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## G = C7×M4(2).C4order 448 = 26·7

### Direct product of C7 and M4(2).C4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C7×M4(2).C4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C28 — C7×M4(2) — C7×C8.C4 — C7×M4(2).C4
 Lower central C1 — C2 — C4 — C7×M4(2).C4
 Upper central C1 — C28 — C22×C28 — C7×M4(2).C4

Generators and relations for C7×M4(2).C4
G = < a,b,c,d | a7=b8=c2=1, d4=b4, ab=ba, ac=ca, ad=da, cbc=b5, dbd-1=b-1, dcd-1=b4c >

Subgroups: 130 in 102 conjugacy classes, 78 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C8, C2×C4, C2×C4, C23, C14, C14, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C28, C28, C2×C14, C2×C14, C2×C14, C8.C4, C2×M4(2), C2×M4(2), C56, C56, C2×C28, C2×C28, C22×C14, M4(2).C4, C2×C56, C2×C56, C7×M4(2), C7×M4(2), C22×C28, C7×C8.C4, C14×M4(2), C14×M4(2), C7×M4(2).C4
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, Q8, C23, C14, C4⋊C4, C22×C4, C2×D4, C2×Q8, C28, C2×C14, C2×C4⋊C4, C2×C28, C7×D4, C7×Q8, C22×C14, M4(2).C4, C7×C4⋊C4, C22×C28, D4×C14, Q8×C14, C14×C4⋊C4, C7×M4(2).C4

Smallest permutation representation of C7×M4(2).C4
On 112 points
Generators in S112
(1 61 18 49 39 41 10)(2 62 19 50 40 42 11)(3 63 20 51 33 43 12)(4 64 21 52 34 44 13)(5 57 22 53 35 45 14)(6 58 23 54 36 46 15)(7 59 24 55 37 47 16)(8 60 17 56 38 48 9)(25 79 108 91 88 71 100)(26 80 109 92 81 72 101)(27 73 110 93 82 65 102)(28 74 111 94 83 66 103)(29 75 112 95 84 67 104)(30 76 105 96 85 68 97)(31 77 106 89 86 69 98)(32 78 107 90 87 70 99)
(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 5)(3 7)(10 14)(12 16)(18 22)(20 24)(25 29)(27 31)(33 37)(35 39)(41 45)(43 47)(49 53)(51 55)(57 61)(59 63)(65 69)(67 71)(73 77)(75 79)(82 86)(84 88)(89 93)(91 95)(98 102)(100 104)(106 110)(108 112)
(1 97 7 99 5 101 3 103)(2 104 8 98 6 100 4 102)(9 69 15 71 13 65 11 67)(10 68 16 70 14 72 12 66)(17 77 23 79 21 73 19 75)(18 76 24 78 22 80 20 74)(25 64 27 62 29 60 31 58)(26 63 28 61 30 59 32 57)(33 94 39 96 37 90 35 92)(34 93 40 95 38 89 36 91)(41 85 47 87 45 81 43 83)(42 84 48 86 46 88 44 82)(49 105 55 107 53 109 51 111)(50 112 56 106 54 108 52 110)

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

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

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

154 conjugacy classes

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

154 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + - - image C1 C2 C2 C4 C7 C14 C14 C28 D4 Q8 Q8 C7×D4 C7×Q8 C7×Q8 M4(2).C4 C7×M4(2).C4 kernel C7×M4(2).C4 C7×C8.C4 C14×M4(2) C7×M4(2) M4(2).C4 C8.C4 C2×M4(2) M4(2) C2×C28 C2×C28 C22×C14 C2×C4 C2×C4 C23 C7 C1 # reps 1 4 3 8 6 24 18 48 2 1 1 12 6 6 2 12

Matrix representation of C7×M4(2).C4 in GL4(𝔽113) generated by

 109 0 0 0 0 109 0 0 0 0 109 0 0 0 0 109
,
 74 83 0 0 5 39 0 0 19 0 1 111 91 54 8 112
,
 112 0 0 0 93 1 0 0 0 0 1 0 0 0 1 112
,
 0 0 1 0 47 0 66 1 98 0 0 0 86 98 66 0
G:=sub<GL(4,GF(113))| [109,0,0,0,0,109,0,0,0,0,109,0,0,0,0,109],[74,5,19,91,83,39,0,54,0,0,1,8,0,0,111,112],[112,93,0,0,0,1,0,0,0,0,1,1,0,0,0,112],[0,47,98,86,0,0,0,98,1,66,0,66,0,1,0,0] >;

C7×M4(2).C4 in GAP, Magma, Sage, TeX

C_7\times M_4(2).C_4
% in TeX

G:=Group("C7xM4(2).C4");
// GroupNames label

G:=SmallGroup(448,838);
// by ID

G=gap.SmallGroup(448,838);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,400,2403,9804,172,124]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^8=c^2=1,d^4=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^5,d*b*d^-1=b^-1,d*c*d^-1=b^4*c>;
// generators/relations

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