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G = C324Q8order 72 = 23·32

2nd semidirect product of C32 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial

Aliases: C324Q8, C12.3S3, C32Dic6, C6.12D6, C4.(C3⋊S3), (C3×C12).1C2, C3⋊Dic3.3C2, (C3×C6).11C22, C2.3(C2×C3⋊S3), SmallGroup(72,31)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C324Q8
C1C3C32C3×C6C3⋊Dic3 — C324Q8
C32C3×C6 — C324Q8
C1C2C4

Generators and relations for C324Q8
 G = < a,b,c,d | a3=b3=c4=1, d2=c2, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c-1 >

9C4
9C4
9Q8
3Dic3
3Dic3
3Dic3
3Dic3
3Dic3
3Dic3
3Dic3
3Dic3
3Dic6
3Dic6
3Dic6
3Dic6

Character table of C324Q8

 class 123A3B3C3D4A4B4C6A6B6C6D12A12B12C12D12E12F12G12H
 size 11222221818222222222222
ρ1111111111111111111111    trivial
ρ21111111-1-1111111111111    linear of order 2
ρ3111111-11-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ4111111-1-111111-1-1-1-1-1-1-1-1    linear of order 2
ρ522-1-12-1200-1-1-12-1-12-12-1-1-1    orthogonal lifted from S3
ρ622-1-1-12200-12-1-1-1-1-1-1-1-122    orthogonal lifted from S3
ρ722-12-1-1-200-1-12-11-2111-211    orthogonal lifted from D6
ρ822-12-1-1200-1-12-1-12-1-1-12-1-1    orthogonal lifted from S3
ρ9222-1-1-1-2002-1-1-1-211-21111    orthogonal lifted from D6
ρ1022-1-12-1-200-1-1-1211-21-2111    orthogonal lifted from D6
ρ11222-1-1-12002-1-1-12-1-12-1-1-1-1    orthogonal lifted from S3
ρ1222-1-1-12-200-12-1-1111111-2-2    orthogonal lifted from D6
ρ132-22222000-2-2-2-200000000    symplectic lifted from Q8, Schur index 2
ρ142-2-1-1-120001-211-3-3-333300    symplectic lifted from Dic6, Schur index 2
ρ152-2-12-1-100011-21-3033-303-3    symplectic lifted from Dic6, Schur index 2
ρ162-2-1-12-1000111-2-33030-3-33    symplectic lifted from Dic6, Schur index 2
ρ172-22-1-1-1000-21110-330-33-33    symplectic lifted from Dic6, Schur index 2
ρ182-22-1-1-1000-211103-303-33-3    symplectic lifted from Dic6, Schur index 2
ρ192-2-1-1-120001-211333-3-3-300    symplectic lifted from Dic6, Schur index 2
ρ202-2-12-1-100011-2130-3-330-33    symplectic lifted from Dic6, Schur index 2
ρ212-2-1-12-1000111-23-30-3033-3    symplectic lifted from Dic6, Schur index 2

Smallest permutation representation of C324Q8
Regular action on 72 points
Generators in S72
(1 40 7)(2 37 8)(3 38 5)(4 39 6)(9 66 46)(10 67 47)(11 68 48)(12 65 45)(13 51 31)(14 52 32)(15 49 29)(16 50 30)(17 42 62)(18 43 63)(19 44 64)(20 41 61)(21 58 33)(22 59 34)(23 60 35)(24 57 36)(25 56 71)(26 53 72)(27 54 69)(28 55 70)
(1 21 29)(2 22 30)(3 23 31)(4 24 32)(5 35 51)(6 36 52)(7 33 49)(8 34 50)(9 69 43)(10 70 44)(11 71 41)(12 72 42)(13 38 60)(14 39 57)(15 40 58)(16 37 59)(17 45 53)(18 46 54)(19 47 55)(20 48 56)(25 61 68)(26 62 65)(27 63 66)(28 64 67)
(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 18 3 20)(2 17 4 19)(5 41 7 43)(6 44 8 42)(9 51 11 49)(10 50 12 52)(13 68 15 66)(14 67 16 65)(21 54 23 56)(22 53 24 55)(25 58 27 60)(26 57 28 59)(29 46 31 48)(30 45 32 47)(33 69 35 71)(34 72 36 70)(37 62 39 64)(38 61 40 63)

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

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

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

C324Q8 is a maximal subgroup of
D12.S3  C323Q16  C242S3  C325Q16  C329SD16  C327Q16  S3×Dic6  D125S3  C12.59D6  C12.D6  Q8×C3⋊S3  He33Q8  C12.D9  C334Q8  C338Q8  A4⋊Dic6  C12.6S4  C15⋊Dic6  C12.D15
C324Q8 is a maximal quotient of
C6.Dic6  C12⋊Dic3  C12.D9  He34Q8  C334Q8  C338Q8  A4⋊Dic6  C15⋊Dic6  C12.D15

Matrix representation of C324Q8 in GL4(𝔽13) generated by

1000
0100
0090
0053
,
0100
121200
0030
0089
,
3600
71000
00120
00012
,
5000
8800
0028
001111
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,9,5,0,0,0,3],[0,12,0,0,1,12,0,0,0,0,3,8,0,0,0,9],[3,7,0,0,6,10,0,0,0,0,12,0,0,0,0,12],[5,8,0,0,0,8,0,0,0,0,2,11,0,0,8,11] >;

C324Q8 in GAP, Magma, Sage, TeX

C_3^2\rtimes_4Q_8
% in TeX

G:=Group("C3^2:4Q8");
// GroupNames label

G:=SmallGroup(72,31);
// by ID

G=gap.SmallGroup(72,31);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-3,20,61,26,323,1204]);
// Polycyclic

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

Export

Subgroup lattice of C324Q8 in TeX
Character table of C324Q8 in TeX

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