Copied to
clipboard

G = C62.79D6order 432 = 24·33

27th non-split extension by C62 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: C62.79D6, (C6×Dic3)⋊4S3, (C3×C6).42D12, C329(D6⋊C4), (C32×C6).43D4, C3312(C22⋊C4), C6.17(C12⋊S3), C2.2(C338D4), C2.2(C337D4), C6.6(C327D4), (C3×C62).9C22, C6.27(C3⋊D12), C31(C6.11D12), C31(C6.D12), C6.10(C6.D6), (C2×C6).33S32, C6.4(C4×C3⋊S3), (Dic3×C3×C6)⋊4C2, (C3×C6).50(C4×S3), (C6×C3⋊Dic3)⋊3C2, (C2×C3⋊Dic3)⋊8S3, C22.7(S3×C3⋊S3), (C2×C33⋊C2)⋊3C4, C2.4(C338(C2×C4)), (C2×Dic3)⋊2(C3⋊S3), (C3×C6).62(C3⋊D4), (C32×C6).40(C2×C4), (C22×C33⋊C2).1C2, (C2×C6).15(C2×C3⋊S3), SmallGroup(432,451)

Series: Derived Chief Lower central Upper central

C1C32×C6 — C62.79D6
C1C3C32C33C32×C6C3×C62Dic3×C3×C6 — C62.79D6
C33C32×C6 — C62.79D6
C1C22

Generators and relations for C62.79D6
 G = < a,b,c,d | a6=b6=d2=1, c6=a3, ab=ba, ac=ca, dad=a-1, cbc-1=dbd=b-1, dcd=b3c5 >

Subgroups: 2456 in 332 conjugacy classes, 72 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C3, C3 [×4], C3 [×4], C4 [×2], C22, C22 [×4], S3 [×26], C6 [×3], C6 [×12], C6 [×12], C2×C4 [×2], C23, C32, C32 [×4], C32 [×4], Dic3 [×5], C12 [×5], D6 [×44], C2×C6, C2×C6 [×4], C2×C6 [×4], C22⋊C4, C3⋊S3 [×26], C3×C6 [×3], C3×C6 [×12], C3×C6 [×12], C2×Dic3, C2×Dic3 [×4], C2×C12 [×5], C22×S3 [×9], C33, C3×Dic3 [×8], C3⋊Dic3, C3×C12, C2×C3⋊S3 [×44], C62, C62 [×4], C62 [×4], D6⋊C4 [×5], C33⋊C2 [×2], C32×C6 [×3], C6×Dic3 [×4], C6×Dic3 [×4], C2×C3⋊Dic3, C6×C12, C22×C3⋊S3 [×9], C32×Dic3, C3×C3⋊Dic3, C2×C33⋊C2 [×2], C2×C33⋊C2 [×2], C3×C62, C6.D12 [×4], C6.11D12, Dic3×C3×C6, C6×C3⋊Dic3, C22×C33⋊C2, C62.79D6
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×5], C2×C4, D4 [×2], D6 [×5], C22⋊C4, C3⋊S3, C4×S3 [×5], D12 [×5], C3⋊D4 [×5], S32 [×4], C2×C3⋊S3, D6⋊C4 [×5], C6.D6 [×4], C3⋊D12 [×8], C4×C3⋊S3, C12⋊S3, C327D4, S3×C3⋊S3, C6.D12 [×4], C6.11D12, C338(C2×C4), C337D4, C338D4, C62.79D6

Smallest permutation representation of C62.79D6
On 72 points
Generators in S72
(1 17 50 7 23 56)(2 18 51 8 24 57)(3 19 52 9 13 58)(4 20 53 10 14 59)(5 21 54 11 15 60)(6 22 55 12 16 49)(25 40 71 31 46 65)(26 41 72 32 47 66)(27 42 61 33 48 67)(28 43 62 34 37 68)(29 44 63 35 38 69)(30 45 64 36 39 70)
(1 30 9 26 5 34)(2 35 6 27 10 31)(3 32 11 28 7 36)(4 25 8 29 12 33)(13 41 21 37 17 45)(14 46 18 38 22 42)(15 43 23 39 19 47)(16 48 20 40 24 44)(49 67 53 71 57 63)(50 64 58 72 54 68)(51 69 55 61 59 65)(52 66 60 62 56 70)
(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 28)(2 8)(3 26)(4 6)(5 36)(7 34)(9 32)(10 12)(11 30)(13 72)(14 55)(15 70)(16 53)(17 68)(18 51)(19 66)(20 49)(21 64)(22 59)(23 62)(24 57)(25 35)(27 33)(29 31)(37 50)(38 65)(39 60)(40 63)(41 58)(42 61)(43 56)(44 71)(45 54)(46 69)(47 52)(48 67)

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

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

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

66 conjugacy classes

class 1 2A2B2C2D2E3A···3E3F3G3H3I4A4B4C4D6A···6O6P···6AA12A···12P12Q12R12S12T
order1222223···3333344446···66···612···1212121212
size111154542···244446618182···24···46···618181818

66 irreducible representations

dim111112222222444
type++++++++++++
imageC1C2C2C2C4S3S3D4D6C4×S3D12C3⋊D4S32C6.D6C3⋊D12
kernelC62.79D6Dic3×C3×C6C6×C3⋊Dic3C22×C33⋊C2C2×C33⋊C2C6×Dic3C2×C3⋊Dic3C32×C6C62C3×C6C3×C6C3×C6C2×C6C6C6
# reps111144125101010448

Matrix representation of C62.79D6 in GL8(𝔽13)

10000000
01000000
00010000
0012120000
000011200
00001000
00000010
00000001
,
120000000
012000000
00100000
00010000
00001000
00000100
000000121
000000120
,
01000000
10000000
00100000
00010000
00000800
00005800
00000001
00000010
,
120000000
01000000
00100000
0012120000
000012100
00000100
00000001
00000010

G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,8,8,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;

C62.79D6 in GAP, Magma, Sage, TeX

C_6^2._{79}D_6
% in TeX

G:=Group("C6^2.79D6");
// GroupNames label

G:=SmallGroup(432,451);
// by ID

G=gap.SmallGroup(432,451);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,141,92,571,2028,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^6=d^2=1,c^6=a^3,a*b=b*a,a*c=c*a,d*a*d=a^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=b^3*c^5>;
// generators/relations

׿
×
𝔽