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G = C62.154C23order 288 = 25·32

149th non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.154C23, C32:102+ 1+4, (C3xD4):20D6, (C2xC12):10D6, (C3xQ8):21D6, C3:5(D4oD12), (C6xC12):16C22, C6.65(S3xC23), (C3xC6).64C24, C12:S3:30C22, C12.59D6:13C2, C12.26D6:10C2, C12.116(C22xS3), (C3xC12).158C23, (D4xC32):27C22, C32:7D4:15C22, C3:Dic3.52C23, (Q8xC32):24C22, C32:4Q8:28C22, D4:8(C2xC3:S3), Q8:8(C2xC3:S3), (D4xC3:S3):10C2, (C3xC4oD4):8S3, C4oD4:5(C3:S3), (C4xC3:S3):9C22, (C32xC4oD4):9C2, (C2xC12:S3):22C2, C4.33(C22xC3:S3), C2.13(C23xC3:S3), (C2xC3:S3).56C23, (C2xC6).18(C22xS3), C22.3(C22xC3:S3), (C22xC3:S3):12C22, (C2xC4):4(C2xC3:S3), SmallGroup(288,1014)

Series: Derived Chief Lower central Upper central

C1C3xC6 — C62.154C23
C1C3C32C3xC6C2xC3:S3C22xC3:S3D4xC3:S3 — C62.154C23
C32C3xC6 — C62.154C23
C1C2C4oD4

Generators and relations for C62.154C23
 G = < a,b,c,d,e | a6=b6=c2=d2=1, e2=b3, ab=ba, cac=a-1, dad=ab3, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, de=ed >

Subgroups: 2052 in 498 conjugacy classes, 153 normal (12 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C6, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C32, Dic3, C12, D6, C2xC6, C2xD4, C4oD4, C4oD4, C3:S3, C3xC6, C3xC6, Dic6, C4xS3, D12, C3:D4, C2xC12, C3xD4, C3xQ8, C22xS3, 2+ 1+4, C3:Dic3, C3xC12, C3xC12, C2xC3:S3, C2xC3:S3, C62, C2xD12, C4oD12, S3xD4, Q8:3S3, C3xC4oD4, C32:4Q8, C4xC3:S3, C12:S3, C32:7D4, C6xC12, D4xC32, Q8xC32, C22xC3:S3, D4oD12, C2xC12:S3, C12.59D6, D4xC3:S3, C12.26D6, C32xC4oD4, C62.154C23
Quotients: C1, C2, C22, S3, C23, D6, C24, C3:S3, C22xS3, 2+ 1+4, C2xC3:S3, S3xC23, C22xC3:S3, D4oD12, C23xC3:S3, C62.154C23

Smallest permutation representation of C62.154C23
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)
(1 35 31 6 11 9)(2 36 32 4 12 7)(3 34 33 5 10 8)(13 23 19 18 29 26)(14 24 20 16 30 27)(15 22 21 17 28 25)(37 72 65 40 69 62)(38 67 66 41 70 63)(39 68 61 42 71 64)(43 49 55 46 52 58)(44 50 56 47 53 59)(45 51 57 48 54 60)
(1 32)(2 31)(3 33)(4 9)(5 8)(6 7)(11 12)(13 16)(14 18)(15 17)(19 24)(20 23)(21 22)(25 28)(26 30)(27 29)(35 36)(37 42)(38 41)(39 40)(43 54)(44 53)(45 52)(46 51)(47 50)(48 49)(55 57)(58 60)(61 72)(62 71)(63 70)(64 69)(65 68)(66 67)
(1 51)(2 49)(3 53)(4 52)(5 50)(6 54)(7 43)(8 47)(9 45)(10 56)(11 60)(12 58)(13 39)(14 37)(15 41)(16 40)(17 38)(18 42)(19 61)(20 65)(21 63)(22 70)(23 68)(24 72)(25 66)(26 64)(27 62)(28 67)(29 71)(30 69)(31 48)(32 46)(33 44)(34 59)(35 57)(36 55)
(1 19 6 26)(2 20 4 27)(3 21 5 25)(7 24 32 30)(8 22 33 28)(9 23 31 29)(10 15 34 17)(11 13 35 18)(12 14 36 16)(37 55 40 58)(38 56 41 59)(39 57 42 60)(43 72 46 69)(44 67 47 70)(45 68 48 71)(49 65 52 62)(50 66 53 63)(51 61 54 64)

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

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

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

57 conjugacy classes

class 1 2A2B2C2D2E···2J3A3B3C3D4A4B4C4D4E4F6A6B6C6D6E···6P12A···12H12I···12T
order122222···2333344444466666···612···1212···12
size1122218···1822222222181822224···42···24···4

57 irreducible representations

dim111111222244
type++++++++++++
imageC1C2C2C2C2C2S3D6D6D62+ 1+4D4oD12
kernelC62.154C23C2xC12:S3C12.59D6D4xC3:S3C12.26D6C32xC4oD4C3xC4oD4C2xC12C3xD4C3xQ8C32C3
# reps13362141212418

Matrix representation of C62.154C23 in GL6(F13)

1200000
0120000
000100
00121200
0000012
000011
,
12120000
100000
001100
0012000
000011
0000120
,
1200000
110000
0012000
001100
0000120
000011
,
100000
010000
000010
000001
001000
000100
,
100000
010000
003600
0071000
000036
0000710

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,0,1,0,0,0,0,12,1],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,0,0,0,0,1,12,0,0,0,0,1,0],[12,1,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,7,0,0,0,0,6,10,0,0,0,0,0,0,3,7,0,0,0,0,6,10] >;

C62.154C23 in GAP, Magma, Sage, TeX

C_6^2._{154}C_2^3
% in TeX

G:=Group("C6^2.154C2^3");
// GroupNames label

G:=SmallGroup(288,1014);
// by ID

G=gap.SmallGroup(288,1014);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,219,675,80,2693,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^6=c^2=d^2=1,e^2=b^3,a*b=b*a,c*a*c=a^-1,d*a*d=a*b^3,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=b^3*c,d*e=e*d>;
// generators/relations

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