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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), (C3xA4):2C8, C4.4(C3:S4), (C4xA4).2S3, (C6xA4).2C4, (C2xA4).Dic3, C6.5(A4:C4), (C12xA4).5C2, C22:(C32:4C8), (C22xC12).7S3, C23.(C3:Dic3), C2.1(C6.7S4), (C22xC6).4Dic3, (C2xC6):2(C3:C8), (C22xC4).1(C3:S3), SmallGroup(288,402)

Series: Derived Chief Lower central Upper central

C1C22C3xA4 — C12.12S4
C1C22C2xC6C3xA4C6xA4C12xA4 — C12.12S4
C3xA4 — 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, C3, C3, C4, C4, C22, C22, C6, C6, C8, C2xC4, C23, C32, C12, C12, A4, C2xC6, C2xC6, C2xC8, C22xC4, C3xC6, C3:C8, C2xC12, C2xA4, C22xC6, C22:C8, C3xC12, C3xA4, C2xC3:C8, C4xA4, C22xC12, C32:4C8, C6xA4, C12.55D4, A4:C8, C12xA4, C12.12S4
Quotients: C1, C2, C4, S3, C8, Dic3, C3:S3, C3:C8, S4, C3:Dic3, A4:C4, C32:4C8, 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)
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(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 30 63)(2 31 64)(3 32 65)(4 33 66)(5 34 67)(6 35 68)(7 36 69)(8 25 70)(9 26 71)(10 27 72)(11 28 61)(12 29 62)(13 41 60)(14 42 49)(15 43 50)(16 44 51)(17 45 52)(18 46 53)(19 47 54)(20 48 55)(21 37 56)(22 38 57)(23 39 58)(24 40 59)
(1 14 10 23 7 20 4 17)(2 19 11 16 8 13 5 22)(3 24 12 21 9 18 6 15)(25 60 34 57 31 54 28 51)(26 53 35 50 32 59 29 56)(27 58 36 55 33 52 30 49)(37 71 46 68 43 65 40 62)(38 64 47 61 44 70 41 67)(39 69 48 66 45 63 42 72)

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

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

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

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
kernelC12.12S4C12xA4C6xA4C3xA4C4xA4C22xC12C2xA4C22xC6A4C2xC6C12C6C3C4C2C1
# 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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