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G = C9×C23⋊C4order 288 = 25·32

Direct product of C9 and C23⋊C4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C9×C23⋊C4, C23⋊C36, (C2×C4)⋊C36, (C2×C36)⋊2C4, (C6×D4).6C6, C22⋊C41C18, (C2×C12).1C12, (C22×C18)⋊1C4, (D4×C18).7C2, (C2×D4).1C18, (C2×C18).22D4, C22.2(D4×C9), C22.2(C2×C36), C23.1(C2×C18), (C22×C6).2C12, C18.21(C22⋊C4), (C22×C18).1C22, C3.(C3×C23⋊C4), (C3×C23⋊C4).C3, (C9×C22⋊C4)⋊2C2, (C2×C6).25(C3×D4), C2.3(C9×C22⋊C4), (C2×C6).23(C2×C12), (C2×C18).19(C2×C4), (C3×C22⋊C4).1C6, C6.21(C3×C22⋊C4), (C22×C6).6(C2×C6), SmallGroup(288,49)

Series: Derived Chief Lower central Upper central

C1C22 — C9×C23⋊C4
C1C2C6C2×C6C22×C6C22×C18C9×C22⋊C4 — C9×C23⋊C4
C1C2C22 — C9×C23⋊C4
C1C18C22×C18 — C9×C23⋊C4

Generators and relations for C9×C23⋊C4
 G = < a,b,c,d,e | a9=b2=c2=d2=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=bcd, ece-1=cd=dc, de=ed >

Subgroups: 150 in 78 conjugacy classes, 36 normal (24 characteristic)
C1, C2, C2 [×4], C3, C4 [×3], C22, C22 [×2], C22 [×3], C6, C6 [×4], C2×C4, C2×C4 [×2], D4 [×2], C23 [×2], C9, C12 [×3], C2×C6, C2×C6 [×2], C2×C6 [×3], C22⋊C4 [×2], C2×D4, C18, C18 [×4], C2×C12, C2×C12 [×2], C3×D4 [×2], C22×C6 [×2], C23⋊C4, C36 [×3], C2×C18, C2×C18 [×2], C2×C18 [×3], C3×C22⋊C4 [×2], C6×D4, C2×C36, C2×C36 [×2], D4×C9 [×2], C22×C18 [×2], C3×C23⋊C4, C9×C22⋊C4 [×2], D4×C18, C9×C23⋊C4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, D4 [×2], C9, C12 [×2], C2×C6, C22⋊C4, C18 [×3], C2×C12, C3×D4 [×2], C23⋊C4, C36 [×2], C2×C18, C3×C22⋊C4, C2×C36, D4×C9 [×2], C3×C23⋊C4, C9×C22⋊C4, C9×C23⋊C4

Smallest permutation representation of C9×C23⋊C4
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 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 10)(8 11)(9 12)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 46)(28 59)(29 60)(30 61)(31 62)(32 63)(33 55)(34 56)(35 57)(36 58)(37 71)(38 72)(39 64)(40 65)(41 66)(42 67)(43 68)(44 69)(45 70)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 37)(7 38)(8 39)(9 40)(10 72)(11 64)(12 65)(13 66)(14 67)(15 68)(16 69)(17 70)(18 71)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)(25 28)(26 29)(27 30)(46 61)(47 62)(48 63)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)
(1 53)(2 54)(3 46)(4 47)(5 48)(6 49)(7 50)(8 51)(9 52)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 19)(17 20)(18 21)(28 66)(29 67)(30 68)(31 69)(32 70)(33 71)(34 72)(35 64)(36 65)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)
(10 72 22 34)(11 64 23 35)(12 65 24 36)(13 66 25 28)(14 67 26 29)(15 68 27 30)(16 69 19 31)(17 70 20 32)(18 71 21 33)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)

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

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

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

99 conjugacy classes

class 1 2A2B2C2D2E3A3B4A···4E6A6B6C···6H6I6J9A···9F12A···12J18A···18F18G···18X18Y···18AD36A···36AD
order122222334···4666···6669···912···1218···1818···1818···1836···36
size112224114···4112···2441···14···41···12···24···44···4

99 irreducible representations

dim111111111111111222444
type+++++
imageC1C2C2C3C4C4C6C6C9C12C12C18C18C36C36D4C3×D4D4×C9C23⋊C4C3×C23⋊C4C9×C23⋊C4
kernelC9×C23⋊C4C9×C22⋊C4D4×C18C3×C23⋊C4C2×C36C22×C18C3×C22⋊C4C6×D4C23⋊C4C2×C12C22×C6C22⋊C4C2×D4C2×C4C23C2×C18C2×C6C22C9C3C1
# reps1212224264412612122412126

Matrix representation of C9×C23⋊C4 in GL4(𝔽37) generated by

7000
0700
0070
0007
,
0010
0001
1000
0100
,
0100
1000
0001
0010
,
36000
03600
00360
00036
,
1000
03600
00036
0010
G:=sub<GL(4,GF(37))| [7,0,0,0,0,7,0,0,0,0,7,0,0,0,0,7],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[36,0,0,0,0,36,0,0,0,0,36,0,0,0,0,36],[1,0,0,0,0,36,0,0,0,0,0,1,0,0,36,0] >;

C9×C23⋊C4 in GAP, Magma, Sage, TeX

C_9\times C_2^3\rtimes C_4
% in TeX

G:=Group("C9xC2^3:C4");
// GroupNames label

G:=SmallGroup(288,49);
// by ID

G=gap.SmallGroup(288,49);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-2,168,197,268,4371,2951]);
// Polycyclic

G:=Group<a,b,c,d,e|a^9=b^2=c^2=d^2=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,e*c*e^-1=c*d=d*c,d*e=e*d>;
// generators/relations

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