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## G = C2×D7×SD16order 448 = 26·7

### Direct product of C2, D7 and SD16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C2×D7×SD16
 Chief series C1 — C7 — C14 — C28 — C4×D7 — C2×C4×D7 — C2×D4×D7 — C2×D7×SD16
 Lower central C7 — C14 — C28 — C2×D7×SD16
 Upper central C1 — C22 — C2×C4 — C2×SD16

Generators and relations for C2×D7×SD16
G = < a,b,c,d,e | a2=b7=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d3 >

Subgroups: 1668 in 298 conjugacy classes, 111 normal (33 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D7, D7, C14, C14, C14, C2×C8, C2×C8, SD16, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C22×C8, C2×SD16, C2×SD16, C22×D4, C22×Q8, C7⋊C8, C56, Dic14, Dic14, C4×D7, C4×D7, D28, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C22×D7, C22×D7, C22×C14, C22×SD16, C8×D7, C56⋊C2, C2×C7⋊C8, D4.D7, Q8⋊D7, C2×C56, C7×SD16, C2×Dic14, C2×Dic14, C2×C4×D7, C2×C4×D7, C2×D28, D4×D7, D4×D7, Q8×D7, Q8×D7, C2×C7⋊D4, D4×C14, Q8×C14, C23×D7, D7×C2×C8, C2×C56⋊C2, D7×SD16, C2×D4.D7, C2×Q8⋊D7, C14×SD16, C2×D4×D7, C2×Q8×D7, C2×D7×SD16
Quotients: C1, C2, C22, D4, C23, D7, SD16, C2×D4, C24, D14, C2×SD16, C22×D4, C22×D7, C22×SD16, D4×D7, C23×D7, D7×SD16, C2×D4×D7, C2×D7×SD16

Smallest permutation representation of C2×D7×SD16
On 112 points
Generators in S112
(1 32)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 40)(10 33)(11 34)(12 35)(13 36)(14 37)(15 38)(16 39)(17 69)(18 70)(19 71)(20 72)(21 65)(22 66)(23 67)(24 68)(41 110)(42 111)(43 112)(44 105)(45 106)(46 107)(47 108)(48 109)(49 89)(50 90)(51 91)(52 92)(53 93)(54 94)(55 95)(56 96)(57 83)(58 84)(59 85)(60 86)(61 87)(62 88)(63 81)(64 82)(73 100)(74 101)(75 102)(76 103)(77 104)(78 97)(79 98)(80 99)
(1 102 14 86 111 68 50)(2 103 15 87 112 69 51)(3 104 16 88 105 70 52)(4 97 9 81 106 71 53)(5 98 10 82 107 72 54)(6 99 11 83 108 65 55)(7 100 12 84 109 66 56)(8 101 13 85 110 67 49)(17 91 25 76 38 61 43)(18 92 26 77 39 62 44)(19 93 27 78 40 63 45)(20 94 28 79 33 64 46)(21 95 29 80 34 57 47)(22 96 30 73 35 58 48)(23 89 31 74 36 59 41)(24 90 32 75 37 60 42)
(1 54)(2 55)(3 56)(4 49)(5 50)(6 51)(7 52)(8 53)(9 110)(10 111)(11 112)(12 105)(13 106)(14 107)(15 108)(16 109)(17 80)(18 73)(19 74)(20 75)(21 76)(22 77)(23 78)(24 79)(25 95)(26 96)(27 89)(28 90)(29 91)(30 92)(31 93)(32 94)(33 42)(34 43)(35 44)(36 45)(37 46)(38 47)(39 48)(40 41)(57 61)(58 62)(59 63)(60 64)(65 103)(66 104)(67 97)(68 98)(69 99)(70 100)(71 101)(72 102)(81 85)(82 86)(83 87)(84 88)
(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 5)(2 8)(4 6)(9 11)(10 14)(13 15)(17 23)(19 21)(20 24)(25 31)(27 29)(28 32)(33 37)(34 40)(36 38)(41 43)(42 46)(45 47)(49 51)(50 54)(53 55)(57 63)(59 61)(60 64)(65 71)(67 69)(68 72)(74 76)(75 79)(78 80)(81 83)(82 86)(85 87)(89 91)(90 94)(93 95)(97 99)(98 102)(101 103)(106 108)(107 111)(110 112)

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

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

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

70 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 4G 4H 7A 7B 7C 8A 8B 8C 8D 8E 8F 8G 8H 14A ··· 14I 14J ··· 14O 28A ··· 28F 28G ··· 28L 56A ··· 56L order 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 7 7 7 8 8 8 8 8 8 8 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 1 1 4 4 7 7 7 7 28 28 2 2 4 4 14 14 28 28 2 2 2 2 2 2 2 14 14 14 14 2 ··· 2 8 ··· 8 4 ··· 4 8 ··· 8 4 ··· 4

70 irreducible representations

 dim 1 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 C2 D4 D4 D4 D7 SD16 D14 D14 D14 D14 D4×D7 D4×D7 D7×SD16 kernel C2×D7×SD16 D7×C2×C8 C2×C56⋊C2 D7×SD16 C2×D4.D7 C2×Q8⋊D7 C14×SD16 C2×D4×D7 C2×Q8×D7 C4×D7 C2×Dic7 C22×D7 C2×SD16 D14 C2×C8 SD16 C2×D4 C2×Q8 C4 C22 C2 # reps 1 1 1 8 1 1 1 1 1 2 1 1 3 8 3 12 3 3 3 3 12

Matrix representation of C2×D7×SD16 in GL4(𝔽113) generated by

 112 0 0 0 0 112 0 0 0 0 112 0 0 0 0 112
,
 0 1 0 0 112 79 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 112 0 0 0 0 112
,
 112 0 0 0 0 112 0 0 0 0 100 100 0 0 13 100
,
 1 0 0 0 0 1 0 0 0 0 112 0 0 0 0 1
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,112,0,0,0,0,112],[0,112,0,0,1,79,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,112,0,0,0,0,112],[112,0,0,0,0,112,0,0,0,0,100,13,0,0,100,100],[1,0,0,0,0,1,0,0,0,0,112,0,0,0,0,1] >;

C2×D7×SD16 in GAP, Magma, Sage, TeX

C_2\times D_7\times {\rm SD}_{16}
% in TeX

G:=Group("C2xD7xSD16");
// GroupNames label

G:=SmallGroup(448,1211);
// by ID

G=gap.SmallGroup(448,1211);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,185,136,438,235,102,18822]);
// Polycyclic

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

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