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G = C327C4≀C2order 288 = 25·32

The semidirect product of C32 and C4≀C2 acting via C4≀C2/Q8=C4

metabelian, soluble, monomial

Aliases: C327C4≀C2, C12⋊S32C4, Q82(C32⋊C4), (Q8×C32)⋊2C4, C3⋊Dic3.53D4, C2.9(C62⋊C4), C12.26D6.1C2, C32⋊M4(2)⋊3C2, (C4×C32⋊C4)⋊2C2, C4.4(C2×C32⋊C4), (C3×C12).4(C2×C4), (C2×C3⋊S3).15D4, (C4×C3⋊S3).8C22, (C3×C6).19(C22⋊C4), SmallGroup(288,433)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C327C4≀C2
C1C32C3×C6C3⋊Dic3C4×C3⋊S3C32⋊M4(2) — C327C4≀C2
C32C3×C6C3×C12 — C327C4≀C2
C1C2C4Q8

Generators and relations for C327C4≀C2
 G = < a,b,c,d,e | a3=b3=c4=d2=e4=1, ab=ba, ac=ca, dad=a-1, eae-1=ab-1, bc=cb, dbd=b-1, ebe-1=a-1b-1, dcd=c-1, ce=ec, ede-1=c-1d >

Subgroups: 520 in 84 conjugacy classes, 16 normal (all characteristic)
C1, C2, C2 [×2], C3 [×2], C4, C4 [×4], C22 [×2], S3 [×6], C6 [×2], C8, C2×C4 [×3], D4 [×2], Q8, C32, Dic3 [×2], C12 [×6], D6 [×6], C42, M4(2), C4○D4, C3⋊S3 [×2], C3×C6, C4×S3 [×6], D12 [×6], C3×Q8 [×2], C4≀C2, C3⋊Dic3, C3×C12, C3×C12, C32⋊C4 [×2], C2×C3⋊S3, C2×C3⋊S3, Q83S3 [×2], C322C8, C4×C3⋊S3, C4×C3⋊S3, C12⋊S3, C12⋊S3, Q8×C32, C2×C32⋊C4, C32⋊M4(2), C4×C32⋊C4, C12.26D6, C327C4≀C2
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, C4≀C2, C32⋊C4, C2×C32⋊C4, C62⋊C4, C327C4≀C2

Character table of C327C4≀C2

 class 12A2B2C3A3B4A4B4C4D4E4F4G4H6A6B8A8B12A12B12C12D12E12F
 size 11183644249918181818443636888888
ρ1111111111111111111111111    trivial
ρ2111-1111-111-1-1-1-11111-1-1-111-1    linear of order 2
ρ31111111111-1-1-1-111-1-1111111    linear of order 2
ρ4111-1111-111111111-1-1-1-1-111-1    linear of order 2
ρ511-1-11111-1-1i-i-ii11i-i111111    linear of order 4
ρ611-11111-1-1-1-iii-i11i-i-1-1-111-1    linear of order 4
ρ711-1-11111-1-1-iii-i11-ii111111    linear of order 4
ρ811-11111-1-1-1i-i-ii11-ii-1-1-111-1    linear of order 4
ρ922-2022-202200002200000-2-20    orthogonal lifted from D4
ρ10222022-20-2-200002200000-2-20    orthogonal lifted from D4
ρ112-20022002i-2i1-i-1-i1+i-1+i-2-200000000    complex lifted from C4≀C2
ρ122-2002200-2i2i1+i-1+i1-i-1-i-2-200000000    complex lifted from C4≀C2
ρ132-20022002i-2i-1+i1+i-1-i1-i-2-200000000    complex lifted from C4≀C2
ρ142-2002200-2i2i-1-i1-i-1+i1+i-2-200000000    complex lifted from C4≀C2
ρ1544001-2440000001-200-2111-2-2    orthogonal lifted from C32⋊C4
ρ1644001-2-400000001-20003-3-120    orthogonal lifted from C62⋊C4
ρ1744001-2-400000001-2000-33-120    orthogonal lifted from C62⋊C4
ρ184400-21-40000000-2100-3002-13    orthogonal lifted from C62⋊C4
ρ194400-21-40000000-21003002-1-3    orthogonal lifted from C62⋊C4
ρ204400-2144000000-21001-2-2-211    orthogonal lifted from C32⋊C4
ρ214400-214-4000000-2100-122-21-1    orthogonal lifted from C2×C32⋊C4
ρ2244001-24-40000001-2002-1-11-22    orthogonal lifted from C2×C32⋊C4
ρ238-8002-400000000-2400000000    orthogonal faithful, Schur index 2
ρ248-800-42000000004-200000000    orthogonal faithful, Schur index 2

Smallest permutation representation of C327C4≀C2
On 48 points
Generators in S48
(5 45 10)(6 46 11)(7 47 12)(8 48 9)(33 44 39)(34 41 40)(35 42 37)(36 43 38)
(1 17 16)(2 18 13)(3 19 14)(4 20 15)(5 45 10)(6 46 11)(7 47 12)(8 48 9)(21 30 25)(22 31 26)(23 32 27)(24 29 28)(33 44 39)(34 41 40)(35 42 37)(36 43 38)
(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 2)(3 4)(5 12)(6 11)(7 10)(8 9)(13 17)(14 20)(15 19)(16 18)(21 24)(22 23)(25 29)(26 32)(27 31)(28 30)(33 35)(37 44)(38 43)(39 42)(40 41)(45 47)
(1 45 23 36)(2 46 24 33)(3 47 21 34)(4 48 22 35)(5 27 43 17)(6 28 44 18)(7 25 41 19)(8 26 42 20)(9 31 37 15)(10 32 38 16)(11 29 39 13)(12 30 40 14)

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

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

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

Matrix representation of C327C4≀C2 in GL6(𝔽73)

100000
010000
001000
000100
0000072
0000172
,
100000
010000
0072100
0072000
0000072
0000172
,
010000
7200000
001000
000100
000010
000001
,
0720000
7200000
000100
001000
000001
000010
,
13600000
13130000
000001
000010
0072000
0007200

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

C327C4≀C2 in GAP, Magma, Sage, TeX

C_3^2\rtimes_7C_4\wr C_2
% in TeX

G:=Group("C3^2:7C4wrC2");
// GroupNames label

G:=SmallGroup(288,433);
// by ID

G=gap.SmallGroup(288,433);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,141,120,100,675,346,80,9413,691,12550,2372]);
// Polycyclic

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

Export

Character table of C327C4≀C2 in TeX

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