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G = M4(2)×D13order 416 = 25·13

Direct product of M4(2) and D13

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: M4(2)×D13, C86D26, C1046C22, C52.38C23, (C8×D13)⋊7C2, C8⋊D135C2, C52.33(C2×C4), (C4×D13).1C4, C4.15(C4×D13), (C2×C4).45D26, C135(C2×M4(2)), C52.4C45C2, D26.11(C2×C4), C22.7(C4×D13), C132C811C22, (C13×M4(2))⋊3C2, (C2×C52).25C22, C26.28(C22×C4), (C2×Dic13).7C4, (C22×D13).5C4, C4.38(C22×D13), Dic13.13(C2×C4), (C4×D13).38C22, (C2×C4×D13).4C2, C2.16(C2×C4×D13), (C2×C26).25(C2×C4), SmallGroup(416,127)

Series: Derived Chief Lower central Upper central

C1C26 — M4(2)×D13
C1C13C26C52C4×D13C2×C4×D13 — M4(2)×D13
C13C26 — M4(2)×D13
C1C4M4(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 2A2B2C2D2E4A4B4C4D4E4F8A8B8C8D8E8F8G8H13A···13F26A···26F26G···26L52A···52L52M···52R104A···104X
order1222224444448888888813···1326···2626···2652···5252···52104···104
size1121313261121313262222262626262···22···24···42···24···44···4

80 irreducible representations

dim1111111112222224
type+++++++++
imageC1C2C2C2C2C2C4C4C4M4(2)D13D26D26C4×D13C4×D13M4(2)×D13
kernelM4(2)×D13C8×D13C8⋊D13C52.4C4C13×M4(2)C2×C4×D13C4×D13C2×Dic13C22×D13D13M4(2)C8C2×C4C4C22C1
# reps12211142246126121212

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

28826400
1402500
002880
000288
,
1000
19731200
0010
0001
,
1000
0100
0001
0031224
,
1000
0100
0001
0010
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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