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G = C62.38D4order 288 = 25·32

22nd non-split extension by C62 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.38D4, (C6×C12)⋊7C4, (C2×C62)⋊3C4, (C6×D4).25S3, (C2×C12)⋊2Dic3, C625C43C2, C329(C23⋊C4), (C22×C6)⋊4Dic3, (C22×C6).57D6, C62.105(C2×C4), C232(C3⋊Dic3), C2.5(C625C4), C32(C23.7D6), (C2×C62).12C22, C6.25(C6.D4), C22.2(C327D4), (C2×C4)⋊(C3⋊Dic3), (D4×C3×C6).12C2, C23.7(C2×C3⋊S3), (C2×D4).3(C3⋊S3), (C2×C6).14(C3⋊D4), (C2×C6).48(C2×Dic3), C22.3(C2×C3⋊Dic3), (C3×C6).73(C22⋊C4), SmallGroup(288,309)

Series: Derived Chief Lower central Upper central

C1C62 — C62.38D4
C1C3C32C3×C6C62C2×C62C625C4 — C62.38D4
C32C3×C6C62 — C62.38D4
C1C2C23C2×D4

Generators and relations for C62.38D4
 G = < a,b,c,d | a6=b6=c4=1, d2=a3, ab=ba, cac-1=a-1b3, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=a3c-1 >

Subgroups: 540 in 156 conjugacy classes, 57 normal (15 characteristic)
C1, C2, C2 [×4], C3 [×4], C4 [×3], C22, C22 [×2], C22 [×3], C6 [×4], C6 [×16], C2×C4, C2×C4 [×2], D4 [×2], C23 [×2], C32, Dic3 [×8], C12 [×4], C2×C6 [×12], C2×C6 [×12], C22⋊C4 [×2], C2×D4, C3×C6, C3×C6 [×4], C2×Dic3 [×8], C2×C12 [×4], C3×D4 [×8], C22×C6 [×8], C23⋊C4, C3⋊Dic3 [×2], C3×C12, C62, C62 [×2], C62 [×3], C6.D4 [×8], C6×D4 [×4], C2×C3⋊Dic3 [×2], C6×C12, D4×C32 [×2], C2×C62 [×2], C23.7D6 [×4], C625C4 [×2], D4×C3×C6, C62.38D4
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], C23⋊C4, C3⋊Dic3 [×2], C2×C3⋊S3, C6.D4 [×4], C2×C3⋊Dic3, C327D4 [×2], C23.7D6 [×4], C625C4, C62.38D4

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

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

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

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

51 conjugacy classes

class 1 2A2B2C2D2E3A3B3C3D4A4B4C4D4E6A···6L6M···6AB12A···12H
order1222223333444446···66···612···12
size11222422224363636362···24···44···4

51 irreducible representations

dim1111122222244
type+++++--++
imageC1C2C2C4C4S3D4Dic3Dic3D6C3⋊D4C23⋊C4C23.7D6
kernelC62.38D4C625C4D4×C3×C6C6×C12C2×C62C6×D4C62C2×C12C22×C6C22×C6C2×C6C32C3
# reps12122424441618

Matrix representation of C62.38D4 in GL8(𝔽13)

120000000
012000000
001120000
00100000
00001000
00000100
000000120
0000129012
,
012000000
112000000
000120000
001120000
000012000
000001200
000000120
000000012
,
08000000
80000000
00800000
00850000
00002804
000057910
00001000
000051284
,
08000000
80000000
00800000
00850000
00001000
000051200
00002804
00009230

G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,0,0,0,1,0,9,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,8,8,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,2,5,1,5,0,0,0,0,8,7,0,12,0,0,0,0,0,9,0,8,0,0,0,0,4,10,0,4],[0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,8,8,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,1,5,2,9,0,0,0,0,0,12,8,2,0,0,0,0,0,0,0,3,0,0,0,0,0,0,4,0] >;

C62.38D4 in GAP, Magma, Sage, TeX

C_6^2._{38}D_4
% in TeX

G:=Group("C6^2.38D4");
// GroupNames label

G:=SmallGroup(288,309);
// by ID

G=gap.SmallGroup(288,309);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,219,675,2693,9414]);
// Polycyclic

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

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