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G = C327D8order 144 = 24·32

2nd semidirect product of C32 and D8 acting via D8/D4=C2

metabelian, supersoluble, monomial

Aliases: C327D8, C12.16D6, D4⋊(C3⋊S3), (C3×D4)⋊1S3, C33(D4⋊S3), (C3×C6).34D4, C12⋊S33C2, C324C83C2, (D4×C32)⋊2C2, C6.22(C3⋊D4), (C3×C12).12C22, C2.4(C327D4), C4.1(C2×C3⋊S3), SmallGroup(144,96)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C327D8
C1C3C32C3×C6C3×C12C12⋊S3 — C327D8
C32C3×C6C3×C12 — C327D8
C1C2C4D4

Generators and relations for C327D8
 G = < a,b,c,d | a3=b3=c8=d2=1, ab=ba, cac-1=dad=a-1, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 258 in 66 conjugacy classes, 27 normal (11 characteristic)
C1, C2, C2 [×2], C3 [×4], C4, C22 [×2], S3 [×4], C6 [×4], C6 [×4], C8, D4, D4, C32, C12 [×4], D6 [×4], C2×C6 [×4], D8, C3⋊S3, C3×C6, C3×C6, C3⋊C8 [×4], D12 [×4], C3×D4 [×4], C3×C12, C2×C3⋊S3, C62, D4⋊S3 [×4], C324C8, C12⋊S3, D4×C32, C327D8
Quotients: C1, C2 [×3], C22, S3 [×4], D4, D6 [×4], D8, C3⋊S3, C3⋊D4 [×4], C2×C3⋊S3, D4⋊S3 [×4], C327D4, C327D8

Character table of C327D8

 class 12A2B2C3A3B3C3D46A6B6C6D6E6F6G6H6I6J6K6L8A8B12A12B12C12D
 size 114362222222224444444418184444
ρ1111111111111111111111111111    trivial
ρ211-1-1111111111-1-1-1-1-1-1-1-1111111    linear of order 2
ρ311-11111111111-1-1-1-1-1-1-1-1-1-11111    linear of order 2
ρ4111-111111111111111111-1-11111    linear of order 2
ρ522-20-12-1-12-1-12-1-2111111-2002-1-1-1    orthogonal lifted from D6
ρ62220-1-12-12-1-1-12-12-1-12-1-1-100-12-1-1    orthogonal lifted from S3
ρ722-202-1-1-12-12-1-1111-21-21100-1-1-12    orthogonal lifted from D6
ρ82220-12-1-12-1-12-12-1-1-1-1-1-12002-1-1-1    orthogonal lifted from S3
ρ922-20-1-12-12-1-1-121-211-211100-12-1-1    orthogonal lifted from D6
ρ1022002222-222220000000000-2-2-2-2    orthogonal lifted from D4
ρ1122-20-1-1-1222-1-1-111-2111-2100-1-12-1    orthogonal lifted from D6
ρ122220-1-1-1222-1-1-1-1-12-1-1-12-100-1-12-1    orthogonal lifted from S3
ρ1322202-1-1-12-12-1-1-1-1-12-12-1-100-1-1-12    orthogonal lifted from S3
ρ142-20022220-2-2-2-2000000002-20000    orthogonal lifted from D8
ρ152-20022220-2-2-2-200000000-220000    orthogonal lifted from D8
ρ1622002-1-1-1-2-12-1-1--3-3--30--30-3-300111-2    complex lifted from C3⋊D4
ρ1722002-1-1-1-2-12-1-1-3--3-30-30--3--300111-2    complex lifted from C3⋊D4
ρ182200-12-1-1-2-1-12-10--3--3--3-3-3-3000-2111    complex lifted from C3⋊D4
ρ192200-1-12-1-2-1-1-12-30--3-30--3-3--3001-211    complex lifted from C3⋊D4
ρ202200-1-1-12-22-1-1-1-3-30--3--3-30--30011-21    complex lifted from C3⋊D4
ρ212200-12-1-1-2-1-12-10-3-3-3--3--3--3000-2111    complex lifted from C3⋊D4
ρ222200-1-12-1-2-1-1-12--30-3--30-3--3-3001-211    complex lifted from C3⋊D4
ρ232200-1-1-12-22-1-1-1--3--30-3-3--30-30011-21    complex lifted from C3⋊D4
ρ244-400-24-2-2022-4200000000000000    orthogonal lifted from D4⋊S3, Schur index 2
ρ254-400-2-2-240-422200000000000000    orthogonal lifted from D4⋊S3, Schur index 2
ρ264-4004-2-2-202-42200000000000000    orthogonal lifted from D4⋊S3, Schur index 2
ρ274-400-2-24-20222-400000000000000    orthogonal lifted from D4⋊S3, Schur index 2

Smallest permutation representation of C327D8
On 72 points
Generators in S72
(1 45 29)(2 30 46)(3 47 31)(4 32 48)(5 41 25)(6 26 42)(7 43 27)(8 28 44)(9 20 38)(10 39 21)(11 22 40)(12 33 23)(13 24 34)(14 35 17)(15 18 36)(16 37 19)(49 61 70)(50 71 62)(51 63 72)(52 65 64)(53 57 66)(54 67 58)(55 59 68)(56 69 60)
(1 57 13)(2 14 58)(3 59 15)(4 16 60)(5 61 9)(6 10 62)(7 63 11)(8 12 64)(17 67 46)(18 47 68)(19 69 48)(20 41 70)(21 71 42)(22 43 72)(23 65 44)(24 45 66)(25 49 38)(26 39 50)(27 51 40)(28 33 52)(29 53 34)(30 35 54)(31 55 36)(32 37 56)
(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)
(2 8)(3 7)(4 6)(9 61)(10 60)(11 59)(12 58)(13 57)(14 64)(15 63)(16 62)(17 52)(18 51)(19 50)(20 49)(21 56)(22 55)(23 54)(24 53)(25 41)(26 48)(27 47)(28 46)(29 45)(30 44)(31 43)(32 42)(33 67)(34 66)(35 65)(36 72)(37 71)(38 70)(39 69)(40 68)

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

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

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

C327D8 is a maximal subgroup of
S3×D4⋊S3  Dic63D6  D12.7D6  Dic6.20D6  D8×C3⋊S3  C248D6  C247D6  C24.40D6  C62.131D4  C62.73D4  C62.74D4  He36D8  C36.18D6  C336D8  C337D8  C3315D8
C327D8 is a maximal quotient of
C12.9Dic6  C62.113D4  C327D16  C328SD32  C3210SD32  C327Q32  C62.116D4  C36.18D6  He37D8  C336D8  C337D8  C3315D8

Matrix representation of C327D8 in GL6(𝔽73)

100000
010000
0072100
0072000
000010
000001
,
7210000
7200000
0072100
0072000
000010
000001
,
60430000
30130000
0072100
000100
00005757
00001657
,
010000
100000
0072100
000100
000001
000010

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,30,0,0,0,0,43,13,0,0,0,0,0,0,72,0,0,0,0,0,1,1,0,0,0,0,0,0,57,16,0,0,0,0,57,57],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,1,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C327D8 in GAP, Magma, Sage, TeX

C_3^2\rtimes_7D_8
% in TeX

G:=Group("C3^2:7D8");
// GroupNames label

G:=SmallGroup(144,96);
// by ID

G=gap.SmallGroup(144,96);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,73,218,116,50,964,3461]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^8=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

Export

Character table of C327D8 in TeX

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