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G = D13⋊M4(2)  order 416 = 25·13

The semidirect product of D13 and M4(2) acting via M4(2)/C2×C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D13⋊M4(2), Dic13.11C23, D13⋊C84C2, C13⋊C82C22, (C2×C52).8C4, C52.20(C2×C4), (C4×D13).8C4, C132(C2×M4(2)), C52.C45C2, D26.14(C2×C4), C26.3(C22×C4), C13⋊M4(2)⋊3C2, (C22×D13).9C4, Dic13.16(C2×C4), (C4×D13).34C22, (C2×Dic13).56C22, C4.21(C2×C13⋊C4), (C2×C4).8(C13⋊C4), (C2×C4×D13).15C2, C2.5(C22×C13⋊C4), C22.6(C2×C13⋊C4), (C2×C26).15(C2×C4), SmallGroup(416,201)

Series: Derived Chief Lower central Upper central

C1C26 — D13⋊M4(2)
C1C13C26Dic13C13⋊C8D13⋊C8 — D13⋊M4(2)
C13C26 — D13⋊M4(2)
C1C4C2×C4

Generators and relations for D13⋊M4(2)
 G = < a,b,c,d | a13=b2=c8=d2=1, bab=a-1, cac-1=a5, ad=da, cbc-1=a4b, bd=db, dcd=c5 >

Subgroups: 436 in 68 conjugacy classes, 36 normal (24 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×2], C22, C22 [×4], C8 [×4], C2×C4, C2×C4 [×5], C23, C13, C2×C8 [×2], M4(2) [×4], C22×C4, D13 [×2], D13, C26, C26, C2×M4(2), Dic13 [×2], C52 [×2], D26 [×2], D26 [×2], C2×C26, C13⋊C8 [×4], C4×D13 [×4], C2×Dic13, C2×C52, C22×D13, D13⋊C8 [×2], C52.C4 [×2], C13⋊M4(2) [×2], C2×C4×D13, D13⋊M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, M4(2) [×2], C22×C4, C2×M4(2), C13⋊C4, C2×C13⋊C4 [×3], C22×C13⋊C4, D13⋊M4(2)

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

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F8A···8H13A13B13C26A···26I52A···52L
order1222224444448···813131326···2652···52
size11213132611213132626···264444···44···4

44 irreducible representations

dim1111111124444
type++++++++
imageC1C2C2C2C2C4C4C4M4(2)C13⋊C4C2×C13⋊C4C2×C13⋊C4D13⋊M4(2)
kernelD13⋊M4(2)D13⋊C8C52.C4C13⋊M4(2)C2×C4×D13C4×D13C2×C52C22×D13D13C2×C4C4C22C1
# reps12221422436312

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

0100
3127700
0049247
0066288
,
0100
1000
0049247
00264264
,
003120
000312
288000
2662500
,
312000
031200
0010
0001
G:=sub<GL(4,GF(313))| [0,312,0,0,1,77,0,0,0,0,49,66,0,0,247,288],[0,1,0,0,1,0,0,0,0,0,49,264,0,0,247,264],[0,0,288,266,0,0,0,25,312,0,0,0,0,312,0,0],[312,0,0,0,0,312,0,0,0,0,1,0,0,0,0,1] >;

D13⋊M4(2) in GAP, Magma, Sage, TeX

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

G:=Group("D13:M4(2)");
// GroupNames label

G:=SmallGroup(416,201);
// by ID

G=gap.SmallGroup(416,201);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,103,362,69,9221,1751]);
// Polycyclic

G:=Group<a,b,c,d|a^13=b^2=c^8=d^2=1,b*a*b=a^-1,c*a*c^-1=a^5,a*d=d*a,c*b*c^-1=a^4*b,b*d=d*b,d*c*d=c^5>;
// generators/relations

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