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G = C326C4≀C2order 288 = 25·32

The semidirect product of C32 and C4≀C2 acting via C4≀C2/D4=C4

metabelian, soluble, monomial

Aliases: C326C4≀C2, D42(C32⋊C4), (D4×C32)⋊2C4, C324Q81C4, C3⋊Dic3.52D4, C2.7(C62⋊C4), C12.D6.1C2, C32⋊M4(2)⋊2C2, (C4×C32⋊C4)⋊1C2, C4.2(C2×C32⋊C4), (C3×C12).2(C2×C4), (C2×C3⋊S3).14D4, (C4×C3⋊S3).6C22, (C3×C6).17(C22⋊C4), SmallGroup(288,431)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C326C4≀C2
C1C32C3×C6C3⋊Dic3C4×C3⋊S3C32⋊M4(2) — C326C4≀C2
C32C3×C6C3×C12 — C326C4≀C2
C1C2C4D4

Generators and relations for C326C4≀C2
 G = < a,b,c,d,e | a3=b3=c4=d2=e4=1, ebe-1=ab=ba, ac=ca, ad=da, eae-1=a-1b, bc=cb, bd=db, dcd=c-1, ce=ec, ede-1=c-1d >

Subgroups: 456 in 84 conjugacy classes, 16 normal (all characteristic)
C1, C2, C2 [×2], C3 [×2], C4, C4 [×4], C22 [×2], S3 [×2], C6 [×6], C8, C2×C4 [×3], D4, D4, Q8, C32, Dic3 [×6], C12 [×2], D6 [×2], C2×C6 [×4], C42, M4(2), C4○D4, C3⋊S3, C3×C6, C3×C6, Dic6 [×2], C4×S3 [×2], C2×Dic3 [×4], C3⋊D4 [×4], C3×D4 [×2], C4≀C2, C3⋊Dic3, C3⋊Dic3, C3×C12, C32⋊C4 [×2], C2×C3⋊S3, C62, D42S3 [×2], C322C8, C324Q8, C4×C3⋊S3, C2×C3⋊Dic3, C327D4, D4×C32, C2×C32⋊C4, C32⋊M4(2), C4×C32⋊C4, C12.D6, C326C4≀C2
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, C4≀C2, C32⋊C4, C2×C32⋊C4, C62⋊C4, C326C4≀C2

Character table of C326C4≀C2

 class 12A2B2C3A3B4A4B4C4D4E4F4G4H6A6B6C6D6E6F8A8B12A12B
 size 11418442991818181836448888363688
ρ1111111111111111111111111    trivial
ρ211-11111111111-111-1-1-1-1-1-111    linear of order 2
ρ3111111111-1-1-1-11111111-1-111    linear of order 2
ρ411-1111111-1-1-1-1-111-1-1-1-11111    linear of order 2
ρ5111-1111-1-1-ii-ii-1111111i-i11    linear of order 4
ρ611-1-1111-1-1-ii-ii111-1-1-1-1-ii11    linear of order 4
ρ7111-1111-1-1i-ii-i-1111111-ii11    linear of order 4
ρ811-1-1111-1-1i-ii-i111-1-1-1-1i-i11    linear of order 4
ρ9220222-2-2-20000022000000-2-2    orthogonal lifted from D4
ρ10220-222-2220000022000000-2-2    orthogonal lifted from D4
ρ112-2002202i-2i-1-i1-i1+i-1+i0-2-200000000    complex lifted from C4≀C2
ρ122-2002202i-2i1+i-1+i-1-i1-i0-2-200000000    complex lifted from C4≀C2
ρ132-200220-2i2i1-i-1-i-1+i1+i0-2-200000000    complex lifted from C4≀C2
ρ142-200220-2i2i-1+i1+i1-i-1-i0-2-200000000    complex lifted from C4≀C2
ρ1544001-2-400000001-230-3000-12    orthogonal lifted from C62⋊C4
ρ1644-401-2400000001-2-12-12001-2    orthogonal lifted from C2×C32⋊C4
ρ1744-40-2140000000-212-12-100-21    orthogonal lifted from C2×C32⋊C4
ρ184440-2140000000-21-21-2100-21    orthogonal lifted from C32⋊C4
ρ194400-21-40000000-210-303002-1    orthogonal lifted from C62⋊C4
ρ2044401-2400000001-21-21-2001-2    orthogonal lifted from C32⋊C4
ρ2144001-2-400000001-2-303000-12    orthogonal lifted from C62⋊C4
ρ224400-21-40000000-21030-3002-1    orthogonal lifted from C62⋊C4
ρ238-8002-400000000-2400000000    symplectic faithful, Schur index 2
ρ248-800-42000000004-200000000    symplectic faithful, Schur index 2

