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G = C7xSD32order 224 = 25·7

Direct product of C7 and SD32

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

Aliases: C7xSD32, D8.C14, C16:2C14, C112:6C2, Q16:1C14, C14.16D8, C28.37D4, C56.25C22, C2.4(C7xD8), C4.2(C7xD4), C8.3(C2xC14), (C7xQ16):5C2, (C7xD8).2C2, SmallGroup(224,61)

Series: Derived Chief Lower central Upper central

C1C8 — C7xSD32
C1C2C4C8C56C7xQ16 — C7xSD32
C1C2C4C8 — C7xSD32
C1C14C28C56 — C7xSD32

Generators and relations for C7xSD32
 G = < a,b,c | a7=b16=c2=1, ab=ba, ac=ca, cbc=b7 >

Subgroups: 56 in 26 conjugacy classes, 16 normal (all characteristic)
Quotients: C1, C2, C22, C7, D4, C14, D8, C2xC14, SD32, C7xD4, C7xD8, C7xSD32
8C2
4C22
4C4
8C14
2D4
2Q8
4C28
4C2xC14
2C7xQ8
2C7xD4

Smallest permutation representation of C7xSD32
On 112 points
Generators in S112
(1 94 67 58 39 22 107)(2 95 68 59 40 23 108)(3 96 69 60 41 24 109)(4 81 70 61 42 25 110)(5 82 71 62 43 26 111)(6 83 72 63 44 27 112)(7 84 73 64 45 28 97)(8 85 74 49 46 29 98)(9 86 75 50 47 30 99)(10 87 76 51 48 31 100)(11 88 77 52 33 32 101)(12 89 78 53 34 17 102)(13 90 79 54 35 18 103)(14 91 80 55 36 19 104)(15 92 65 56 37 20 105)(16 93 66 57 38 21 106)
(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 15)(4 6)(5 13)(7 11)(10 16)(12 14)(17 19)(18 26)(20 24)(21 31)(23 29)(25 27)(28 32)(33 45)(34 36)(35 43)(37 41)(38 48)(40 46)(42 44)(49 59)(51 57)(52 64)(53 55)(54 62)(56 60)(61 63)(65 69)(66 76)(68 74)(70 72)(71 79)(73 77)(78 80)(81 83)(82 90)(84 88)(85 95)(87 93)(89 91)(92 96)(97 101)(98 108)(100 106)(102 104)(103 111)(105 109)(110 112)

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

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

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

C7xSD32 is a maximal subgroup of   D112:C2  SD32:D7  SD32:3D7

77 conjugacy classes

class 1 2A2B4A4B7A···7F8A8B14A···14F14G···14L16A16B16C16D28A···28F28G···28L56A···56L112A···112X
order122447···78814···1414···141616161628···2828···2856···56112···112
size118281···1221···18···822222···28···82···22···2

77 irreducible representations

dim11111111222222
type++++++
imageC1C2C2C2C7C14C14C14D4D8SD32C7xD4C7xD8C7xSD32
kernelC7xSD32C112C7xD8C7xQ16SD32C16D8Q16C28C14C7C4C2C1
# reps1111666612461224

Matrix representation of C7xSD32 in GL4(F113) generated by

109000
010900
0010
0001
,
823100
828200
005369
004453
,
1000
011200
0010
000112
G:=sub<GL(4,GF(113))| [109,0,0,0,0,109,0,0,0,0,1,0,0,0,0,1],[82,82,0,0,31,82,0,0,0,0,53,44,0,0,69,53],[1,0,0,0,0,112,0,0,0,0,1,0,0,0,0,112] >;

C7xSD32 in GAP, Magma, Sage, TeX

C_7\times {\rm SD}_{32}
% in TeX

G:=Group("C7xSD32");
// GroupNames label

G:=SmallGroup(224,61);
// by ID

G=gap.SmallGroup(224,61);
# by ID

G:=PCGroup([6,-2,-2,-7,-2,-2,-2,672,361,2019,1017,165,5044,2530,88]);
// Polycyclic

G:=Group<a,b,c|a^7=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^7>;
// generators/relations

Export

Subgroup lattice of C7xSD32 in TeX

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