metabelian, supersoluble, monomial
Aliases: C32⋊7D8, C12.16D6, D4⋊(C3⋊S3), (C3×D4)⋊1S3, C3⋊3(D4⋊S3), (C3×C6).34D4, C12⋊S3⋊3C2, C32⋊4C8⋊3C2, (D4×C32)⋊2C2, C6.22(C3⋊D4), (C3×C12).12C22, C2.4(C32⋊7D4), C4.1(C2×C3⋊S3), SmallGroup(144,96)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32⋊7D8
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], C32⋊4C8, C12⋊S3, D4×C32, C32⋊7D8
Quotients: C1, C2 [×3], C22, S3 [×4], D4, D6 [×4], D8, C3⋊S3, C3⋊D4 [×4], C2×C3⋊S3, D4⋊S3 [×4], C32⋊7D4, C32⋊7D8
Character table of C32⋊7D8
class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 8A | 8B | 12A | 12B | 12C | 12D | |
size | 1 | 1 | 4 | 36 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 18 | 18 | 4 | 4 | 4 | 4 | |
ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
ρ2 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
ρ3 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
ρ4 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
ρ5 | 2 | 2 | -2 | 0 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | -2 | 1 | 1 | 1 | 1 | 1 | 1 | -2 | 0 | 0 | 2 | -1 | -1 | -1 | orthogonal lifted from D6 |
ρ6 | 2 | 2 | 2 | 0 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | -1 | 0 | 0 | -1 | 2 | -1 | -1 | orthogonal lifted from S3 |
ρ7 | 2 | 2 | -2 | 0 | 2 | -1 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | 1 | 1 | 1 | -2 | 1 | -2 | 1 | 1 | 0 | 0 | -1 | -1 | -1 | 2 | orthogonal lifted from D6 |
ρ8 | 2 | 2 | 2 | 0 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | orthogonal lifted from S3 |
ρ9 | 2 | 2 | -2 | 0 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | 2 | 1 | -2 | 1 | 1 | -2 | 1 | 1 | 1 | 0 | 0 | -1 | 2 | -1 | -1 | orthogonal lifted from D6 |
ρ10 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | -2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | orthogonal lifted from D4 |
ρ11 | 2 | 2 | -2 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | 1 | 1 | -2 | 1 | 1 | 1 | -2 | 1 | 0 | 0 | -1 | -1 | 2 | -1 | orthogonal lifted from D6 |
ρ12 | 2 | 2 | 2 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | -1 | 0 | 0 | -1 | -1 | 2 | -1 | orthogonal lifted from S3 |
ρ13 | 2 | 2 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | 2 | orthogonal lifted from S3 |
ρ14 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
ρ15 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
ρ16 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | -2 | -1 | 2 | -1 | -1 | -√-3 | √-3 | -√-3 | 0 | -√-3 | 0 | √-3 | √-3 | 0 | 0 | 1 | 1 | 1 | -2 | complex lifted from C3⋊D4 |
ρ17 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | -2 | -1 | 2 | -1 | -1 | √-3 | -√-3 | √-3 | 0 | √-3 | 0 | -√-3 | -√-3 | 0 | 0 | 1 | 1 | 1 | -2 | complex lifted from C3⋊D4 |
ρ18 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | -1 | -2 | -1 | -1 | 2 | -1 | 0 | -√-3 | -√-3 | -√-3 | √-3 | √-3 | √-3 | 0 | 0 | 0 | -2 | 1 | 1 | 1 | complex lifted from C3⋊D4 |
ρ19 | 2 | 2 | 0 | 0 | -1 | -1 | 2 | -1 | -2 | -1 | -1 | -1 | 2 | √-3 | 0 | -√-3 | √-3 | 0 | -√-3 | √-3 | -√-3 | 0 | 0 | 1 | -2 | 1 | 1 | complex lifted from C3⋊D4 |
ρ20 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | -2 | 2 | -1 | -1 | -1 | √-3 | √-3 | 0 | -√-3 | -√-3 | √-3 | 0 | -√-3 | 0 | 0 | 1 | 1 | -2 | 1 | complex lifted from C3⋊D4 |
ρ21 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | -1 | -2 | -1 | -1 | 2 | -1 | 0 | √-3 | √-3 | √-3 | -√-3 | -√-3 | -√-3 | 0 | 0 | 0 | -2 | 1 | 1 | 1 | complex lifted from C3⋊D4 |
ρ22 | 2 | 2 | 0 | 0 | -1 | -1 | 2 | -1 | -2 | -1 | -1 | -1 | 2 | -√-3 | 0 | √-3 | -√-3 | 0 | √-3 | -√-3 | √-3 | 0 | 0 | 1 | -2 | 1 | 1 | complex lifted from C3⋊D4 |
ρ23 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | -2 | 2 | -1 | -1 | -1 | -√-3 | -√-3 | 0 | √-3 | √-3 | -√-3 | 0 | √-3 | 0 | 0 | 1 | 1 | -2 | 1 | complex lifted from C3⋊D4 |
ρ24 | 4 | -4 | 0 | 0 | -2 | 4 | -2 | -2 | 0 | 2 | 2 | -4 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4⋊S3, Schur index 2 |
ρ25 | 4 | -4 | 0 | 0 | -2 | -2 | -2 | 4 | 0 | -4 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4⋊S3, Schur index 2 |
ρ26 | 4 | -4 | 0 | 0 | 4 | -2 | -2 | -2 | 0 | 2 | -4 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4⋊S3, Schur index 2 |
ρ27 | 4 | -4 | 0 | 0 | -2 | -2 | 4 | -2 | 0 | 2 | 2 | 2 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4⋊S3, Schur index 2 |
(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)])
C32⋊7D8 is a maximal subgroup of
S3×D4⋊S3 Dic6⋊3D6 D12.7D6 Dic6.20D6 D8×C3⋊S3 C24⋊8D6 C24⋊7D6 C24.40D6 C62.131D4 C62.73D4 C62.74D4 He3⋊6D8 C36.18D6 C33⋊6D8 C33⋊7D8 C33⋊15D8
C32⋊7D8 is a maximal quotient of
C12.9Dic6 C62.113D4 C32⋊7D16 C32⋊8SD32 C32⋊10SD32 C32⋊7Q32 C62.116D4 C36.18D6 He3⋊7D8 C33⋊6D8 C33⋊7D8 C33⋊15D8
Matrix representation of C32⋊7D8 ►in GL6(𝔽73)
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 72 | 1 | 0 | 0 |
0 | 0 | 72 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
72 | 1 | 0 | 0 | 0 | 0 |
72 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 72 | 1 | 0 | 0 |
0 | 0 | 72 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
60 | 43 | 0 | 0 | 0 | 0 |
30 | 13 | 0 | 0 | 0 | 0 |
0 | 0 | 72 | 1 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 57 | 57 |
0 | 0 | 0 | 0 | 16 | 57 |
0 | 1 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 72 | 1 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 | 1 | 0 |
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] >;
C32⋊7D8 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
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