Smallest permutation representation of C326C4≀C2
On 48 points
Generators in S48
(5 10 47)(6 11 48)(7 12 45)(8 9 46)(33 37 42)(34 38 43)(35 39 44)(36 40 41)
(1 14 19)(2 15 20)(3 16 17)(4 13 18)(5 10 47)(6 11 48)(7 12 45)(8 9 46)(21 27 32)(22 28 29)(23 25 30)(24 26 31)(33 37 42)(34 38 43)(35 39 44)(36 40 41)
(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)
(1 21)(2 24)(3 23)(4 22)(5 39)(6 38)(7 37)(8 40)(9 41)(10 44)(11 43)(12 42)(13 28)(14 27)(15 26)(16 25)(17 30)(18 29)(19 32)(20 31)(33 45)(34 48)(35 47)(36 46)
(1 48 3 46)(2 45 4 47)(5 20 12 13)(6 17 9 14)(7 18 10 15)(8 19 11 16)(21 33)(22 34)(23 35)(24 36)(25 39 30 44)(26 40 31 41)(27 37 32 42)(28 38 29 43)

G:=sub<Sym(48)| (5,10,47)(6,11,48)(7,12,45)(8,9,46)(33,37,42)(34,38,43)(35,39,44)(36,40,41), (1,14,19)(2,15,20)(3,16,17)(4,13,18)(5,10,47)(6,11,48)(7,12,45)(8,9,46)(21,27,32)(22,28,29)(23,25,30)(24,26,31)(33,37,42)(34,38,43)(35,39,44)(36,40,41), (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), (1,21)(2,24)(3,23)(4,22)(5,39)(6,38)(7,37)(8,40)(9,41)(10,44)(11,43)(12,42)(13,28)(14,27)(15,26)(16,25)(17,30)(18,29)(19,32)(20,31)(33,45)(34,48)(35,47)(36,46), (1,48,3,46)(2,45,4,47)(5,20,12,13)(6,17,9,14)(7,18,10,15)(8,19,11,16)(21,33)(22,34)(23,35)(24,36)(25,39,30,44)(26,40,31,41)(27,37,32,42)(28,38,29,43)>;

G:=Group( (5,10,47)(6,11,48)(7,12,45)(8,9,46)(33,37,42)(34,38,43)(35,39,44)(36,40,41), (1,14,19)(2,15,20)(3,16,17)(4,13,18)(5,10,47)(6,11,48)(7,12,45)(8,9,46)(21,27,32)(22,28,29)(23,25,30)(24,26,31)(33,37,42)(34,38,43)(35,39,44)(36,40,41), (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), (1,21)(2,24)(3,23)(4,22)(5,39)(6,38)(7,37)(8,40)(9,41)(10,44)(11,43)(12,42)(13,28)(14,27)(15,26)(16,25)(17,30)(18,29)(19,32)(20,31)(33,45)(34,48)(35,47)(36,46), (1,48,3,46)(2,45,4,47)(5,20,12,13)(6,17,9,14)(7,18,10,15)(8,19,11,16)(21,33)(22,34)(23,35)(24,36)(25,39,30,44)(26,40,31,41)(27,37,32,42)(28,38,29,43) );

G=PermutationGroup([(5,10,47),(6,11,48),(7,12,45),(8,9,46),(33,37,42),(34,38,43),(35,39,44),(36,40,41)], [(1,14,19),(2,15,20),(3,16,17),(4,13,18),(5,10,47),(6,11,48),(7,12,45),(8,9,46),(21,27,32),(22,28,29),(23,25,30),(24,26,31),(33,37,42),(34,38,43),(35,39,44),(36,40,41)], [(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)], [(1,21),(2,24),(3,23),(4,22),(5,39),(6,38),(7,37),(8,40),(9,41),(10,44),(11,43),(12,42),(13,28),(14,27),(15,26),(16,25),(17,30),(18,29),(19,32),(20,31),(33,45),(34,48),(35,47),(36,46)], [(1,48,3,46),(2,45,4,47),(5,20,12,13),(6,17,9,14),(7,18,10,15),(8,19,11,16),(21,33),(22,34),(23,35),(24,36),(25,39,30,44),(26,40,31,41),(27,37,32,42),(28,38,29,43)])

Matrix representation of C326C4≀C2 in GL6(𝔽73)

100000
010000
001000
000100
00007272
000010
,
100000
010000
000100
00727200
00007272
000010
,
2700000
46460000
001000
000100
000010
000001
,
27540000
46460000
001000
000100
000010
000001
,
4600000
1410000
000010
000001
001000
00727200

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,72,1,0,0,0,0,72,0],[27,46,0,0,0,0,0,46,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[27,46,0,0,0,0,54,46,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[46,14,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,1,0,0,0,0,0,0,1,0,0] >;

C326C4≀C2 in GAP, Magma, Sage, TeX

C_3^2\rtimes_6C_4\wr C_2
% in TeX

G:=Group("C3^2:6C4wrC2");
// GroupNames label

G:=SmallGroup(288,431);
// by ID

G=gap.SmallGroup(288,431);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,141,100,675,346,80,9413,691,12550,2372]);
// Polycyclic

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

Export

Character table of C326C4≀C2 in TeX

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