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## G = C7×C22.SD16order 448 = 26·7

### Direct product of C7 and C22.SD16

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C7×C22.SD16
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C22×C28 — C7×C2.C42 — C7×C22.SD16
 Lower central C1 — C22 — C2×C4 — C7×C22.SD16
 Upper central C1 — C2×C14 — C22×C28 — C7×C22.SD16

Generators and relations for C7×C22.SD16
G = < a,b,c,d,e | a7=b2=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=bcd3 >

Subgroups: 210 in 90 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, C23, C14, C14, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C28, C2×C14, C2×C14, C2.C42, C22⋊C8, C4⋊D4, C56, C2×C28, C2×C28, C7×D4, C22×C14, C22×C14, C22.SD16, C7×C22⋊C4, C7×C4⋊C4, C2×C56, C22×C28, C22×C28, D4×C14, D4×C14, C7×C2.C42, C7×C22⋊C8, C7×C4⋊D4, C7×C22.SD16
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C14, C22⋊C4, D8, SD16, C28, C2×C14, C23⋊C4, D4⋊C4, C4≀C2, C2×C28, C7×D4, C22.SD16, C7×C22⋊C4, C7×D8, C7×SD16, C7×C23⋊C4, C7×D4⋊C4, C7×C4≀C2, C7×C22.SD16

Smallest permutation representation of C7×C22.SD16
On 112 points
Generators in S112
(1 63 109 55 101 47 93)(2 64 110 56 102 48 94)(3 57 111 49 103 41 95)(4 58 112 50 104 42 96)(5 59 105 51 97 43 89)(6 60 106 52 98 44 90)(7 61 107 53 99 45 91)(8 62 108 54 100 46 92)(9 66 87 36 26 17 78)(10 67 88 37 27 18 79)(11 68 81 38 28 19 80)(12 69 82 39 29 20 73)(13 70 83 40 30 21 74)(14 71 84 33 31 22 75)(15 72 85 34 32 23 76)(16 65 86 35 25 24 77)
(1 5)(2 70)(3 7)(4 72)(6 66)(8 68)(9 90)(10 14)(11 92)(12 16)(13 94)(15 96)(17 98)(18 22)(19 100)(20 24)(21 102)(23 104)(25 29)(26 52)(27 31)(28 54)(30 56)(32 50)(33 37)(34 112)(35 39)(36 106)(38 108)(40 110)(41 45)(42 76)(43 47)(44 78)(46 80)(48 74)(49 53)(51 55)(57 61)(58 85)(59 63)(60 87)(62 81)(64 83)(65 69)(67 71)(73 77)(75 79)(82 86)(84 88)(89 93)(91 95)(97 101)(99 103)(105 109)(107 111)
(1 65)(2 66)(3 67)(4 68)(5 69)(6 70)(7 71)(8 72)(9 94)(10 95)(11 96)(12 89)(13 90)(14 91)(15 92)(16 93)(17 102)(18 103)(19 104)(20 97)(21 98)(22 99)(23 100)(24 101)(25 55)(26 56)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 107)(34 108)(35 109)(36 110)(37 111)(38 112)(39 105)(40 106)(41 79)(42 80)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(57 88)(58 81)(59 82)(60 83)(61 84)(62 85)(63 86)(64 87)
(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 6)(2 5)(3 68)(4 67)(7 72)(8 71)(9 12)(10 96)(11 95)(13 16)(14 92)(15 91)(17 20)(18 104)(19 103)(21 24)(22 100)(23 99)(25 30)(26 29)(27 50)(28 49)(31 54)(32 53)(33 108)(34 107)(35 40)(36 39)(37 112)(38 111)(41 80)(42 79)(43 48)(44 47)(45 76)(46 75)(51 56)(52 55)(57 81)(58 88)(59 64)(60 63)(61 85)(62 84)(65 70)(66 69)(73 78)(74 77)(82 87)(83 86)(89 94)(90 93)(97 102)(98 101)(105 110)(106 109)

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

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

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

133 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C ··· 4G 4H 7A ··· 7F 8A 8B 8C 8D 14A ··· 14R 14S ··· 14AD 14AE ··· 14AJ 28A ··· 28L 28M ··· 28AP 28AQ ··· 28AV 56A ··· 56X order 1 2 2 2 2 2 2 4 4 4 ··· 4 4 7 ··· 7 8 8 8 8 14 ··· 14 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 1 1 2 2 8 2 2 4 ··· 4 8 1 ··· 1 4 4 4 4 1 ··· 1 2 ··· 2 8 ··· 8 2 ··· 2 4 ··· 4 8 ··· 8 4 ··· 4

133 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + image C1 C2 C2 C2 C4 C4 C7 C14 C14 C14 C28 C28 D4 D4 D8 SD16 C4≀C2 C7×D4 C7×D4 C7×D8 C7×SD16 C7×C4≀C2 C23⋊C4 C7×C23⋊C4 kernel C7×C22.SD16 C7×C2.C42 C7×C22⋊C8 C7×C4⋊D4 C7×C4⋊C4 D4×C14 C22.SD16 C2.C42 C22⋊C8 C4⋊D4 C4⋊C4 C2×D4 C2×C28 C22×C14 C2×C14 C2×C14 C14 C2×C4 C23 C22 C22 C2 C14 C2 # reps 1 1 1 1 2 2 6 6 6 6 12 12 1 1 2 2 4 6 6 12 12 24 1 6

Matrix representation of C7×C22.SD16 in GL4(𝔽113) generated by

 109 0 0 0 0 109 0 0 0 0 1 0 0 0 0 1
,
 112 0 0 0 0 112 0 0 0 0 112 0 0 0 88 1
,
 1 0 0 0 0 1 0 0 0 0 112 0 0 0 0 112
,
 31 82 0 0 31 31 0 0 0 0 18 98 0 0 0 95
,
 82 82 0 0 82 31 0 0 0 0 69 112 0 0 14 44
G:=sub<GL(4,GF(113))| [109,0,0,0,0,109,0,0,0,0,1,0,0,0,0,1],[112,0,0,0,0,112,0,0,0,0,112,88,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,112,0,0,0,0,112],[31,31,0,0,82,31,0,0,0,0,18,0,0,0,98,95],[82,82,0,0,82,31,0,0,0,0,69,14,0,0,112,44] >;

C7×C22.SD16 in GAP, Magma, Sage, TeX

C_7\times C_2^2.{\rm SD}_{16}
% in TeX

G:=Group("C7xC2^2.SD16");
// GroupNames label

G:=SmallGroup(448,131);
// by ID

G=gap.SmallGroup(448,131);
# by ID

G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,3923,3538,248,6871]);
// Polycyclic

G:=Group<a,b,c,d,e|a^7=b^2=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e=b*c*d^3>;
// generators/relations

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