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## G = (C22×D7)⋊C8order 448 = 26·7

### The semidirect product of C22×D7 and C8 acting via C8/C2=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — (C22×D7)⋊C8
 Chief series C1 — C7 — C14 — C2×C14 — C2×C28 — C22×C28 — C2×D14⋊C4 — (C22×D7)⋊C8
 Lower central C7 — C14 — C2×C14 — (C22×D7)⋊C8
 Upper central C1 — C22 — C22×C4 — C22⋊C8

Generators and relations for (C22×D7)⋊C8
G = < a,b,c,d,e | a2=b2=c7=d2=e8=1, eae-1=ab=ba, ac=ca, ede-1=ad=da, bc=cb, bd=db, be=eb, dcd=c-1, ce=ec >

Subgroups: 668 in 98 conjugacy classes, 31 normal (29 characteristic)
C1, C2, C2, C4, C22, C22, C7, C8, C2×C4, C2×C4, C23, C23, D7, C14, C14, C22⋊C4, C2×C8, C22×C4, C22×C4, C24, Dic7, C28, D14, C2×C14, C2×C14, C22⋊C8, C22⋊C8, C2×C22⋊C4, C7⋊C8, C56, C2×Dic7, C2×C28, C2×C28, C22×D7, C22×D7, C22×C14, C23⋊C8, C2×C7⋊C8, D14⋊C4, C2×C56, C22×Dic7, C22×C28, C23×D7, C28.55D4, C7×C22⋊C8, C2×D14⋊C4, (C22×D7)⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, D7, C22⋊C4, C2×C8, M4(2), D14, C22⋊C8, C23⋊C4, C4.D4, C4×D7, D28, C7⋊D4, C23⋊C8, C8×D7, C8⋊D7, D14⋊C4, C23.1D14, D14⋊C8, C28.46D4, (C22×D7)⋊C8

Smallest permutation representation of (C22×D7)⋊C8
On 112 points
Generators in S112
(1 62)(2 6)(3 64)(4 8)(5 58)(7 60)(9 13)(10 55)(11 15)(12 49)(14 51)(16 53)(17 30)(18 22)(19 32)(20 24)(21 26)(23 28)(25 29)(27 31)(33 42)(34 38)(35 44)(36 40)(37 46)(39 48)(41 45)(43 47)(50 54)(52 56)(57 61)(59 63)(65 69)(66 107)(67 71)(68 109)(70 111)(72 105)(73 88)(74 78)(75 82)(76 80)(77 84)(79 86)(81 85)(83 87)(89 93)(90 104)(91 95)(92 98)(94 100)(96 102)(97 101)(99 103)(106 110)(108 112)
(1 58)(2 59)(3 60)(4 61)(5 62)(6 63)(7 64)(8 57)(9 50)(10 51)(11 52)(12 53)(13 54)(14 55)(15 56)(16 49)(17 26)(18 27)(19 28)(20 29)(21 30)(22 31)(23 32)(24 25)(33 46)(34 47)(35 48)(36 41)(37 42)(38 43)(39 44)(40 45)(65 110)(66 111)(67 112)(68 105)(69 106)(70 107)(71 108)(72 109)(73 84)(74 85)(75 86)(76 87)(77 88)(78 81)(79 82)(80 83)(89 99)(90 100)(91 101)(92 102)(93 103)(94 104)(95 97)(96 98)
(1 26 79 10 111 42 96)(2 27 80 11 112 43 89)(3 28 73 12 105 44 90)(4 29 74 13 106 45 91)(5 30 75 14 107 46 92)(6 31 76 15 108 47 93)(7 32 77 16 109 48 94)(8 25 78 9 110 41 95)(17 82 51 66 37 98 58)(18 83 52 67 38 99 59)(19 84 53 68 39 100 60)(20 85 54 69 40 101 61)(21 86 55 70 33 102 62)(22 87 56 71 34 103 63)(23 88 49 72 35 104 64)(24 81 50 65 36 97 57)
(1 98)(2 93)(3 90)(4 97)(5 102)(6 89)(7 94)(8 101)(9 54)(10 51)(11 15)(13 50)(14 55)(17 42)(18 34)(19 39)(20 41)(21 46)(22 38)(23 35)(24 45)(25 40)(26 37)(27 47)(28 44)(29 36)(30 33)(31 43)(32 48)(52 56)(57 91)(58 96)(59 103)(60 100)(61 95)(62 92)(63 99)(64 104)(65 74)(66 79)(67 87)(68 84)(69 78)(70 75)(71 83)(72 88)(73 105)(76 112)(77 109)(80 108)(81 106)(82 111)(85 110)(86 107)
(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)

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

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

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

82 conjugacy classes

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

82 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C4 C4 C8 D4 D7 M4(2) D14 D28 C7⋊D4 C4×D7 C8×D7 C8⋊D7 C23⋊C4 C4.D4 C23.1D14 C28.46D4 kernel (C22×D7)⋊C8 C28.55D4 C7×C22⋊C8 C2×D14⋊C4 C22×Dic7 C23×D7 C22×D7 C2×C28 C22⋊C8 C2×C14 C22×C4 C2×C4 C2×C4 C23 C22 C22 C14 C14 C2 C2 # reps 1 1 1 1 2 2 8 2 3 2 3 6 6 6 12 12 1 1 6 6

Matrix representation of (C22×D7)⋊C8 in GL6(𝔽113)

 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 13 10 112 0 0 0 44 26 0 112
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 112 0 0 0 0 0 0 112 0 0 0 0 0 0 112 0 0 0 0 0 0 112
,
 12 14 0 0 0 0 99 12 0 0 0 0 0 0 103 112 0 0 0 0 2 34 0 0 0 0 37 0 24 112 0 0 82 76 1 0
,
 14 101 0 0 0 0 101 99 0 0 0 0 0 0 11 10 0 0 0 0 101 102 0 0 0 0 47 16 24 112 0 0 97 0 10 89
,
 44 0 0 0 0 0 0 44 0 0 0 0 0 0 80 37 0 58 0 0 29 33 55 98 0 0 95 59 49 76 0 0 61 70 37 64

G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,13,44,0,0,0,1,10,26,0,0,0,0,112,0,0,0,0,0,0,112],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[12,99,0,0,0,0,14,12,0,0,0,0,0,0,103,2,37,82,0,0,112,34,0,76,0,0,0,0,24,1,0,0,0,0,112,0],[14,101,0,0,0,0,101,99,0,0,0,0,0,0,11,101,47,97,0,0,10,102,16,0,0,0,0,0,24,10,0,0,0,0,112,89],[44,0,0,0,0,0,0,44,0,0,0,0,0,0,80,29,95,61,0,0,37,33,59,70,0,0,0,55,49,37,0,0,58,98,76,64] >;

(C22×D7)⋊C8 in GAP, Magma, Sage, TeX

(C_2^2\times D_7)\rtimes C_8
% in TeX

G:=Group("(C2^2xD7):C8");
// GroupNames label

G:=SmallGroup(448,25);
// by ID

G=gap.SmallGroup(448,25);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,141,36,758,100,570,18822]);
// Polycyclic

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

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