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## G = M4(2)×D13order 416 = 25·13

### Direct product of M4(2) and D13

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — M4(2)×D13
 Chief series C1 — C13 — C26 — C52 — C4×D13 — C2×C4×D13 — M4(2)×D13
 Lower central C13 — C26 — M4(2)×D13
 Upper central C1 — C4 — M4(2)

Generators and relations for M4(2)×D13
G = < a,b,c,d | a8=b2=c13=d2=1, bab=a5, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 400 in 68 conjugacy classes, 39 normal (27 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×2], C22, C22 [×4], C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], C23, C13, C2×C8 [×2], M4(2), M4(2) [×3], C22×C4, D13 [×2], D13, C26, C26, C2×M4(2), Dic13 [×2], C52 [×2], D26 [×2], D26 [×2], C2×C26, C132C8 [×2], C104 [×2], C4×D13 [×4], C2×Dic13, C2×C52, C22×D13, C8×D13 [×2], C8⋊D13 [×2], C52.4C4, C13×M4(2), C2×C4×D13, M4(2)×D13
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, M4(2) [×2], C22×C4, D13, C2×M4(2), D26 [×3], C4×D13 [×2], C22×D13, C2×C4×D13, M4(2)×D13

Smallest permutation representation of M4(2)×D13
On 104 points
Generators in S104
(1 83 34 77 26 92 52 64)(2 84 35 78 14 93 40 65)(3 85 36 66 15 94 41 53)(4 86 37 67 16 95 42 54)(5 87 38 68 17 96 43 55)(6 88 39 69 18 97 44 56)(7 89 27 70 19 98 45 57)(8 90 28 71 20 99 46 58)(9 91 29 72 21 100 47 59)(10 79 30 73 22 101 48 60)(11 80 31 74 23 102 49 61)(12 81 32 75 24 103 50 62)(13 82 33 76 25 104 51 63)
(53 66)(54 67)(55 68)(56 69)(57 70)(58 71)(59 72)(60 73)(61 74)(62 75)(63 76)(64 77)(65 78)(79 101)(80 102)(81 103)(82 104)(83 92)(84 93)(85 94)(86 95)(87 96)(88 97)(89 98)(90 99)(91 100)
(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)
(1 13)(2 12)(3 11)(4 10)(5 9)(6 8)(14 24)(15 23)(16 22)(17 21)(18 20)(25 26)(28 39)(29 38)(30 37)(31 36)(32 35)(33 34)(40 50)(41 49)(42 48)(43 47)(44 46)(51 52)(53 61)(54 60)(55 59)(56 58)(62 65)(63 64)(66 74)(67 73)(68 72)(69 71)(75 78)(76 77)(79 86)(80 85)(81 84)(82 83)(87 91)(88 90)(92 104)(93 103)(94 102)(95 101)(96 100)(97 99)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 8A 8B 8C 8D 8E 8F 8G 8H 13A ··· 13F 26A ··· 26F 26G ··· 26L 52A ··· 52L 52M ··· 52R 104A ··· 104X order 1 2 2 2 2 2 4 4 4 4 4 4 8 8 8 8 8 8 8 8 13 ··· 13 26 ··· 26 26 ··· 26 52 ··· 52 52 ··· 52 104 ··· 104 size 1 1 2 13 13 26 1 1 2 13 13 26 2 2 2 2 26 26 26 26 2 ··· 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 M4(2) D13 D26 D26 C4×D13 C4×D13 M4(2)×D13 kernel M4(2)×D13 C8×D13 C8⋊D13 C52.4C4 C13×M4(2) C2×C4×D13 C4×D13 C2×Dic13 C22×D13 D13 M4(2) C8 C2×C4 C4 C22 C1 # reps 1 2 2 1 1 1 4 2 2 4 6 12 6 12 12 12

Matrix representation of M4(2)×D13 in GL4(𝔽313) generated by

 288 264 0 0 140 25 0 0 0 0 288 0 0 0 0 288
,
 1 0 0 0 197 312 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 312 24
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(313))| [288,140,0,0,264,25,0,0,0,0,288,0,0,0,0,288],[1,197,0,0,0,312,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,312,0,0,1,24],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0] >;

M4(2)×D13 in GAP, Magma, Sage, TeX

M_4(2)\times D_{13}
% in TeX

G:=Group("M4(2)xD13");
// GroupNames label

G:=SmallGroup(416,127);
// by ID

G=gap.SmallGroup(416,127);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,188,50,69,13829]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^13=d^2=1,b*a*b=a^5,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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