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## G = C7×D8⋊C22order 448 = 26·7

### Direct product of C7 and D8⋊C22

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C7×D8⋊C22
 Chief series C1 — C2 — C4 — C28 — C7×D4 — C7×D8 — C7×C8⋊C22 — C7×D8⋊C22
 Lower central C1 — C2 — C4 — C7×D8⋊C22
 Upper central C1 — C28 — C22×C28 — C7×D8⋊C22

Generators and relations for C7×D8⋊C22
G = < a,b,c,d,e | a7=b8=c2=d2=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, dbd=b5, be=eb, dcd=ece=b4c, de=ed >

Subgroups: 402 in 262 conjugacy classes, 158 normal (22 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C14, C14, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C28, C28, C28, C2×C14, C2×C14, C2×C14, C2×M4(2), C4○D8, C8⋊C22, C8.C22, C2×C4○D4, C56, C2×C28, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C22×C14, C22×C14, D8⋊C22, C2×C56, C7×M4(2), C7×D8, C7×SD16, C7×Q16, C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C7×C4○D4, C7×C4○D4, C14×M4(2), C7×C4○D8, C7×C8⋊C22, C7×C8.C22, C14×C4○D4, C7×D8⋊C22
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C24, C2×C14, C22×D4, C7×D4, C22×C14, D8⋊C22, D4×C14, C23×C14, D4×C2×C14, C7×D8⋊C22

Smallest permutation representation of C7×D8⋊C22
On 112 points
Generators in S112
(1 79 42 22 91 71 34)(2 80 43 23 92 72 35)(3 73 44 24 93 65 36)(4 74 45 17 94 66 37)(5 75 46 18 95 67 38)(6 76 47 19 96 68 39)(7 77 48 20 89 69 40)(8 78 41 21 90 70 33)(9 111 83 54 26 103 58)(10 112 84 55 27 104 59)(11 105 85 56 28 97 60)(12 106 86 49 29 98 61)(13 107 87 50 30 99 62)(14 108 88 51 31 100 63)(15 109 81 52 32 101 64)(16 110 82 53 25 102 57)
(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 8)(2 7)(3 6)(4 5)(9 16)(10 15)(11 14)(12 13)(17 18)(19 24)(20 23)(21 22)(25 26)(27 32)(28 31)(29 30)(33 34)(35 40)(36 39)(37 38)(41 42)(43 48)(44 47)(45 46)(49 50)(51 56)(52 55)(53 54)(57 58)(59 64)(60 63)(61 62)(65 68)(66 67)(69 72)(70 71)(73 76)(74 75)(77 80)(78 79)(81 84)(82 83)(85 88)(86 87)(89 92)(90 91)(93 96)(94 95)(97 100)(98 99)(101 104)(102 103)(105 108)(106 107)(109 112)(110 111)
(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)(25 29)(27 31)(33 37)(35 39)(41 45)(43 47)(49 53)(51 55)(57 61)(59 63)(66 70)(68 72)(74 78)(76 80)(82 86)(84 88)(90 94)(92 96)(98 102)(100 104)(106 110)(108 112)
(1 97)(2 98)(3 99)(4 100)(5 101)(6 102)(7 103)(8 104)(9 48)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(17 108)(18 109)(19 110)(20 111)(21 112)(22 105)(23 106)(24 107)(25 39)(26 40)(27 33)(28 34)(29 35)(30 36)(31 37)(32 38)(49 72)(50 65)(51 66)(52 67)(53 68)(54 69)(55 70)(56 71)(57 76)(58 77)(59 78)(60 79)(61 80)(62 73)(63 74)(64 75)(81 95)(82 96)(83 89)(84 90)(85 91)(86 92)(87 93)(88 94)

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

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

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

154 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C 4D 4E 4F 4G 4H 4I 7A ··· 7F 8A 8B 8C 8D 14A ··· 14F 14G ··· 14X 14Y ··· 14AV 28A ··· 28L 28M ··· 28AD 28AE ··· 28BB 56A ··· 56X order 1 2 2 2 2 2 2 2 2 4 4 4 4 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 2 2 2 4 4 4 4 1 1 2 2 2 4 4 4 4 1 ··· 1 4 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4 4 ··· 4

154 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C7 C14 C14 C14 C14 C14 D4 D4 C7×D4 C7×D4 D8⋊C22 C7×D8⋊C22 kernel C7×D8⋊C22 C14×M4(2) C7×C4○D8 C7×C8⋊C22 C7×C8.C22 C14×C4○D4 D8⋊C22 C2×M4(2) C4○D8 C8⋊C22 C8.C22 C2×C4○D4 C2×C28 C22×C14 C2×C4 C23 C7 C1 # reps 1 1 4 4 4 2 6 6 24 24 24 12 3 1 18 6 2 12

Matrix representation of C7×D8⋊C22 in GL4(𝔽113) generated by

 109 0 0 0 0 109 0 0 0 0 109 0 0 0 0 109
,
 20 0 111 0 20 0 112 112 30 1 93 0 30 0 93 0
,
 20 0 111 0 0 0 112 1 30 0 93 0 30 1 93 0
,
 1 0 0 0 0 1 0 0 20 0 112 0 20 0 0 112
,
 15 83 0 0 15 98 0 0 74 39 0 98 0 39 15 0
G:=sub<GL(4,GF(113))| [109,0,0,0,0,109,0,0,0,0,109,0,0,0,0,109],[20,20,30,30,0,0,1,0,111,112,93,93,0,112,0,0],[20,0,30,30,0,0,0,1,111,112,93,93,0,1,0,0],[1,0,20,20,0,1,0,0,0,0,112,0,0,0,0,112],[15,15,74,0,83,98,39,39,0,0,0,15,0,0,98,0] >;

C7×D8⋊C22 in GAP, Magma, Sage, TeX

C_7\times D_8\rtimes C_2^2
% in TeX

G:=Group("C7xD8:C2^2");
// GroupNames label

G:=SmallGroup(448,1358);
// by ID

G=gap.SmallGroup(448,1358);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1597,4790,808,14117,7068,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^7=b^8=c^2=d^2=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,d*b*d=b^5,b*e=e*b,d*c*d=e*c*e=b^4*c,d*e=e*d>;
// generators/relations

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