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G = C62.39D4order 288 = 25·32

23rd non-split extension by C62 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.39D4, C3213C4≀C2, (C3×D4)⋊2Dic3, (C3×Q8)⋊4Dic3, (C2×C12).97D6, (D4×C32)⋊7C4, (Q8×C32)⋊7C4, Q83(C3⋊Dic3), D42(C3⋊Dic3), (C3×C12).173D4, C12.58D66C2, (C6×C12).64C22, C12.15(C2×Dic3), C33(Q83Dic3), C12.134(C3⋊D4), C2.8(C625C4), C4.31(C327D4), C6.28(C6.D4), C22.3(C327D4), (C4×C3⋊Dic3)⋊3C2, C4.3(C2×C3⋊Dic3), (C3×C12).54(C2×C4), C4○D4.3(C3⋊S3), (C3×C4○D4).14S3, (C2×C6).15(C3⋊D4), (C32×C4○D4).2C2, (C3×C6).76(C22⋊C4), (C2×C4).42(C2×C3⋊S3), SmallGroup(288,312)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C62.39D4
C1C3C32C3×C6C3×C12C6×C12C12.58D6 — C62.39D4
C32C3×C6C3×C12 — C62.39D4
C1C4C2×C4C4○D4

Generators and relations for C62.39D4
 G = < a,b,c,d | a6=b6=1, c4=b3, d2=a3b3, ab=ba, cac-1=a-1b3, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=a3c3 >

Subgroups: 380 in 132 conjugacy classes, 57 normal (21 characteristic)
C1, C2, C2 [×2], C3 [×4], C4 [×2], C4 [×3], C22, C22, C6 [×4], C6 [×8], C8, C2×C4, C2×C4 [×2], D4, D4, Q8, C32, Dic3 [×8], C12 [×8], C12 [×4], C2×C6 [×4], C2×C6 [×4], C42, M4(2), C4○D4, C3×C6, C3×C6 [×2], C3⋊C8 [×4], C2×Dic3 [×4], C2×C12 [×4], C2×C12 [×4], C3×D4 [×4], C3×D4 [×4], C3×Q8 [×4], C4≀C2, C3⋊Dic3 [×2], C3×C12 [×2], C3×C12, C62, C62, C4.Dic3 [×4], C4×Dic3 [×4], C3×C4○D4 [×4], C324C8, C2×C3⋊Dic3, C6×C12, C6×C12, D4×C32, D4×C32, Q8×C32, Q83Dic3 [×4], C12.58D6, C4×C3⋊Dic3, C32×C4○D4, C62.39D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, D4 [×2], Dic3 [×8], D6 [×4], C22⋊C4, C3⋊S3, C2×Dic3 [×4], C3⋊D4 [×8], C4≀C2, C3⋊Dic3 [×2], C2×C3⋊S3, C6.D4 [×4], C2×C3⋊Dic3, C327D4 [×2], Q83Dic3 [×4], C625C4, C62.39D4

Smallest permutation representation of C62.39D4
On 72 points
Generators in S72
(1 61 50 5 57 54)(2 51 58)(3 63 52 7 59 56)(4 53 60)(6 55 62)(8 49 64)(9 30 39)(10 36 31 14 40 27)(11 32 33)(12 38 25 16 34 29)(13 26 35)(15 28 37)(17 72 46)(18 43 65 22 47 69)(19 66 48)(20 45 67 24 41 71)(21 68 42)(23 70 44)
(1 22 25 5 18 29)(2 30 19 6 26 23)(3 24 27 7 20 31)(4 32 21 8 28 17)(9 48 62 13 44 58)(10 59 45 14 63 41)(11 42 64 15 46 60)(12 61 47 16 57 43)(33 68 49 37 72 53)(34 54 65 38 50 69)(35 70 51 39 66 55)(36 56 67 40 52 71)
(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 5)(2 8 6 4)(3 7)(9 68 13 72)(10 67)(11 70 15 66)(12 69)(14 71)(16 65)(17 30 21 26)(18 29)(19 32 23 28)(20 31)(22 25)(24 27)(33 44 37 48)(34 43)(35 46 39 42)(36 45)(38 47)(40 41)(49 62 53 58)(50 61)(51 64 55 60)(52 63)(54 57)(56 59)

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

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

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

54 conjugacy classes

class 1 2A2B2C3A3B3C3D4A4B4C4D4E4F4G4H6A6B6C6D6E···6P8A8B12A···12H12I···12T
order122233334444444466666···68812···1212···12
size1124222211241818181822224···436362···24···4

54 irreducible representations

dim1111112222222224
type++++++++--
imageC1C2C2C2C4C4S3D4D4D6Dic3Dic3C3⋊D4C3⋊D4C4≀C2Q83Dic3
kernelC62.39D4C12.58D6C4×C3⋊Dic3C32×C4○D4D4×C32Q8×C32C3×C4○D4C3×C12C62C2×C12C3×D4C3×Q8C12C2×C6C32C3
# reps1111224114448848

Matrix representation of C62.39D4 in GL6(𝔽73)

7200000
010000
001000
000100
00007170
000011
,
7200000
0720000
0007200
0017200
000010
000001
,
0270000
7200000
00434300
00133000
00001734
00004356
,
7200000
0270000
0017200
0007200
000010
00007272

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,71,1,0,0,0,0,70,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,72,0,0,0,0,27,0,0,0,0,0,0,0,43,13,0,0,0,0,43,30,0,0,0,0,0,0,17,43,0,0,0,0,34,56],[72,0,0,0,0,0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72] >;

C62.39D4 in GAP, Magma, Sage, TeX

C_6^2._{39}D_4
% in TeX

G:=Group("C6^2.39D4");
// GroupNames label

G:=SmallGroup(288,312);
// by ID

G=gap.SmallGroup(288,312);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,100,675,346,80,2693,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^6=1,c^4=b^3,d^2=a^3*b^3,a*b=b*a,c*a*c^-1=a^-1*b^3,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^3*c^3>;
// generators/relations

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