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## G = C7×SD32order 224 = 25·7

### Direct product of C7 and SD32

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

Aliases: C7×SD32, D8.C14, C162C14, C1126C2, Q161C14, C14.16D8, C28.37D4, C56.25C22, C2.4(C7×D8), C4.2(C7×D4), C8.3(C2×C14), (C7×Q16)⋊5C2, (C7×D8).2C2, SmallGroup(224,61)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8 — C7×SD32
 Chief series C1 — C2 — C4 — C8 — C56 — C7×Q16 — C7×SD32
 Lower central C1 — C2 — C4 — C8 — C7×SD32
 Upper central C1 — C14 — C28 — C56 — C7×SD32

Generators and relations for C7×SD32
G = < a,b,c | a7=b16=c2=1, ab=ba, ac=ca, cbc=b7 >

Smallest permutation representation of C7×SD32
On 112 points
Generators in S112
(1 35 25 59 100 95 69)(2 36 26 60 101 96 70)(3 37 27 61 102 81 71)(4 38 28 62 103 82 72)(5 39 29 63 104 83 73)(6 40 30 64 105 84 74)(7 41 31 49 106 85 75)(8 42 32 50 107 86 76)(9 43 17 51 108 87 77)(10 44 18 52 109 88 78)(11 45 19 53 110 89 79)(12 46 20 54 111 90 80)(13 47 21 55 112 91 65)(14 48 22 56 97 92 66)(15 33 23 57 98 93 67)(16 34 24 58 99 94 68)
(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 37)(34 44)(36 42)(38 40)(39 47)(41 45)(46 48)(49 53)(50 60)(52 58)(54 56)(55 63)(57 61)(62 64)(65 73)(66 80)(67 71)(68 78)(70 76)(72 74)(75 79)(81 93)(82 84)(83 91)(85 89)(86 96)(88 94)(90 92)(97 111)(98 102)(99 109)(101 107)(103 105)(104 112)(106 110)

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

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

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

C7×SD32 is a maximal subgroup of   D112⋊C2  SD32⋊D7  SD323D7

77 conjugacy classes

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

77 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C7 C14 C14 C14 D4 D8 SD32 C7×D4 C7×D8 C7×SD32 kernel C7×SD32 C112 C7×D8 C7×Q16 SD32 C16 D8 Q16 C28 C14 C7 C4 C2 C1 # reps 1 1 1 1 6 6 6 6 1 2 4 6 12 24

Matrix representation of C7×SD32 in GL4(𝔽113) generated by

 109 0 0 0 0 109 0 0 0 0 1 0 0 0 0 1
,
 82 31 0 0 82 82 0 0 0 0 53 69 0 0 44 53
,
 1 0 0 0 0 112 0 0 0 0 1 0 0 0 0 112
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] >;

C7×SD32 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

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