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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

C1C8 — C7×SD32
C1C2C4C8C56C7×Q16 — C7×SD32
C1C2C4C8 — C7×SD32
C1C14C28C56 — C7×SD32

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

8C2
4C22
4C4
8C14
2D4
2Q8
4C28
4C2×C14
2C7×Q8
2C7×D4

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 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++++++
imageC1C2C2C2C7C14C14C14D4D8SD32C7×D4C7×D8C7×SD32
kernelC7×SD32C112C7×D8C7×Q16SD32C16D8Q16C28C14C7C4C2C1
# reps1111666612461224

Matrix representation of C7×SD32 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] >;

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

Export

Subgroup lattice of C7×SD32 in TeX

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