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G = C7xD8:2C4order 448 = 26·7

Direct product of C7 and D8:2C4

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Aliases: C7xD8:2C4, D8:2C28, Q16:2C28, C56.102D4, M5(2):5C14, C28.43SD16, (C7xD8):8C4, C8.1(C2xC28), (C7xQ16):8C4, C4.Q8:1C14, C8.22(C7xD4), C56.42(C2xC4), C4oD8.2C14, (C2xC14).25D8, C4.8(C7xSD16), C22.3(C7xD8), (C2xC28).279D4, (C7xM5(2)):13C2, C28.72(C22:C4), (C2xC56).265C22, C14.40(D4:C4), (C7xC4oD8).7C2, (C7xC4.Q8):10C2, (C2xC4).10(C7xD4), C4.4(C7xC22:C4), (C2xC8).12(C2xC14), C2.9(C7xD4:C4), SmallGroup(448,164)

Series: Derived Chief Lower central Upper central

C1C8 — C7xD8:2C4
C1C2C4C2xC4C2xC8C2xC56C7xC4.Q8 — C7xD8:2C4
C1C2C4C8 — C7xD8:2C4
C1C14C2xC28C2xC56 — C7xD8:2C4

Generators and relations for C7xD8:2C4
 G = < a,b,c,d | a7=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b3, dcd-1=b5c >

Subgroups: 130 in 58 conjugacy classes, 30 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2xC4, C2xC4, D4, Q8, C14, C14, C16, C4:C4, C2xC8, D8, SD16, Q16, C4oD4, C28, C28, C2xC14, C2xC14, C4.Q8, M5(2), C4oD8, C56, C2xC28, C2xC28, C7xD4, C7xQ8, D8:2C4, C112, C7xC4:C4, C2xC56, C7xD8, C7xSD16, C7xQ16, C7xC4oD4, C7xC4.Q8, C7xM5(2), C7xC4oD8, C7xD8:2C4
Quotients: C1, C2, C4, C22, C7, C2xC4, D4, C14, C22:C4, D8, SD16, C28, C2xC14, D4:C4, C2xC28, C7xD4, D8:2C4, C7xC22:C4, C7xD8, C7xSD16, C7xD4:C4, C7xD8:2C4

Smallest permutation representation of C7xD8:2C4
On 112 points
Generators in S112
(1 55 47 39 31 23 15)(2 56 48 40 32 24 16)(3 49 41 33 25 17 9)(4 50 42 34 26 18 10)(5 51 43 35 27 19 11)(6 52 44 36 28 20 12)(7 53 45 37 29 21 13)(8 54 46 38 30 22 14)(57 105 97 89 81 73 65)(58 106 98 90 82 74 66)(59 107 99 91 83 75 67)(60 108 100 92 84 76 68)(61 109 101 93 85 77 69)(62 110 102 94 86 78 70)(63 111 103 95 87 79 71)(64 112 104 96 88 80 72)
(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 64)(2 63)(3 62)(4 61)(5 60)(6 59)(7 58)(8 57)(9 70)(10 69)(11 68)(12 67)(13 66)(14 65)(15 72)(16 71)(17 78)(18 77)(19 76)(20 75)(21 74)(22 73)(23 80)(24 79)(25 86)(26 85)(27 84)(28 83)(29 82)(30 81)(31 88)(32 87)(33 94)(34 93)(35 92)(36 91)(37 90)(38 89)(39 96)(40 95)(41 102)(42 101)(43 100)(44 99)(45 98)(46 97)(47 104)(48 103)(49 110)(50 109)(51 108)(52 107)(53 106)(54 105)(55 112)(56 111)
(2 4)(3 7)(6 8)(9 13)(10 16)(12 14)(17 21)(18 24)(20 22)(25 29)(26 32)(28 30)(33 37)(34 40)(36 38)(41 45)(42 48)(44 46)(49 53)(50 56)(52 54)(57 58 61 62)(59 64 63 60)(65 66 69 70)(67 72 71 68)(73 74 77 78)(75 80 79 76)(81 82 85 86)(83 88 87 84)(89 90 93 94)(91 96 95 92)(97 98 101 102)(99 104 103 100)(105 106 109 110)(107 112 111 108)

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

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

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

112 conjugacy classes

class 1 2A2B2C4A4B4C4D4E7A···7F8A8B8C14A···14F14G···14L14M···14R16A16B16C16D28A···28L28M···28AD56A···56L56M···56R112A···112X
order1222444447···788814···1414···1414···141616161628···2828···2856···5656···56112···112
size1128228881···12241···12···28···844442···28···82···24···44···4

112 irreducible representations

dim1111111111112222222244
type+++++++
imageC1C2C2C2C4C4C7C14C14C14C28C28D4D4SD16D8C7xD4C7xD4C7xSD16C7xD8D8:2C4C7xD8:2C4
kernelC7xD8:2C4C7xC4.Q8C7xM5(2)C7xC4oD8C7xD8C7xQ16D8:2C4C4.Q8M5(2)C4oD8D8Q16C56C2xC28C28C2xC14C8C2xC4C4C22C7C1
# reps111122666612121122661212212

Matrix representation of C7xD8:2C4 in GL6(F113)

3000000
0300000
001000
000100
000010
000001
,
11200000
01120000
001001300
0010010000
00001313
000010013
,
01120000
11200000
00001313
000010013
001001300
0010010000
,
9800000
0150000
001000
00011200
000013100
0000100100

G:=sub<GL(6,GF(113))| [30,0,0,0,0,0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,100,100,0,0,0,0,13,100,0,0,0,0,0,0,13,100,0,0,0,0,13,13],[0,112,0,0,0,0,112,0,0,0,0,0,0,0,0,0,100,100,0,0,0,0,13,100,0,0,13,100,0,0,0,0,13,13,0,0],[98,0,0,0,0,0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,13,100,0,0,0,0,100,100] >;

C7xD8:2C4 in GAP, Magma, Sage, TeX

C_7\times D_8\rtimes_2C_4
% in TeX

G:=Group("C7xD8:2C4");
// GroupNames label

G:=SmallGroup(448,164);
// by ID

G=gap.SmallGroup(448,164);
# by ID

G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,3923,5106,136,4911,14117,7068,124]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^8=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^3,d*c*d^-1=b^5*c>;
// generators/relations

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