C7×C4○D4, C7×C4○D4, C7×C4○D4, C7×C8○D4, C14×SD16, C7×C4○D8, C7×C8⋊C22, C7×C8.C22, C7×2+ 1+4, C7×2- 1+4, C7×D4○SD16
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C24, C2×C14, C22×D4, C7×D4, C22×C14, D4○SD16, D4×C14, C23×C14, D4×C2×C14, C7×D4○SD16

Smallest permutation representation of C7×D4○SD16
On 112 points
Generators in S112
(1 77 42 20 91 69 34)(2 78 43 21 92 70 35)(3 79 44 22 93 71 36)(4 80 45 23 94 72 37)(5 73 46 24 95 65 38)(6 74 47 17 96 66 39)(7 75 48 18 89 67 40)(8 76 41 19 90 68 33)(9 112 82 55 25 104 59)(10 105 83 56 26 97 60)(11 106 84 49 27 98 61)(12 107 85 50 28 99 62)(13 108 86 51 29 100 63)(14 109 87 52 30 101 64)(15 110 88 53 31 102 57)(16 111 81 54 32 103 58)
(1 99 5 103)(2 100 6 104)(3 101 7 97)(4 102 8 98)(9 43 13 47)(10 44 14 48)(11 45 15 41)(12 46 16 42)(17 112 21 108)(18 105 22 109)(19 106 23 110)(20 107 24 111)(25 35 29 39)(26 36 30 40)(27 37 31 33)(28 38 32 34)(49 72 53 68)(50 65 54 69)(51 66 55 70)(52 67 56 71)(57 76 61 80)(58 77 62 73)(59 78 63 74)(60 79 64 75)(81 91 85 95)(82 92 86 96)(83 93 87 89)(84 94 88 90)
(9 13)(10 14)(11 15)(12 16)(25 29)(26 30)(27 31)(28 32)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(81 85)(82 86)(83 87)(84 88)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)
(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 4)(3 7)(6 8)(9 11)(10 14)(13 15)(17 19)(18 22)(21 23)(25 27)(26 30)(29 31)(33 39)(35 37)(36 40)(41 47)(43 45)(44 48)(49 55)(51 53)(52 56)(57 63)(59 61)(60 64)(66 68)(67 71)(70 72)(74 76)(75 79)(78 80)(82 84)(83 87)(86 88)(89 93)(90 96)(92 94)(97 101)(98 104)(100 102)(105 109)(106 112)(108 110)

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

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

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

154 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H7A···7F8A8B8C8D8E14A···14F14G···14X14Y···14AV28A···28X28Y···28AV56A···56L56M···56AD
order122222222444444447···78888814···1414···1414···1428···2828···2856···5656···56
size112224444222244441···1224441···12···24···42···24···42···24···4

154 irreducible representations

dim1111111111111111222244
type++++++++++
imageC1C2C2C2C2C2C2C2C7C14C14C14C14C14C14C14D4D4C7×D4C7×D4D4○SD16C7×D4○SD16
kernelC7×D4○SD16C7×C8○D4C14×SD16C7×C4○D8C7×C8⋊C22C7×C8.C22C7×2+ 1+4C7×2- 1+4D4○SD16C8○D4C2×SD16C4○D8C8⋊C22C8.C222+ 1+42- 1+4C7×D4C7×Q8D4Q8C7C1
# reps1133331166181818186631186212

Matrix representation of C7×D4○SD16 in GL4(𝔽113) generated by

30000
03000
00300
00030
,
001120
11211122
1000
11120112
,
1000
0100
001120
11120112
,
1001300
10010000
13100087
1301387
,
1000
011200
0010
101112
G:=sub<GL(4,GF(113))| [30,0,0,0,0,30,0,0,0,0,30,0,0,0,0,30],[0,112,1,1,0,1,0,112,112,112,0,0,0,2,0,112],[1,0,0,1,0,1,0,112,0,0,112,0,0,0,0,112],[100,100,13,13,13,100,100,0,0,0,0,13,0,0,87,87],[1,0,0,1,0,112,0,0,0,0,1,1,0,0,0,112] >;

C7×D4○SD16 in GAP, Magma, Sage, TeX

C_7\times D_4\circ {\rm SD}_{16}
% in TeX

G:=Group("C7xD4oSD16");
// GroupNames label

G:=SmallGroup(448,1360);
// by ID

G=gap.SmallGroup(448,1360);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1568,1597,1641,14117,7068,124]);
// Polycyclic

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

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