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G = C32×C22⋊C4order 144 = 24·32

Direct product of C32 and C22⋊C4

direct product, metabelian, nilpotent (class 2), monomial

Aliases: C32×C22⋊C4, C623C4, C22.2C62, C62.36C22, (C2×C6)⋊3C12, (C2×C12)⋊2C6, (C6×C12)⋊3C2, C2.1(C6×C12), C6.18(C3×D4), (C3×C6).39D4, C6.13(C2×C12), C222(C3×C12), (C22×C6).7C6, C23.2(C3×C6), (C2×C62).1C2, C2.1(D4×C32), (C2×C4)⋊1(C3×C6), (C2×C6).19(C2×C6), (C3×C6).35(C2×C4), SmallGroup(144,102)

Series: Derived Chief Lower central Upper central

C1C2 — C32×C22⋊C4
C1C2C22C2×C6C62C6×C12 — C32×C22⋊C4
C1C2 — C32×C22⋊C4
C1C62 — C32×C22⋊C4

Generators and relations for C32×C22⋊C4
 G = < a,b,c,d,e | a3=b3=c2=d2=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, de=ed >

Subgroups: 138 in 102 conjugacy classes, 66 normal (10 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×4], C4 [×2], C22, C22 [×2], C22 [×2], C6 [×12], C6 [×8], C2×C4 [×2], C23, C32, C12 [×8], C2×C6 [×12], C2×C6 [×8], C22⋊C4, C3×C6, C3×C6 [×2], C3×C6 [×2], C2×C12 [×8], C22×C6 [×4], C3×C12 [×2], C62, C62 [×2], C62 [×2], C3×C22⋊C4 [×4], C6×C12 [×2], C2×C62, C32×C22⋊C4
Quotients: C1, C2 [×3], C3 [×4], C4 [×2], C22, C6 [×12], C2×C4, D4 [×2], C32, C12 [×8], C2×C6 [×4], C22⋊C4, C3×C6 [×3], C2×C12 [×4], C3×D4 [×8], C3×C12 [×2], C62, C3×C22⋊C4 [×4], C6×C12, D4×C32 [×2], C32×C22⋊C4

Smallest permutation representation of C32×C22⋊C4
On 72 points
Generators in S72
(1 6 51)(2 7 52)(3 8 49)(4 5 50)(9 53 17)(10 54 18)(11 55 19)(12 56 20)(13 45 41)(14 46 42)(15 47 43)(16 48 44)(21 39 29)(22 40 30)(23 37 31)(24 38 32)(25 69 61)(26 70 62)(27 71 63)(28 72 64)(33 65 57)(34 66 58)(35 67 59)(36 68 60)
(1 47 11)(2 48 12)(3 45 9)(4 46 10)(5 42 54)(6 43 55)(7 44 56)(8 41 53)(13 17 49)(14 18 50)(15 19 51)(16 20 52)(21 25 57)(22 26 58)(23 27 59)(24 28 60)(29 61 65)(30 62 66)(31 63 67)(32 64 68)(33 39 69)(34 40 70)(35 37 71)(36 38 72)
(1 3)(2 22)(4 24)(5 38)(6 8)(7 40)(9 11)(10 60)(12 58)(13 15)(14 64)(16 62)(17 19)(18 68)(20 66)(21 23)(25 27)(26 48)(28 46)(29 31)(30 52)(32 50)(33 35)(34 56)(36 54)(37 39)(41 43)(42 72)(44 70)(45 47)(49 51)(53 55)(57 59)(61 63)(65 67)(69 71)
(1 23)(2 24)(3 21)(4 22)(5 40)(6 37)(7 38)(8 39)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(25 45)(26 46)(27 47)(28 48)(29 49)(30 50)(31 51)(32 52)(33 53)(34 54)(35 55)(36 56)(41 69)(42 70)(43 71)(44 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)

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

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

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

C32×C22⋊C4 is a maximal subgroup of
C62.110D4  C62.221C23  C626Q8  C62.223C23  C62.225C23  C6212D4  C62.227C23  C62.228C23  C62.229C23  C62.69D4  D4×C3×C12

90 conjugacy classes

class 1 2A2B2C2D2E3A···3H4A4B4C4D6A···6X6Y···6AN12A···12AF
order1222223···344446···66···612···12
size1111221···122221···12···22···2

90 irreducible representations

dim1111111122
type++++
imageC1C2C2C3C4C6C6C12D4C3×D4
kernelC32×C22⋊C4C6×C12C2×C62C3×C22⋊C4C62C2×C12C22×C6C2×C6C3×C6C6
# reps1218416832216

Matrix representation of C32×C22⋊C4 in GL4(𝔽13) generated by

3000
0100
0030
0003
,
3000
0100
0090
0009
,
12000
0100
0010
00012
,
1000
0100
00120
00012
,
1000
0800
0001
0010
G:=sub<GL(4,GF(13))| [3,0,0,0,0,1,0,0,0,0,3,0,0,0,0,3],[3,0,0,0,0,1,0,0,0,0,9,0,0,0,0,9],[12,0,0,0,0,1,0,0,0,0,1,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,8,0,0,0,0,0,1,0,0,1,0] >;

C32×C22⋊C4 in GAP, Magma, Sage, TeX

C_3^2\times C_2^2\rtimes C_4
% in TeX

G:=Group("C3^2xC2^2:C4");
// GroupNames label

G:=SmallGroup(144,102);
// by ID

G=gap.SmallGroup(144,102);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-2,-2,432,457]);
// Polycyclic

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

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