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## G = C2×C8⋊D7order 224 = 25·7

### Direct product of C2 and C8⋊D7

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×C8⋊D7
G = < a,b,c,d | a2=b8=c7=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >

Subgroups: 238 in 68 conjugacy classes, 41 normal (19 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, C23, D7, C14, C14, C2×C8, C2×C8, M4(2), C22×C4, Dic7, C28, D14, D14, C2×C14, C2×M4(2), C7⋊C8, C56, C4×D7, C2×Dic7, C2×C28, C22×D7, C8⋊D7, C2×C7⋊C8, C2×C56, C2×C4×D7, C2×C8⋊D7
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, M4(2), C22×C4, D14, C2×M4(2), C4×D7, C22×D7, C8⋊D7, C2×C4×D7, C2×C8⋊D7

Smallest permutation representation of C2×C8⋊D7
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 62)(18 63)(19 64)(20 57)(21 58)(22 59)(23 60)(24 61)(41 103)(42 104)(43 97)(44 98)(45 99)(46 100)(47 101)(48 102)(49 89)(50 90)(51 91)(52 92)(53 93)(54 94)(55 95)(56 96)(65 84)(66 85)(67 86)(68 87)(69 88)(70 81)(71 82)(72 83)(73 107)(74 108)(75 109)(76 110)(77 111)(78 112)(79 105)(80 106)
(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 47 50 86 19 10 105)(2 48 51 87 20 11 106)(3 41 52 88 21 12 107)(4 42 53 81 22 13 108)(5 43 54 82 23 14 109)(6 44 55 83 24 15 110)(7 45 56 84 17 16 111)(8 46 49 85 18 9 112)(25 102 91 68 57 34 80)(26 103 92 69 58 35 73)(27 104 93 70 59 36 74)(28 97 94 71 60 37 75)(29 98 95 72 61 38 76)(30 99 96 65 62 39 77)(31 100 89 66 63 40 78)(32 101 90 67 64 33 79)
(1 105)(2 110)(3 107)(4 112)(5 109)(6 106)(7 111)(8 108)(9 42)(10 47)(11 44)(12 41)(13 46)(14 43)(15 48)(16 45)(17 56)(18 53)(19 50)(20 55)(21 52)(22 49)(23 54)(24 51)(25 76)(26 73)(27 78)(28 75)(29 80)(30 77)(31 74)(32 79)(33 101)(34 98)(35 103)(36 100)(37 97)(38 102)(39 99)(40 104)(57 95)(58 92)(59 89)(60 94)(61 91)(62 96)(63 93)(64 90)(66 70)(68 72)(81 85)(83 87)

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

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

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

68 conjugacy classes

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

68 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 D7 M4(2) D14 D14 C4×D7 C4×D7 C8⋊D7 kernel C2×C8⋊D7 C8⋊D7 C2×C7⋊C8 C2×C56 C2×C4×D7 C4×D7 C2×Dic7 C22×D7 C2×C8 C14 C8 C2×C4 C4 C22 C2 # reps 1 4 1 1 1 4 2 2 3 4 6 3 6 6 24

Matrix representation of C2×C8⋊D7 in GL4(𝔽113) generated by

 112 0 0 0 0 112 0 0 0 0 1 0 0 0 0 1
,
 98 0 0 0 0 98 0 0 0 0 49 76 0 0 37 64
,
 9 112 0 0 1 0 0 0 0 0 79 112 0 0 1 0
,
 9 112 0 0 80 104 0 0 0 0 34 1 0 0 88 79
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[98,0,0,0,0,98,0,0,0,0,49,37,0,0,76,64],[9,1,0,0,112,0,0,0,0,0,79,1,0,0,112,0],[9,80,0,0,112,104,0,0,0,0,34,88,0,0,1,79] >;

C2×C8⋊D7 in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes D_7
% in TeX

G:=Group("C2xC8:D7");
// GroupNames label

G:=SmallGroup(224,95);
// by ID

G=gap.SmallGroup(224,95);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,362,50,69,6917]);
// Polycyclic

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

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