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## G = C7×C8○D4order 224 = 25·7

### Direct product of C7 and C8○D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C7×C8○D4
 Chief series C1 — C2 — C4 — C28 — C56 — C2×C56 — C7×C8○D4
 Lower central C1 — C2 — C7×C8○D4
 Upper central C1 — C56 — C7×C8○D4

Generators and relations for C7×C8○D4
G = < a,b,c,d | a7=b8=d2=1, c2=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c >

Subgroups: 68 in 62 conjugacy classes, 56 normal (14 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C8, C2×C4, D4, Q8, C14, C14, C2×C8, M4(2), C4○D4, C28, C28, C2×C14, C8○D4, C56, C56, C2×C28, C7×D4, C7×Q8, C2×C56, C7×M4(2), C7×C4○D4, C7×C8○D4
Quotients: C1, C2, C4, C22, C7, C2×C4, C23, C14, C22×C4, C28, C2×C14, C8○D4, C2×C28, C22×C14, C22×C28, C7×C8○D4

Smallest permutation representation of C7×C8○D4
On 112 points
Generators in S112
(1 31 47 79 95 23 39)(2 32 48 80 96 24 40)(3 25 41 73 89 17 33)(4 26 42 74 90 18 34)(5 27 43 75 91 19 35)(6 28 44 76 92 20 36)(7 29 45 77 93 21 37)(8 30 46 78 94 22 38)(9 106 66 50 82 98 62)(10 107 67 51 83 99 63)(11 108 68 52 84 100 64)(12 109 69 53 85 101 57)(13 110 70 54 86 102 58)(14 111 71 55 87 103 59)(15 112 72 56 88 104 60)(16 105 65 49 81 97 61)
(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 99 5 103)(2 100 6 104)(3 101 7 97)(4 102 8 98)(9 42 13 46)(10 43 14 47)(11 44 15 48)(12 45 16 41)(17 53 21 49)(18 54 22 50)(19 55 23 51)(20 56 24 52)(25 57 29 61)(26 58 30 62)(27 59 31 63)(28 60 32 64)(33 85 37 81)(34 86 38 82)(35 87 39 83)(36 88 40 84)(65 89 69 93)(66 90 70 94)(67 91 71 95)(68 92 72 96)(73 109 77 105)(74 110 78 106)(75 111 79 107)(76 112 80 108)
(1 103)(2 104)(3 97)(4 98)(5 99)(6 100)(7 101)(8 102)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 41)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(25 61)(26 62)(27 63)(28 64)(29 57)(30 58)(31 59)(32 60)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(65 89)(66 90)(67 91)(68 92)(69 93)(70 94)(71 95)(72 96)(73 105)(74 106)(75 107)(76 108)(77 109)(78 110)(79 111)(80 112)

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

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

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

C7×C8○D4 is a maximal subgroup of
C56.92D4  C56.70C23  D4.3D28  D4.4D28  D4.5D28  C56.93D4  C56.50D4  C56.49C23  D4.11D28  D4.12D28  D4.13D28
C7×C8○D4 is a maximal quotient of
D4×C56  Q8×C56

140 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 7A ··· 7F 8A 8B 8C 8D 8E ··· 8J 14A ··· 14F 14G ··· 14X 28A ··· 28L 28M ··· 28AD 56A ··· 56X 56Y ··· 56BH order 1 2 2 2 2 4 4 4 4 4 7 ··· 7 8 8 8 8 8 ··· 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 56 ··· 56 size 1 1 2 2 2 1 1 2 2 2 1 ··· 1 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

140 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C2 C4 C4 C7 C14 C14 C14 C28 C28 C8○D4 C7×C8○D4 kernel C7×C8○D4 C2×C56 C7×M4(2) C7×C4○D4 C7×D4 C7×Q8 C8○D4 C2×C8 M4(2) C4○D4 D4 Q8 C7 C1 # reps 1 3 3 1 6 2 6 18 18 6 36 12 4 24

Matrix representation of C7×C8○D4 in GL3(𝔽113) generated by

 106 0 0 0 1 0 0 0 1
,
 1 0 0 0 44 0 0 0 44
,
 1 0 0 0 101 111 0 16 12
,
 112 0 0 0 101 111 0 15 12
G:=sub<GL(3,GF(113))| [106,0,0,0,1,0,0,0,1],[1,0,0,0,44,0,0,0,44],[1,0,0,0,101,16,0,111,12],[112,0,0,0,101,15,0,111,12] >;

C7×C8○D4 in GAP, Magma, Sage, TeX

C_7\times C_8\circ D_4
% in TeX

G:=Group("C7xC8oD4");
// GroupNames label

G:=SmallGroup(224,166);
// by ID

G=gap.SmallGroup(224,166);
# by ID

G:=PCGroup([6,-2,-2,-2,-7,-2,-2,336,1052,88]);
// Polycyclic

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

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