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## G = C7×C4○D8order 224 = 25·7

### Direct product of C7 and C4○D8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C7×C4○D8
 Chief series C1 — C2 — C4 — C28 — C7×D4 — C7×D8 — C7×C4○D8
 Lower central C1 — C2 — C4 — C7×C4○D8
 Upper central C1 — C28 — C2×C28 — C7×C4○D8

Generators and relations for C7×C4○D8
G = < a,b,c,d | a7=b4=d2=1, c4=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c3 >

Subgroups: 92 in 62 conjugacy classes, 40 normal (24 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C22, C22 [×2], C7, C8 [×2], C2×C4, C2×C4 [×2], D4 [×2], D4 [×2], Q8 [×2], C14, C14 [×3], C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], C28 [×2], C28 [×2], C2×C14, C2×C14 [×2], C4○D8, C56 [×2], C2×C28, C2×C28 [×2], C7×D4 [×2], C7×D4 [×2], C7×Q8 [×2], C2×C56, C7×D8, C7×SD16 [×2], C7×Q16, C7×C4○D4 [×2], C7×C4○D8
Quotients: C1, C2 [×7], C22 [×7], C7, D4 [×2], C23, C14 [×7], C2×D4, C2×C14 [×7], C4○D8, C7×D4 [×2], C22×C14, D4×C14, C7×C4○D8

Smallest permutation representation of C7×C4○D8
On 112 points
Generators in S112
(1 73 43 23 93 65 35)(2 74 44 24 94 66 36)(3 75 45 17 95 67 37)(4 76 46 18 96 68 38)(5 77 47 19 89 69 39)(6 78 48 20 90 70 40)(7 79 41 21 91 71 33)(8 80 42 22 92 72 34)(9 105 85 55 27 97 58)(10 106 86 56 28 98 59)(11 107 87 49 29 99 60)(12 108 88 50 30 100 61)(13 109 81 51 31 101 62)(14 110 82 52 32 102 63)(15 111 83 53 25 103 64)(16 112 84 54 26 104 57)
(1 97 5 101)(2 98 6 102)(3 99 7 103)(4 100 8 104)(9 47 13 43)(10 48 14 44)(11 41 15 45)(12 42 16 46)(17 107 21 111)(18 108 22 112)(19 109 23 105)(20 110 24 106)(25 37 29 33)(26 38 30 34)(27 39 31 35)(28 40 32 36)(49 71 53 67)(50 72 54 68)(51 65 55 69)(52 66 56 70)(57 76 61 80)(58 77 62 73)(59 78 63 74)(60 79 64 75)(81 93 85 89)(82 94 86 90)(83 95 87 91)(84 96 88 92)
(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)
(2 8)(3 7)(4 6)(10 16)(11 15)(12 14)(17 21)(18 20)(22 24)(25 29)(26 28)(30 32)(33 37)(34 36)(38 40)(41 45)(42 44)(46 48)(49 53)(50 52)(54 56)(57 59)(60 64)(61 63)(66 72)(67 71)(68 70)(74 80)(75 79)(76 78)(82 88)(83 87)(84 86)(90 96)(91 95)(92 94)(98 104)(99 103)(100 102)(106 112)(107 111)(108 110)

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

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

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

C7×C4○D8 is a maximal subgroup of
D82Dic7  C28.58D8  Q16⋊D14  C56.30C23  C56.31C23  D85Dic7  D84Dic7  D810D14  D815D14  D811D14  D8.10D14
C7×C4○D8 is a maximal quotient of
D8×C28  SD16×C28  Q16×C28

98 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 7A ··· 7F 8A 8B 8C 8D 14A ··· 14F 14G ··· 14L 14M ··· 14X 28A ··· 28L 28M ··· 28R 28S ··· 28AD 56A ··· 56X order 1 2 2 2 2 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 4 4 1 1 2 4 4 1 ··· 1 2 2 2 2 1 ··· 1 2 ··· 2 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2

98 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C7 C14 C14 C14 C14 C14 D4 D4 C4○D8 C7×D4 C7×D4 C7×C4○D8 kernel C7×C4○D8 C2×C56 C7×D8 C7×SD16 C7×Q16 C7×C4○D4 C4○D8 C2×C8 D8 SD16 Q16 C4○D4 C28 C2×C14 C7 C4 C22 C1 # reps 1 1 1 2 1 2 6 6 6 12 6 12 1 1 4 6 6 24

Matrix representation of C7×C4○D8 in GL2(𝔽113) generated by

 16 0 0 16
,
 98 0 0 98
,
 51 82 62 0
,
 1 1 0 112
G:=sub<GL(2,GF(113))| [16,0,0,16],[98,0,0,98],[51,62,82,0],[1,0,1,112] >;

C7×C4○D8 in GAP, Magma, Sage, TeX

C_7\times C_4\circ D_8
% in TeX

G:=Group("C7xC4oD8");
// GroupNames label

G:=SmallGroup(224,170);
// by ID

G=gap.SmallGroup(224,170);
# by ID

G:=PCGroup([6,-2,-2,-2,-7,-2,-2,697,518,5044,2530,88]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^4=d^2=1,c^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^3>;
// generators/relations

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