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G = C36.48D4order 288 = 25·32

4th non-split extension by C36 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C36.48D4, C4.12D36, C12.4D12, M4(2)⋊3D9, (C2×C4).2D18, (C22×D9).C4, (C2×D36).6C2, (C2×C12).43D6, C91(C4.D4), C4.Dic92C2, C22.5(C4×D9), C6.16(D6⋊C4), C4.22(C9⋊D4), (C9×M4(2))⋊7C2, C2.10(D18⋊C4), C18.9(C22⋊C4), (C2×C36).21C22, C3.(C12.46D4), (C3×M4(2)).7S3, C12.109(C3⋊D4), (C2×C6).4(C4×S3), (C2×C18).3(C2×C4), SmallGroup(288,31)

Series: Derived Chief Lower central Upper central

C1C2×C18 — C36.48D4
C1C3C9C18C36C2×C36C2×D36 — C36.48D4
C9C18C2×C18 — C36.48D4
C1C2C2×C4M4(2)

Generators and relations for C36.48D4
 G = < a,b,c,d | a8=b2=c9=d2=1, bab=a5, ac=ca, dad=ab, bc=cb, bd=db, dcd=c-1 >

Subgroups: 436 in 69 conjugacy classes, 26 normal (24 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C22, C22 [×4], S3 [×2], C6, C6, C8 [×2], C2×C4, D4 [×2], C23 [×2], C9, C12 [×2], D6 [×4], C2×C6, M4(2), M4(2), C2×D4, D9 [×2], C18, C18, C3⋊C8, C24, D12 [×2], C2×C12, C22×S3 [×2], C4.D4, C36 [×2], D18 [×4], C2×C18, C4.Dic3, C3×M4(2), C2×D12, C9⋊C8, C72, D36 [×2], C2×C36, C22×D9 [×2], C12.46D4, C4.Dic9, C9×M4(2), C2×D36, C36.48D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D6, C22⋊C4, D9, C4×S3, D12, C3⋊D4, C4.D4, D18, D6⋊C4, C4×D9, D36, C9⋊D4, C12.46D4, D18⋊C4, C36.48D4

Smallest permutation representation of C36.48D4
On 72 points
Generators in S72
(1 59 23 50 14 68 32 41)(2 60 24 51 15 69 33 42)(3 61 25 52 16 70 34 43)(4 62 26 53 17 71 35 44)(5 63 27 54 18 72 36 45)(6 55 19 46 10 64 28 37)(7 56 20 47 11 65 29 38)(8 57 21 48 12 66 30 39)(9 58 22 49 13 67 31 40)
(37 46)(38 47)(39 48)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(61 70)(62 71)(63 72)
(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)
(1 22)(2 21)(3 20)(4 19)(5 27)(6 26)(7 25)(8 24)(9 23)(10 35)(11 34)(12 33)(13 32)(14 31)(15 30)(16 29)(17 28)(18 36)(37 62)(38 61)(39 60)(40 59)(41 58)(42 57)(43 56)(44 55)(45 63)(46 71)(47 70)(48 69)(49 68)(50 67)(51 66)(52 65)(53 64)(54 72)

G:=sub<Sym(72)| (1,59,23,50,14,68,32,41)(2,60,24,51,15,69,33,42)(3,61,25,52,16,70,34,43)(4,62,26,53,17,71,35,44)(5,63,27,54,18,72,36,45)(6,55,19,46,10,64,28,37)(7,56,20,47,11,65,29,38)(8,57,21,48,12,66,30,39)(9,58,22,49,13,67,31,40), (37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72), (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), (1,22)(2,21)(3,20)(4,19)(5,27)(6,26)(7,25)(8,24)(9,23)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,36)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,63)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,72)>;

G:=Group( (1,59,23,50,14,68,32,41)(2,60,24,51,15,69,33,42)(3,61,25,52,16,70,34,43)(4,62,26,53,17,71,35,44)(5,63,27,54,18,72,36,45)(6,55,19,46,10,64,28,37)(7,56,20,47,11,65,29,38)(8,57,21,48,12,66,30,39)(9,58,22,49,13,67,31,40), (37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72), (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), (1,22)(2,21)(3,20)(4,19)(5,27)(6,26)(7,25)(8,24)(9,23)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,36)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,63)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,72) );

G=PermutationGroup([(1,59,23,50,14,68,32,41),(2,60,24,51,15,69,33,42),(3,61,25,52,16,70,34,43),(4,62,26,53,17,71,35,44),(5,63,27,54,18,72,36,45),(6,55,19,46,10,64,28,37),(7,56,20,47,11,65,29,38),(8,57,21,48,12,66,30,39),(9,58,22,49,13,67,31,40)], [(37,46),(38,47),(39,48),(40,49),(41,50),(42,51),(43,52),(44,53),(45,54),(55,64),(56,65),(57,66),(58,67),(59,68),(60,69),(61,70),(62,71),(63,72)], [(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)], [(1,22),(2,21),(3,20),(4,19),(5,27),(6,26),(7,25),(8,24),(9,23),(10,35),(11,34),(12,33),(13,32),(14,31),(15,30),(16,29),(17,28),(18,36),(37,62),(38,61),(39,60),(40,59),(41,58),(42,57),(43,56),(44,55),(45,63),(46,71),(47,70),(48,69),(49,68),(50,67),(51,66),(52,65),(53,64),(54,72)])

51 conjugacy classes

class 1 2A2B2C2D 3 4A4B6A6B8A8B8C8D9A9B9C12A12B12C18A18B18C18D18E18F24A24B24C24D36A···36F36G36H36I72A···72L
order122223446688889991212121818181818182424242436···3636363672···72
size11236362222444363622222422244444442···24444···4

51 irreducible representations

dim1111122222222222444
type++++++++++++++
imageC1C2C2C2C4S3D4D6D9D12C3⋊D4C4×S3D18D36C9⋊D4C4×D9C4.D4C12.46D4C36.48D4
kernelC36.48D4C4.Dic9C9×M4(2)C2×D36C22×D9C3×M4(2)C36C2×C12M4(2)C12C12C2×C6C2×C4C4C4C22C9C3C1
# reps1111412132223666126

Matrix representation of C36.48D4 in GL6(𝔽73)

100000
0720000
00720710
00072071
0046610
0077001
,
7200000
0720000
001000
000100
00720720
00072072
,
1600000
0320000
0072100
0072000
0000721
0000720
,
0320000
1600000
00661400
007700
00006614
000077

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,4,7,0,0,0,72,66,70,0,0,71,0,1,0,0,0,0,71,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,72,0,0,0,0,1,0,72,0,0,0,0,72,0,0,0,0,0,0,72],[16,0,0,0,0,0,0,32,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[0,16,0,0,0,0,32,0,0,0,0,0,0,0,66,7,0,0,0,0,14,7,0,0,0,0,0,0,66,7,0,0,0,0,14,7] >;

C36.48D4 in GAP, Magma, Sage, TeX

C_{36}._{48}D_4
% in TeX

G:=Group("C36.48D4");
// GroupNames label

G:=SmallGroup(288,31);
// by ID

G=gap.SmallGroup(288,31);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,422,100,346,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^9=d^2=1,b*a*b=a^5,a*c=c*a,d*a*d=a*b,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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