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G = C324C8order 72 = 23·32

2nd semidirect product of C32 and C8 acting via C8/C4=C2

metabelian, supersoluble, monomial, A-group

Aliases: C324C8, C12.6S3, C6.3Dic3, C3⋊(C3⋊C8), (C3×C6).3C4, C4.2(C3⋊S3), (C3×C12).4C2, C2.(C3⋊Dic3), SmallGroup(72,13)

Series: Derived Chief Lower central Upper central

C1C32 — C324C8
C1C3C32C3×C6C3×C12 — C324C8
C32 — C324C8
C1C4

Generators and relations for C324C8
 G = < a,b,c | a3=b3=c8=1, ab=ba, cac-1=a-1, cbc-1=b-1 >

9C8
3C3⋊C8
3C3⋊C8
3C3⋊C8
3C3⋊C8

Character table of C324C8

 class 123A3B3C3D4A4B6A6B6C6D8A8B8C8D12A12B12C12D12E12F12G12H
 size 112222112222999922222222
ρ1111111111111111111111111    trivial
ρ2111111111111-1-1-1-111111111    linear of order 2
ρ3111111-1-11111-ii-ii-1-1-1-1-1-1-1-1    linear of order 4
ρ4111111-1-11111i-ii-i-1-1-1-1-1-1-1-1    linear of order 4
ρ51-11111-ii-1-1-1-1ζ83ζ8ζ87ζ85-i-i-i-iiiii    linear of order 8
ρ61-11111i-i-1-1-1-1ζ85ζ87ζ8ζ83iiii-i-i-i-i    linear of order 8
ρ71-11111-ii-1-1-1-1ζ87ζ85ζ83ζ8-i-i-i-iiiii    linear of order 8
ρ81-11111i-i-1-1-1-1ζ8ζ83ζ85ζ87iiii-i-i-i-i    linear of order 8
ρ922-12-1-1222-1-1-10000-1-12-1-12-1-1    orthogonal lifted from S3
ρ10222-1-1-122-1-1-120000-12-1-12-1-1-1    orthogonal lifted from S3
ρ1122-1-1-1222-1-12-100002-1-1-1-1-1-12    orthogonal lifted from S3
ρ1222-1-12-122-12-1-10000-1-1-12-1-12-1    orthogonal lifted from S3
ρ1322-1-12-1-2-2-12-1-10000111-211-21    symplectic lifted from Dic3, Schur index 2
ρ14222-1-1-1-2-2-1-1-1200001-211-2111    symplectic lifted from Dic3, Schur index 2
ρ1522-1-1-12-2-2-1-12-10000-2111111-2    symplectic lifted from Dic3, Schur index 2
ρ1622-12-1-1-2-22-1-1-1000011-211-211    symplectic lifted from Dic3, Schur index 2
ρ172-22-1-1-12i-2i111-20000-i2i-i-i-2iiii    complex lifted from C3⋊C8
ρ182-2-1-12-1-2i2i1-2110000iii-2i-i-i2i-i    complex lifted from C3⋊C8
ρ192-2-12-1-12i-2i-21110000-i-i2i-ii-2iii    complex lifted from C3⋊C8
ρ202-2-12-1-1-2i2i-21110000ii-2ii-i2i-i-i    complex lifted from C3⋊C8
ρ212-2-1-12-12i-2i1-2110000-i-i-i2iii-2ii    complex lifted from C3⋊C8
ρ222-22-1-1-1-2i2i111-20000i-2iii2i-i-i-i    complex lifted from C3⋊C8
ρ232-2-1-1-12-2i2i11-210000-2iiii-i-i-i2i    complex lifted from C3⋊C8
ρ242-2-1-1-122i-2i11-2100002i-i-i-iiii-2i    complex lifted from C3⋊C8

Smallest permutation representation of C324C8
Regular action on 72 points
Generators in S72
(1 68 47)(2 48 69)(3 70 41)(4 42 71)(5 72 43)(6 44 65)(7 66 45)(8 46 67)(9 60 21)(10 22 61)(11 62 23)(12 24 63)(13 64 17)(14 18 57)(15 58 19)(16 20 59)(25 56 37)(26 38 49)(27 50 39)(28 40 51)(29 52 33)(30 34 53)(31 54 35)(32 36 55)
(1 24 31)(2 32 17)(3 18 25)(4 26 19)(5 20 27)(6 28 21)(7 22 29)(8 30 23)(9 44 40)(10 33 45)(11 46 34)(12 35 47)(13 48 36)(14 37 41)(15 42 38)(16 39 43)(49 58 71)(50 72 59)(51 60 65)(52 66 61)(53 62 67)(54 68 63)(55 64 69)(56 70 57)
(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)

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

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

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

C324C8 is a maximal subgroup of
C322C16  S3×C3⋊C8  D6.Dic3  C322D8  Dic6⋊S3  C322Q16  C8×C3⋊S3  C24⋊S3  C12.58D6  C327D8  C329SD16  C3211SD16  C327Q16  He33C8  C36.S3  C337C8  C12.12S4  C3⋊U2(𝔽3)  C60.S3  C30.Dic3
C324C8 is a maximal quotient of
C24.S3  C36.S3  He34C8  C337C8  C12.12S4  C60.S3  C30.Dic3

Matrix representation of C324C8 in GL5(𝔽73)

10000
00100
0727200
00013
0007271
,
10000
00100
0727200
00010
00001
,
630000
0606200
021300
0004934
0001124

G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,0,72,0,0,0,1,72,0,0,0,0,0,1,72,0,0,0,3,71],[1,0,0,0,0,0,0,72,0,0,0,1,72,0,0,0,0,0,1,0,0,0,0,0,1],[63,0,0,0,0,0,60,2,0,0,0,62,13,0,0,0,0,0,49,11,0,0,0,34,24] >;

C324C8 in GAP, Magma, Sage, TeX

C_3^2\rtimes_4C_8
% in TeX

G:=Group("C3^2:4C8");
// GroupNames label

G:=SmallGroup(72,13);
// by ID

G=gap.SmallGroup(72,13);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-3,10,26,323,1204]);
// Polycyclic

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

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Subgroup lattice of C324C8 in TeX
Character table of C324C8 in TeX

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