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## G = D7×SD32order 448 = 26·7

### Direct product of D7 and SD32

Series: Derived Chief Lower central Upper central

 Derived series C1 — C56 — D7×SD32
 Chief series C1 — C7 — C14 — C28 — C56 — C8×D7 — D7×D8 — D7×SD32
 Lower central C7 — C14 — C28 — C56 — D7×SD32
 Upper central C1 — C2 — C4 — C8 — SD32

Generators and relations for D7×SD32
G = < a,b,c,d | a7=b2=c16=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c7 >

Subgroups: 672 in 90 conjugacy classes, 33 normal (31 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C8, C2×C4, D4, Q8, C23, D7, D7, C14, C14, C16, C16, C2×C8, D8, D8, Q16, Q16, C2×D4, C2×Q8, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C16, SD32, SD32, C2×D8, C2×Q16, C7⋊C8, C56, Dic14, C4×D7, C4×D7, D28, C7⋊D4, C7×D4, C7×Q8, C22×D7, C2×SD32, C7⋊C16, C112, C8×D7, D56, Dic28, D4⋊D7, C7⋊Q16, C7×D8, C7×Q16, D4×D7, Q8×D7, D7×C16, C112⋊C2, D8.D7, C7⋊SD32, C7×SD32, D7×D8, D7×Q16, D7×SD32
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, D14, SD32, C2×D8, C22×D7, C2×SD32, D4×D7, D7×D8, D7×SD32

Smallest permutation representation of D7×SD32
On 112 points
Generators in S112
(1 71 97 50 42 90 25)(2 72 98 51 43 91 26)(3 73 99 52 44 92 27)(4 74 100 53 45 93 28)(5 75 101 54 46 94 29)(6 76 102 55 47 95 30)(7 77 103 56 48 96 31)(8 78 104 57 33 81 32)(9 79 105 58 34 82 17)(10 80 106 59 35 83 18)(11 65 107 60 36 84 19)(12 66 108 61 37 85 20)(13 67 109 62 38 86 21)(14 68 110 63 39 87 22)(15 69 111 64 40 88 23)(16 70 112 49 41 89 24)
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)(33 104)(34 105)(35 106)(36 107)(37 108)(38 109)(39 110)(40 111)(41 112)(42 97)(43 98)(44 99)(45 100)(46 101)(47 102)(48 103)(65 84)(66 85)(67 86)(68 87)(69 88)(70 89)(71 90)(72 91)(73 92)(74 93)(75 94)(76 95)(77 96)(78 81)(79 82)(80 83)
(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)(18 24)(19 31)(20 22)(21 29)(23 27)(26 32)(28 30)(33 43)(35 41)(36 48)(37 39)(38 46)(40 44)(45 47)(49 59)(51 57)(52 64)(53 55)(54 62)(56 60)(61 63)(65 77)(66 68)(67 75)(69 73)(70 80)(72 78)(74 76)(81 91)(83 89)(84 96)(85 87)(86 94)(88 92)(93 95)(98 104)(99 111)(100 102)(101 109)(103 107)(106 112)(108 110)

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

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

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

55 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 7A 7B 7C 8A 8B 8C 8D 14A 14B 14C 14D 14E 14F 16A 16B 16C 16D 16E 16F 16G 16H 28A 28B 28C 28D 28E 28F 56A ··· 56F 112A ··· 112L order 1 2 2 2 2 2 4 4 4 4 7 7 7 8 8 8 8 14 14 14 14 14 14 16 16 16 16 16 16 16 16 28 28 28 28 28 28 56 ··· 56 112 ··· 112 size 1 1 7 7 8 56 2 8 14 56 2 2 2 2 2 14 14 2 2 2 16 16 16 2 2 2 2 14 14 14 14 4 4 4 16 16 16 4 ··· 4 4 ··· 4

55 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D7 D8 D8 D14 D14 D14 SD32 D4×D7 D7×D8 D7×SD32 kernel D7×SD32 D7×C16 C112⋊C2 D8.D7 C7⋊SD32 C7×SD32 D7×D8 D7×Q16 C7⋊C8 C4×D7 SD32 Dic7 D14 C16 D8 Q16 D7 C4 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 3 2 2 3 3 3 8 3 6 12

Matrix representation of D7×SD32 in GL4(𝔽113) generated by

 1 0 0 0 0 1 0 0 0 0 88 1 0 0 111 104
,
 1 0 0 0 0 1 0 0 0 0 34 79 0 0 24 79
,
 104 44 0 0 25 16 0 0 0 0 112 0 0 0 0 112
,
 1 1 0 0 0 112 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,88,111,0,0,1,104],[1,0,0,0,0,1,0,0,0,0,34,24,0,0,79,79],[104,25,0,0,44,16,0,0,0,0,112,0,0,0,0,112],[1,0,0,0,1,112,0,0,0,0,1,0,0,0,0,1] >;

D7×SD32 in GAP, Magma, Sage, TeX

D_7\times {\rm SD}_{32}
% in TeX

G:=Group("D7xSD32");
// GroupNames label

G:=SmallGroup(448,447);
// by ID

G=gap.SmallGroup(448,447);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,135,184,346,185,192,851,438,102,18822]);
// Polycyclic

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

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