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## G = (C7×D4).31D4order 448 = 26·7

### 1st non-split extension by C7×D4 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — (C7×D4).31D4
 Chief series C1 — C7 — C14 — C2×C14 — C2×C28 — C2×Dic14 — C2×D4.D7 — (C7×D4).31D4
 Lower central C7 — C14 — C2×C28 — (C7×D4).31D4
 Upper central C1 — C22 — C22×C4 — C22×D4

Generators and relations for (C7×D4).31D4
G = < a,b,c,d,e | a7=b4=c2=1, d4=e2=b2, ab=ba, ac=ca, dad-1=eae-1=a-1, cbc=ebe-1=b-1, bd=db, dcd-1=bc, ece-1=b-1c, ede-1=d3 >

Subgroups: 660 in 188 conjugacy classes, 51 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C14, C14, C22⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C24, Dic7, C28, C28, C2×C14, C2×C14, C2×C14, C22⋊C8, D4⋊C4, C22⋊Q8, C2×SD16, C22×D4, C7⋊C8, Dic14, C2×Dic7, C2×C28, C2×C28, C7×D4, C7×D4, C22×C14, C22×C14, C22⋊SD16, C2×C7⋊C8, Dic7⋊C4, C4⋊Dic7, D4.D7, C23.D7, C2×Dic14, C22×C28, D4×C14, D4×C14, C23×C14, C28.55D4, D4⋊Dic7, C28.48D4, C2×D4.D7, D4×C2×C14, (C7×D4).31D4
Quotients: C1, C2, C22, D4, C23, D7, SD16, C2×D4, D14, C22≀C2, C2×SD16, C8⋊C22, C7⋊D4, C22×D7, C22⋊SD16, D4.D7, C2×C7⋊D4, D4.D14, C2×D4.D7, C24⋊D7, (C7×D4).31D4

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

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

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

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

79 conjugacy classes

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

79 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 D4 D4 D4 D7 SD16 D14 D14 C7⋊D4 C7⋊D4 C7⋊D4 C8⋊C22 D4.D7 D4.D14 kernel (C7×D4).31D4 C28.55D4 D4⋊Dic7 C28.48D4 C2×D4.D7 D4×C2×C14 C2×C28 C7×D4 C22×C14 C22×D4 C2×C14 C22×C4 C2×D4 C2×C4 D4 C23 C14 C22 C2 # reps 1 1 2 1 2 1 1 4 1 3 4 3 6 6 24 6 1 6 6

Matrix representation of (C7×D4).31D4 in GL6(𝔽113)

 73 112 0 0 0 0 75 64 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
,
 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 0 112 111 0 0 0 0 1 1
,
 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 1 1
,
 36 76 0 0 0 0 35 77 0 0 0 0 0 0 12 112 0 0 0 0 32 101 0 0 0 0 0 0 0 26 0 0 0 0 100 87
,
 36 76 0 0 0 0 35 77 0 0 0 0 0 0 101 1 0 0 0 0 83 12 0 0 0 0 0 0 87 87 0 0 0 0 13 26

G:=sub<GL(6,GF(113))| [73,75,0,0,0,0,112,64,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,1,0,0,0,0,111,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,1,0,0,0,0,0,1],[36,35,0,0,0,0,76,77,0,0,0,0,0,0,12,32,0,0,0,0,112,101,0,0,0,0,0,0,0,100,0,0,0,0,26,87],[36,35,0,0,0,0,76,77,0,0,0,0,0,0,101,83,0,0,0,0,1,12,0,0,0,0,0,0,87,13,0,0,0,0,87,26] >;

(C7×D4).31D4 in GAP, Magma, Sage, TeX

(C_7\times D_4)._{31}D_4
% in TeX

G:=Group("(C7xD4).31D4");
// GroupNames label

G:=SmallGroup(448,752);
// by ID

G=gap.SmallGroup(448,752);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,253,254,1684,851,102,18822]);
// Polycyclic

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

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