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## G = D7×C22⋊Q8order 448 = 26·7

### Direct product of D7 and C22⋊Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — D7×C22⋊Q8
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C23×D7 — D7×C22×C4 — D7×C22⋊Q8
 Lower central C7 — C2×C14 — D7×C22⋊Q8
 Upper central C1 — C22 — C22⋊Q8

Generators and relations for D7×C22⋊Q8
G = < a,b,c,d,e,f | a7=b2=c2=d2=e4=1, f2=e2, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=e-1 >

Subgroups: 1500 in 322 conjugacy classes, 121 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, Q8, C23, C23, D7, D7, C14, C14, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C24, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C2×C4⋊C4, C22⋊Q8, C22⋊Q8, C23×C4, C22×Q8, Dic14, C4×D7, C4×D7, C2×Dic7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C2×C28, C7×Q8, C22×D7, C22×D7, C22×D7, C22×C14, C2×C22⋊Q8, Dic7⋊C4, C4⋊Dic7, C4⋊Dic7, D14⋊C4, C23.D7, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C2×Dic14, C2×Dic14, C2×C4×D7, C2×C4×D7, C2×C4×D7, Q8×D7, C22×Dic7, C22×C28, Q8×C14, C23×D7, C22⋊Dic14, D7×C22⋊C4, D7×C4⋊C4, D7×C4⋊C4, D14⋊Q8, D142Q8, C28.48D4, D143Q8, C7×C22⋊Q8, D7×C22×C4, C2×Q8×D7, D7×C22⋊Q8
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, C4○D4, C24, D14, C22⋊Q8, C22×D4, C22×Q8, C2×C4○D4, C22×D7, C2×C22⋊Q8, D4×D7, Q8×D7, C23×D7, C2×D4×D7, C2×Q8×D7, D7×C4○D4, D7×C22⋊Q8

Smallest permutation representation of D7×C22⋊Q8
On 112 points
Generators in S112
(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 12)(2 11)(3 10)(4 9)(5 8)(6 14)(7 13)(15 24)(16 23)(17 22)(18 28)(19 27)(20 26)(21 25)(29 38)(30 37)(31 36)(32 42)(33 41)(34 40)(35 39)(43 52)(44 51)(45 50)(46 56)(47 55)(48 54)(49 53)(57 66)(58 65)(59 64)(60 70)(61 69)(62 68)(63 67)(71 80)(72 79)(73 78)(74 84)(75 83)(76 82)(77 81)(85 94)(86 93)(87 92)(88 98)(89 97)(90 96)(91 95)(99 108)(100 107)(101 106)(102 112)(103 111)(104 110)(105 109)
(29 50)(30 51)(31 52)(32 53)(33 54)(34 55)(35 56)(36 43)(37 44)(38 45)(39 46)(40 47)(41 48)(42 49)(85 106)(86 107)(87 108)(88 109)(89 110)(90 111)(91 112)(92 99)(93 100)(94 101)(95 102)(96 103)(97 104)(98 105)
(1 27)(2 28)(3 22)(4 23)(5 24)(6 25)(7 26)(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)(29 50)(30 51)(31 52)(32 53)(33 54)(34 55)(35 56)(36 43)(37 44)(38 45)(39 46)(40 47)(41 48)(42 49)(57 78)(58 79)(59 80)(60 81)(61 82)(62 83)(63 84)(64 71)(65 72)(66 73)(67 74)(68 75)(69 76)(70 77)(85 106)(86 107)(87 108)(88 109)(89 110)(90 111)(91 112)(92 99)(93 100)(94 101)(95 102)(96 103)(97 104)(98 105)
(1 76 20 62)(2 77 21 63)(3 71 15 57)(4 72 16 58)(5 73 17 59)(6 74 18 60)(7 75 19 61)(8 78 22 64)(9 79 23 65)(10 80 24 66)(11 81 25 67)(12 82 26 68)(13 83 27 69)(14 84 28 70)(29 92 43 106)(30 93 44 107)(31 94 45 108)(32 95 46 109)(33 96 47 110)(34 97 48 111)(35 98 49 112)(36 85 50 99)(37 86 51 100)(38 87 52 101)(39 88 53 102)(40 89 54 103)(41 90 55 104)(42 91 56 105)
(1 111 20 97)(2 112 21 98)(3 106 15 92)(4 107 16 93)(5 108 17 94)(6 109 18 95)(7 110 19 96)(8 99 22 85)(9 100 23 86)(10 101 24 87)(11 102 25 88)(12 103 26 89)(13 104 27 90)(14 105 28 91)(29 71 43 57)(30 72 44 58)(31 73 45 59)(32 74 46 60)(33 75 47 61)(34 76 48 62)(35 77 49 63)(36 78 50 64)(37 79 51 65)(38 80 52 66)(39 81 53 67)(40 82 54 68)(41 83 55 69)(42 84 56 70)

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

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

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

70 conjugacy classes

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

70 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + - + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 Q8 D7 C4○D4 D14 D14 D14 D14 D4×D7 Q8×D7 D7×C4○D4 kernel D7×C22⋊Q8 C22⋊Dic14 D7×C22⋊C4 D7×C4⋊C4 D14⋊Q8 D14⋊2Q8 C28.48D4 D14⋊3Q8 C7×C22⋊Q8 D7×C22×C4 C2×Q8×D7 C4×D7 C22×D7 C22⋊Q8 D14 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 C4 C22 C2 # reps 1 2 2 3 2 1 1 1 1 1 1 4 4 3 4 6 9 3 3 6 6 6

Matrix representation of D7×C22⋊Q8 in GL6(𝔽29)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 25 1 0 0 0 0 27 22 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 28 0 0 0 0 0 0 28 0 0 0 0 0 0 11 18 0 0 0 0 3 18 0 0 0 0 0 0 28 0 0 0 0 0 0 28
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 28
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 28 0 0 0 0 0 0 28
,
 9 11 0 0 0 0 11 20 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28
,
 2 22 0 0 0 0 9 27 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 0 28 0 0 0 0 28 0

G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,25,27,0,0,0,0,1,22,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,11,3,0,0,0,0,18,18,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[9,11,0,0,0,0,11,20,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[2,9,0,0,0,0,22,27,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,28,0,0,0,0,28,0] >;

D7×C22⋊Q8 in GAP, Magma, Sage, TeX

D_7\times C_2^2\rtimes Q_8
% in TeX

G:=Group("D7xC2^2:Q8");
// GroupNames label

G:=SmallGroup(448,1079);
// by ID

G=gap.SmallGroup(448,1079);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,100,794,297,136,18822]);
// Polycyclic

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

׿
×
𝔽