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G = C7×Q8○M4(2)  order 448 = 26·7

Direct product of C7 and Q8○M4(2)

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C7×Q8○M4(2)
 Chief series C1 — C2 — C4 — C28 — C56 — C2×C56 — C7×C8○D4 — C7×Q8○M4(2)
 Lower central C1 — C2 — C7×Q8○M4(2)
 Upper central C1 — C28 — C7×Q8○M4(2)

Generators and relations for C7×Q8○M4(2)
G = < a,b,c,d,e | a7=b4=e2=1, c2=d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >

Subgroups: 290 in 258 conjugacy classes, 238 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, C14, C14, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C28, C28, C2×C14, C2×C14, C2×C14, C2×M4(2), C8○D4, C2×C4○D4, C56, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, Q8○M4(2), C2×C56, C7×M4(2), C22×C28, D4×C14, Q8×C14, C7×C4○D4, C14×M4(2), C7×C8○D4, C14×C4○D4, C7×Q8○M4(2)
Quotients: C1, C2, C4, C22, C7, C2×C4, C23, C14, C22×C4, C24, C28, C2×C14, C23×C4, C2×C28, C22×C14, Q8○M4(2), C22×C28, C23×C14, C23×C28, C7×Q8○M4(2)

Smallest permutation representation of C7×Q8○M4(2)
On 112 points
Generators in S112
(1 77 42 21 91 16 34)(2 78 43 22 92 9 35)(3 79 44 23 93 10 36)(4 80 45 24 94 11 37)(5 73 46 17 95 12 38)(6 74 47 18 96 13 39)(7 75 48 19 89 14 40)(8 76 41 20 90 15 33)(25 104 59 66 112 82 55)(26 97 60 67 105 83 56)(27 98 61 68 106 84 49)(28 99 62 69 107 85 50)(29 100 63 70 108 86 51)(30 101 64 71 109 87 52)(31 102 57 72 110 88 53)(32 103 58 65 111 81 54)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 31 29 27)(26 32 30 28)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 55 53 51)(50 56 54 52)(57 63 61 59)(58 64 62 60)(65 71 69 67)(66 72 70 68)(73 75 77 79)(74 76 78 80)(81 87 85 83)(82 88 86 84)(89 91 93 95)(90 92 94 96)(97 103 101 99)(98 104 102 100)(105 111 109 107)(106 112 110 108)
(1 99 5 103)(2 100 6 104)(3 101 7 97)(4 102 8 98)(9 51 13 55)(10 52 14 56)(11 53 15 49)(12 54 16 50)(17 111 21 107)(18 112 22 108)(19 105 23 109)(20 106 24 110)(25 35 29 39)(26 36 30 40)(27 37 31 33)(28 38 32 34)(41 68 45 72)(42 69 46 65)(43 70 47 66)(44 71 48 67)(57 76 61 80)(58 77 62 73)(59 78 63 74)(60 79 64 75)(81 91 85 95)(82 92 86 96)(83 93 87 89)(84 94 88 90)
(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 6)(4 8)(9 13)(11 15)(18 22)(20 24)(25 29)(27 31)(33 37)(35 39)(41 45)(43 47)(49 53)(51 55)(57 61)(59 63)(66 70)(68 72)(74 78)(76 80)(82 86)(84 88)(90 94)(92 96)(98 102)(100 104)(106 110)(108 112)

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

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

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

238 conjugacy classes

 class 1 2A 2B ··· 2H 4A 4B 4C ··· 4I 7A ··· 7F 8A ··· 8P 14A ··· 14F 14G ··· 14AV 28A ··· 28L 28M ··· 28BB 56A ··· 56CR order 1 2 2 ··· 2 4 4 4 ··· 4 7 ··· 7 8 ··· 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 2 ··· 2 1 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

238 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4 4 type + + + + image C1 C2 C2 C2 C4 C4 C4 C7 C14 C14 C14 C28 C28 C28 Q8○M4(2) C7×Q8○M4(2) kernel C7×Q8○M4(2) C14×M4(2) C7×C8○D4 C14×C4○D4 D4×C14 Q8×C14 C7×C4○D4 Q8○M4(2) C2×M4(2) C8○D4 C2×C4○D4 C2×D4 C2×Q8 C4○D4 C7 C1 # reps 1 6 8 1 6 2 8 6 36 48 6 36 12 48 2 12

Matrix representation of C7×Q8○M4(2) in GL4(𝔽113) generated by

 28 0 0 0 0 28 0 0 0 0 28 0 0 0 0 28
,
 15 0 0 0 0 98 0 0 0 98 15 0 95 0 0 98
,
 0 1 0 0 112 0 0 0 78 47 0 112 66 78 1 0
,
 94 1 111 0 69 0 0 2 38 47 19 1 34 86 69 0
,
 1 0 0 0 0 1 0 0 94 1 112 0 44 0 0 112
G:=sub<GL(4,GF(113))| [28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[15,0,0,95,0,98,98,0,0,0,15,0,0,0,0,98],[0,112,78,66,1,0,47,78,0,0,0,1,0,0,112,0],[94,69,38,34,1,0,47,86,111,0,19,69,0,2,1,0],[1,0,94,44,0,1,1,0,0,0,112,0,0,0,0,112] >;

C7×Q8○M4(2) in GAP, Magma, Sage, TeX

C_7\times Q_8\circ M_4(2)
% in TeX

G:=Group("C7xQ8oM4(2)");
// GroupNames label

G:=SmallGroup(448,1351);
// by ID

G=gap.SmallGroup(448,1351);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,784,2403,6499,124]);
// Polycyclic

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

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