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G = C2×C9⋊D4order 144 = 24·32

Direct product of C2 and C9⋊D4

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C9⋊D4, C182D4, C232D9, C223D18, D183C22, C18.10C23, Dic92C22, C93(C2×D4), (C2×C6).29D6, (C2×C18)⋊3C22, (C22×C18)⋊2C2, (C2×Dic9)⋊4C2, (C22×D9)⋊3C2, (C22×C6).8S3, C6.19(C3⋊D4), C6.28(C22×S3), C2.10(C22×D9), C3.(C2×C3⋊D4), SmallGroup(144,46)

Series: Derived Chief Lower central Upper central

C1C18 — C2×C9⋊D4
C1C3C9C18D18C22×D9 — C2×C9⋊D4
C9C18 — C2×C9⋊D4
C1C22C23

Generators and relations for C2×C9⋊D4
 G = < a,b,c,d | a2=b9=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 295 in 81 conjugacy classes, 35 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C22, C22 [×2], C22 [×6], S3 [×2], C6, C6 [×2], C6 [×2], C2×C4, D4 [×4], C23, C23, C9, Dic3 [×2], D6 [×4], C2×C6, C2×C6 [×2], C2×C6 [×2], C2×D4, D9 [×2], C18, C18 [×2], C18 [×2], C2×Dic3, C3⋊D4 [×4], C22×S3, C22×C6, Dic9 [×2], D18 [×2], D18 [×2], C2×C18, C2×C18 [×2], C2×C18 [×2], C2×C3⋊D4, C2×Dic9, C9⋊D4 [×4], C22×D9, C22×C18, C2×C9⋊D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, D9, C3⋊D4 [×2], C22×S3, D18 [×3], C2×C3⋊D4, C9⋊D4 [×2], C22×D9, C2×C9⋊D4

Smallest permutation representation of C2×C9⋊D4
On 72 points
Generators in S72
(1 40)(2 41)(3 42)(4 43)(5 44)(6 45)(7 37)(8 38)(9 39)(10 46)(11 47)(12 48)(13 49)(14 50)(15 51)(16 52)(17 53)(18 54)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(25 61)(26 62)(27 63)(28 64)(29 65)(30 66)(31 67)(32 68)(33 69)(34 70)(35 71)(36 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 22 13 31)(2 21 14 30)(3 20 15 29)(4 19 16 28)(5 27 17 36)(6 26 18 35)(7 25 10 34)(8 24 11 33)(9 23 12 32)(37 61 46 70)(38 60 47 69)(39 59 48 68)(40 58 49 67)(41 57 50 66)(42 56 51 65)(43 55 52 64)(44 63 53 72)(45 62 54 71)
(2 9)(3 8)(4 7)(5 6)(10 16)(11 15)(12 14)(17 18)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)(25 28)(26 36)(27 35)(37 43)(38 42)(39 41)(44 45)(46 52)(47 51)(48 50)(53 54)(55 70)(56 69)(57 68)(58 67)(59 66)(60 65)(61 64)(62 72)(63 71)

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

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

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

C2×C9⋊D4 is a maximal subgroup of
C22.D36  Dic94D4  C223D36  C23.9D18  D18⋊D4  Dic9.D4  C22.4D36  C23.28D18  C367D4  C232D18  C362D4  Dic9⋊D4  C36⋊D4  C244D9  C2×D4×D9  D46D18
C2×C9⋊D4 is a maximal quotient of
C36.49D4  C23.28D18  C367D4  D366C22  C23.23D18  C36.17D4  C232D18  C362D4  Dic9⋊D4  C36⋊D4  C36.C23  Dic9⋊Q8  D183Q8  C36.23D4  D4.D18  D4⋊D18  D4.9D18  C244D9

42 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B6A···6G9A9B9C18A···18U
order122222223446···699918···18
size1111221818218182···22222···2

42 irreducible representations

dim111112222222
type++++++++++
imageC1C2C2C2C2S3D4D6D9C3⋊D4D18C9⋊D4
kernelC2×C9⋊D4C2×Dic9C9⋊D4C22×D9C22×C18C22×C6C18C2×C6C23C6C22C2
# reps1141112334912

Matrix representation of C2×C9⋊D4 in GL4(𝔽37) generated by

36000
03600
0010
0001
,
363600
1000
001726
00116
,
36000
1100
001335
001124
,
1000
363600
001120
003126
G:=sub<GL(4,GF(37))| [36,0,0,0,0,36,0,0,0,0,1,0,0,0,0,1],[36,1,0,0,36,0,0,0,0,0,17,11,0,0,26,6],[36,1,0,0,0,1,0,0,0,0,13,11,0,0,35,24],[1,36,0,0,0,36,0,0,0,0,11,31,0,0,20,26] >;

C2×C9⋊D4 in GAP, Magma, Sage, TeX

C_2\times C_9\rtimes D_4
% in TeX

G:=Group("C2xC9:D4");
// GroupNames label

G:=SmallGroup(144,46);
// by ID

G=gap.SmallGroup(144,46);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,218,2404,208,3461]);
// Polycyclic

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

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