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G = C62.154C23order 288 = 25·32

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C62.154C23
 Chief series C1 — C3 — C32 — C3×C6 — C2×C3⋊S3 — C22×C3⋊S3 — D4×C3⋊S3 — C62.154C23
 Lower central C32 — C3×C6 — C62.154C23
 Upper central C1 — C2 — C4○D4

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 [×9], C3 [×4], C4, C4 [×3], C4 [×2], C22 [×3], C22 [×12], S3 [×24], C6 [×4], C6 [×12], C2×C4 [×3], C2×C4 [×6], D4 [×3], D4 [×15], Q8, Q8, C23 [×6], C32, Dic3 [×8], C12 [×16], D6 [×48], C2×C6 [×12], C2×D4 [×9], C4○D4, C4○D4 [×5], C3⋊S3 [×6], C3×C6, C3×C6 [×3], Dic6 [×4], C4×S3 [×24], D12 [×36], C3⋊D4 [×24], C2×C12 [×12], C3×D4 [×12], C3×Q8 [×4], C22×S3 [×24], 2+ 1+4, C3⋊Dic3 [×2], C3×C12, C3×C12 [×3], C2×C3⋊S3 [×6], C2×C3⋊S3 [×6], C62 [×3], C2×D12 [×12], C4○D12 [×12], S3×D4 [×24], Q83S3 [×8], C3×C4○D4 [×4], C324Q8, C4×C3⋊S3 [×6], C12⋊S3 [×9], C327D4 [×6], C6×C12 [×3], D4×C32 [×3], Q8×C32, C22×C3⋊S3 [×6], D4○D12 [×4], C2×C12⋊S3 [×3], C12.59D6 [×3], D4×C3⋊S3 [×6], C12.26D6 [×2], C32×C4○D4, C62.154C23
Quotients: C1, C2 [×15], C22 [×35], S3 [×4], C23 [×15], D6 [×28], C24, C3⋊S3, C22×S3 [×28], 2+ 1+4, C2×C3⋊S3 [×7], S3×C23 [×4], C22×C3⋊S3 [×7], D4○D12 [×4], C23×C3⋊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 32 5 29 8 12)(2 33 6 30 9 10)(3 31 4 28 7 11)(13 17 35 22 26 19)(14 18 36 23 27 20)(15 16 34 24 25 21)(37 44 50 40 47 53)(38 45 51 41 48 54)(39 46 52 42 43 49)(55 63 72 58 66 69)(56 64 67 59 61 70)(57 65 68 60 62 71)
(2 3)(4 9)(5 8)(6 7)(10 31)(11 33)(12 32)(13 20)(14 19)(15 21)(16 25)(17 27)(18 26)(22 36)(23 35)(24 34)(28 30)(38 42)(39 41)(43 54)(44 53)(45 52)(46 51)(47 50)(48 49)(55 56)(57 60)(58 59)(61 72)(62 71)(63 70)(64 69)(65 68)(66 67)
(1 37)(2 41)(3 39)(4 52)(5 50)(6 54)(7 43)(8 47)(9 45)(10 51)(11 49)(12 53)(13 69)(14 67)(15 71)(16 57)(17 55)(18 59)(19 66)(20 64)(21 62)(22 72)(23 70)(24 68)(25 60)(26 58)(27 56)(28 42)(29 40)(30 38)(31 46)(32 44)(33 48)(34 65)(35 63)(36 61)
(1 16 29 25)(2 17 30 26)(3 18 28 27)(4 23 11 14)(5 24 12 15)(6 22 10 13)(7 20 31 36)(8 21 32 34)(9 19 33 35)(37 57 40 60)(38 58 41 55)(39 59 42 56)(43 64 46 61)(44 65 47 62)(45 66 48 63)(49 67 52 70)(50 68 53 71)(51 69 54 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), (1,32,5,29,8,12)(2,33,6,30,9,10)(3,31,4,28,7,11)(13,17,35,22,26,19)(14,18,36,23,27,20)(15,16,34,24,25,21)(37,44,50,40,47,53)(38,45,51,41,48,54)(39,46,52,42,43,49)(55,63,72,58,66,69)(56,64,67,59,61,70)(57,65,68,60,62,71), (2,3)(4,9)(5,8)(6,7)(10,31)(11,33)(12,32)(13,20)(14,19)(15,21)(16,25)(17,27)(18,26)(22,36)(23,35)(24,34)(28,30)(38,42)(39,41)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)(55,56)(57,60)(58,59)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67), (1,37)(2,41)(3,39)(4,52)(5,50)(6,54)(7,43)(8,47)(9,45)(10,51)(11,49)(12,53)(13,69)(14,67)(15,71)(16,57)(17,55)(18,59)(19,66)(20,64)(21,62)(22,72)(23,70)(24,68)(25,60)(26,58)(27,56)(28,42)(29,40)(30,38)(31,46)(32,44)(33,48)(34,65)(35,63)(36,61), (1,16,29,25)(2,17,30,26)(3,18,28,27)(4,23,11,14)(5,24,12,15)(6,22,10,13)(7,20,31,36)(8,21,32,34)(9,19,33,35)(37,57,40,60)(38,58,41,55)(39,59,42,56)(43,64,46,61)(44,65,47,62)(45,66,48,63)(49,67,52,70)(50,68,53,71)(51,69,54,72)>;`

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

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

57 conjugacy classes

 class 1 2A 2B 2C 2D 2E ··· 2J 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E ··· 6P 12A ··· 12H 12I ··· 12T order 1 2 2 2 2 2 ··· 2 3 3 3 3 4 4 4 4 4 4 6 6 6 6 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 2 2 2 18 ··· 18 2 2 2 2 2 2 2 2 18 18 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

57 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D6 D6 D6 2+ 1+4 D4○D12 kernel C62.154C23 C2×C12⋊S3 C12.59D6 D4×C3⋊S3 C12.26D6 C32×C4○D4 C3×C4○D4 C2×C12 C3×D4 C3×Q8 C32 C3 # reps 1 3 3 6 2 1 4 12 12 4 1 8

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

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 12 12 0 0 0 0 0 0 0 12 0 0 0 0 1 1
,
 12 12 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 12 0 0 0 0 0 0 0 1 1 0 0 0 0 12 0
,
 12 0 0 0 0 0 1 1 0 0 0 0 0 0 12 0 0 0 0 0 1 1 0 0 0 0 0 0 12 0 0 0 0 0 1 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 6 0 0 0 0 7 10 0 0 0 0 0 0 3 6 0 0 0 0 7 10

`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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