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G = C12.12S4order 288 = 25·32

12nd non-split extension by C12 of S4 acting via S4/A4=C2

non-abelian, soluble, monomial

Aliases: C12.12S4, A4⋊(C3⋊C8), C3⋊(A4⋊C8), (C3×A4)⋊2C8, C4.4(C3⋊S4), (C4×A4).2S3, (C6×A4).2C4, (C2×A4).Dic3, C6.5(A4⋊C4), (C12×A4).5C2, C22⋊(C324C8), (C22×C12).7S3, C23.(C3⋊Dic3), C2.1(C6.7S4), (C22×C6).4Dic3, (C2×C6)⋊2(C3⋊C8), (C22×C4).1(C3⋊S3), SmallGroup(288,402)

Series: Derived Chief Lower central Upper central

C1C22C3×A4 — C12.12S4
C1C22C2×C6C3×A4C6×A4C12×A4 — C12.12S4
C3×A4 — C12.12S4
C1C4

Generators and relations for C12.12S4
 G = < a,b,c,d,e | a12=b2=c2=d3=1, e2=a9, ab=ba, ac=ca, ad=da, eae-1=a5, dbd-1=ebe-1=bc=cb, dcd-1=b, ce=ec, ede-1=d-1 >

Subgroups: 248 in 68 conjugacy classes, 25 normal (16 characteristic)
C1, C2, C2 [×2], C3, C3 [×3], C4, C4, C22, C22 [×2], C6, C6 [×5], C8 [×2], C2×C4 [×2], C23, C32, C12, C12 [×4], A4 [×3], C2×C6, C2×C6 [×2], C2×C8 [×2], C22×C4, C3×C6, C3⋊C8 [×5], C2×C12 [×2], C2×A4 [×3], C22×C6, C22⋊C8, C3×C12, C3×A4, C2×C3⋊C8 [×2], C4×A4 [×3], C22×C12, C324C8, C6×A4, C12.55D4, A4⋊C8 [×3], C12×A4, C12.12S4
Quotients: C1, C2, C4, S3 [×4], C8, Dic3 [×4], C3⋊S3, C3⋊C8 [×4], S4, C3⋊Dic3, A4⋊C4, C324C8, C3⋊S4, A4⋊C8, C6.7S4, C12.12S4

Smallest permutation representation of C12.12S4
On 72 points
Generators in S72
(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)
(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)
(1 27 37)(2 28 38)(3 29 39)(4 30 40)(5 31 41)(6 32 42)(7 33 43)(8 34 44)(9 35 45)(10 36 46)(11 25 47)(12 26 48)(13 54 61)(14 55 62)(15 56 63)(16 57 64)(17 58 65)(18 59 66)(19 60 67)(20 49 68)(21 50 69)(22 51 70)(23 52 71)(24 53 72)
(1 64 10 61 7 70 4 67)(2 69 11 66 8 63 5 72)(3 62 12 71 9 68 6 65)(13 43 22 40 19 37 16 46)(14 48 23 45 20 42 17 39)(15 41 24 38 21 47 18 44)(25 59 34 56 31 53 28 50)(26 52 35 49 32 58 29 55)(27 57 36 54 33 51 30 60)

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

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

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

36 conjugacy classes

class 1 2A2B2C3A3B3C3D4A4B4C4D6A6B6C6D6E6F8A···8H12A12B12C12D12E···12J
order1222333344446666668···81212121212···12
size11332888113326688818···1822668···8

36 irreducible representations

dim1111222222333666
type++++--++-
imageC1C2C4C8S3S3Dic3Dic3C3⋊C8C3⋊C8S4A4⋊C4A4⋊C8C3⋊S4C6.7S4C12.12S4
kernelC12.12S4C12×A4C6×A4C3×A4C4×A4C22×C12C2×A4C22×C6A4C2×C6C12C6C3C4C2C1
# reps1124313162224112

Matrix representation of C12.12S4 in GL5(𝔽73)

490000
770000
004600
000460
000046
,
10000
01000
00100
000720
000072
,
10000
01000
007200
00010
000072
,
640000
308000
00001
00100
00010
,
1957000
3054000
000051
000510
005100

G:=sub<GL(5,GF(73))| [49,7,0,0,0,0,70,0,0,0,0,0,46,0,0,0,0,0,46,0,0,0,0,0,46],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,72],[1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,72],[64,30,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[19,30,0,0,0,57,54,0,0,0,0,0,0,0,51,0,0,0,51,0,0,0,51,0,0] >;

C12.12S4 in GAP, Magma, Sage, TeX

C_{12}._{12}S_4
% in TeX

G:=Group("C12.12S4");
// GroupNames label

G:=SmallGroup(288,402);
// by ID

G=gap.SmallGroup(288,402);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,14,36,451,1684,6053,782,3534,1350]);
// Polycyclic

G:=Group<a,b,c,d,e|a^12=b^2=c^2=d^3=1,e^2=a^9,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^5,d*b*d^-1=e*b*e^-1=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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