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G = C2×C327D4order 144 = 24·32

Direct product of C2 and C327D4

direct product, metabelian, supersoluble, monomial

Aliases: C2×C327D4, C628C22, (C3×C6)⋊7D4, (C2×C6)⋊10D6, C63(C3⋊D4), (C2×C62)⋊3C2, (C22×C6)⋊4S3, C3213(C2×D4), C232(C3⋊S3), C6.39(C22×S3), (C3×C6).38C23, C3⋊Dic37C22, C34(C2×C3⋊D4), C223(C2×C3⋊S3), (C22×C3⋊S3)⋊5C2, (C2×C3⋊S3)⋊7C22, (C2×C3⋊Dic3)⋊8C2, C2.10(C22×C3⋊S3), SmallGroup(144,177)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C2×C327D4
C1C3C32C3×C6C2×C3⋊S3C22×C3⋊S3 — C2×C327D4
C32C3×C6 — C2×C327D4
C1C22C23

Generators and relations for C2×C327D4
 G = < a,b,c,d,e | a2=b3=c3=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ebe=b-1, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 514 in 162 conjugacy classes, 59 normal (11 characteristic)
C1, C2, C2 [×2], C2 [×4], C3 [×4], C4 [×2], C22, C22 [×2], C22 [×6], S3 [×8], C6 [×12], C6 [×8], C2×C4, D4 [×4], C23, C23, C32, Dic3 [×8], D6 [×16], C2×C6 [×12], C2×C6 [×8], C2×D4, C3⋊S3 [×2], C3×C6, C3×C6 [×2], C3×C6 [×2], C2×Dic3 [×4], C3⋊D4 [×16], C22×S3 [×4], C22×C6 [×4], C3⋊Dic3 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C62 [×2], C62 [×2], C2×C3⋊D4 [×4], C2×C3⋊Dic3, C327D4 [×4], C22×C3⋊S3, C2×C62, C2×C327D4
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], C23, D6 [×12], C2×D4, C3⋊S3, C3⋊D4 [×8], C22×S3 [×4], C2×C3⋊S3 [×3], C2×C3⋊D4 [×4], C327D4 [×2], C22×C3⋊S3, C2×C327D4

Smallest permutation representation of C2×C327D4
On 72 points
Generators in S72
(1 44)(2 41)(3 42)(4 43)(5 67)(6 68)(7 65)(8 66)(9 30)(10 31)(11 32)(12 29)(13 72)(14 69)(15 70)(16 71)(17 35)(18 36)(19 33)(20 34)(21 39)(22 40)(23 37)(24 38)(25 48)(26 45)(27 46)(28 47)(49 56)(50 53)(51 54)(52 55)(57 63)(58 64)(59 61)(60 62)
(1 71 36)(2 33 72)(3 69 34)(4 35 70)(5 37 12)(6 9 38)(7 39 10)(8 11 40)(13 41 19)(14 20 42)(15 43 17)(16 18 44)(21 31 65)(22 66 32)(23 29 67)(24 68 30)(25 63 54)(26 55 64)(27 61 56)(28 53 62)(45 52 58)(46 59 49)(47 50 60)(48 57 51)
(1 8 27)(2 28 5)(3 6 25)(4 26 7)(9 63 69)(10 70 64)(11 61 71)(12 72 62)(13 60 29)(14 30 57)(15 58 31)(16 32 59)(17 52 21)(18 22 49)(19 50 23)(20 24 51)(33 53 37)(34 38 54)(35 55 39)(36 40 56)(41 47 67)(42 68 48)(43 45 65)(44 66 46)
(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 44)(2 43)(3 42)(4 41)(5 45)(6 48)(7 47)(8 46)(9 51)(10 50)(11 49)(12 52)(13 35)(14 34)(15 33)(16 36)(17 72)(18 71)(19 70)(20 69)(21 62)(22 61)(23 64)(24 63)(25 68)(26 67)(27 66)(28 65)(29 55)(30 54)(31 53)(32 56)(37 58)(38 57)(39 60)(40 59)

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

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

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

C2×C327D4 is a maximal subgroup of
C62.32D4  C62.110D4  (C2×C62)⋊C4  (C2×C62).C4  C62.94C23  C62.95C23  C62.100C23  C62.60D4  C62.113C23  C62.117C23  C625D4  C626D4  C62.121C23  C62.125C23  C62.225C23  C6212D4  C62.227C23  C62.228C23  C62.229C23  C62.69D4  C62.129D4  C6219D4  C6213D4  C62.256C23  C6214D4  C62.258C23  C6224D4  C2×S3×C3⋊D4  C32⋊2+ 1+4  C2×D4×C3⋊S3  C3282+ 1+4
C2×C327D4 is a maximal quotient of
C6210Q8  C62.129D4  C6219D4  C62.131D4  C62.72D4  C62.254C23  C6213D4  C62.256C23  C6214D4  C62.258C23  C62.134D4  C62.259C23  C62.261C23  C62.262C23  C62.73D4  C62.74D4  C62.75D4  C6224D4

42 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D4A4B6A···6AB
order122222223333446···6
size1111221818222218182···2

42 irreducible representations

dim111112222
type++++++++
imageC1C2C2C2C2S3D4D6C3⋊D4
kernelC2×C327D4C2×C3⋊Dic3C327D4C22×C3⋊S3C2×C62C22×C6C3×C6C2×C6C6
# reps11411421216

Matrix representation of C2×C327D4 in GL4(𝔽13) generated by

1000
0100
00120
00012
,
0100
121200
0010
0001
,
1000
0100
00121
00120
,
11900
11200
0092
00114
,
1000
121200
00012
00120
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[0,12,0,0,1,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,12,12,0,0,1,0],[11,11,0,0,9,2,0,0,0,0,9,11,0,0,2,4],[1,12,0,0,0,12,0,0,0,0,0,12,0,0,12,0] >;

C2×C327D4 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes_7D_4
% in TeX

G:=Group("C2xC3^2:7D4");
// GroupNames label

G:=SmallGroup(144,177);
// by ID

G=gap.SmallGroup(144,177);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,218,964,3461]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=e*b*e=b^-1,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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