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## G = C32×C4○D4order 144 = 24·32

### Direct product of C32 and C4○D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C32×C4○D4
 Chief series C1 — C2 — C6 — C3×C6 — C62 — D4×C32 — C32×C4○D4
 Lower central C1 — C2 — C32×C4○D4
 Upper central C1 — C3×C12 — C32×C4○D4

Generators and relations for C32×C4○D4
G = < a,b,c,d,e | a3=b3=c4=e2=1, d2=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d >

Subgroups: 138 in 120 conjugacy classes, 102 normal (10 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C2×C4, D4, Q8, C32, C12, C2×C6, C4○D4, C3×C6, C3×C6, C2×C12, C3×D4, C3×Q8, C3×C12, C3×C12, C62, C3×C4○D4, C6×C12, D4×C32, Q8×C32, C32×C4○D4
Quotients: C1, C2, C3, C22, C6, C23, C32, C2×C6, C4○D4, C3×C6, C22×C6, C62, C3×C4○D4, C2×C62, C32×C4○D4

Smallest permutation representation of C32×C4○D4
On 72 points
Generators in S72
(1 37 31)(2 38 32)(3 39 29)(4 40 30)(5 33 24)(6 34 21)(7 35 22)(8 36 23)(9 15 25)(10 16 26)(11 13 27)(12 14 28)(17 55 51)(18 56 52)(19 53 49)(20 54 50)(41 72 45)(42 69 46)(43 70 47)(44 71 48)(57 66 61)(58 67 62)(59 68 63)(60 65 64)
(1 13 7)(2 14 8)(3 15 5)(4 16 6)(9 24 29)(10 21 30)(11 22 31)(12 23 32)(17 64 69)(18 61 70)(19 62 71)(20 63 72)(25 33 39)(26 34 40)(27 35 37)(28 36 38)(41 50 68)(42 51 65)(43 52 66)(44 49 67)(45 54 59)(46 55 60)(47 56 57)(48 53 58)
(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 4 3 2)(5 8 7 6)(9 12 11 10)(13 16 15 14)(17 18 19 20)(21 24 23 22)(25 28 27 26)(29 32 31 30)(33 36 35 34)(37 40 39 38)(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 52)(2 49)(3 50)(4 51)(5 41)(6 42)(7 43)(8 44)(9 59)(10 60)(11 57)(12 58)(13 66)(14 67)(15 68)(16 65)(17 40)(18 37)(19 38)(20 39)(21 46)(22 47)(23 48)(24 45)(25 63)(26 64)(27 61)(28 62)(29 54)(30 55)(31 56)(32 53)(33 72)(34 69)(35 70)(36 71)

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

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

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

C32×C4○D4 is a maximal subgroup of
C62.39D4  D4.(C3⋊Dic3)  C62.73D4  C62.74D4  C62.75D4  C62.154C23  C3292- 1+4  Q8⋊C94C6  C4○D4⋊He3
C32×C4○D4 is a maximal quotient of
D4×C3×C12  Q8×C3×C12

90 conjugacy classes

 class 1 2A 2B 2C 2D 3A ··· 3H 4A 4B 4C 4D 4E 6A ··· 6H 6I ··· 6AF 12A ··· 12P 12Q ··· 12AN order 1 2 2 2 2 3 ··· 3 4 4 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 2 2 2 1 ··· 1 1 1 2 2 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C2 C3 C6 C6 C6 C4○D4 C3×C4○D4 kernel C32×C4○D4 C6×C12 D4×C32 Q8×C32 C3×C4○D4 C2×C12 C3×D4 C3×Q8 C32 C3 # reps 1 3 3 1 8 24 24 8 2 16

Matrix representation of C32×C4○D4 in GL3(𝔽13) generated by

 1 0 0 0 9 0 0 0 9
,
 3 0 0 0 1 0 0 0 1
,
 12 0 0 0 5 0 0 0 5
,
 1 0 0 0 8 0 0 5 5
,
 12 0 0 0 8 3 0 5 5
G:=sub<GL(3,GF(13))| [1,0,0,0,9,0,0,0,9],[3,0,0,0,1,0,0,0,1],[12,0,0,0,5,0,0,0,5],[1,0,0,0,8,5,0,0,5],[12,0,0,0,8,5,0,3,5] >;

C32×C4○D4 in GAP, Magma, Sage, TeX

C_3^2\times C_4\circ D_4
% in TeX

G:=Group("C3^2xC4oD4");
// GroupNames label

G:=SmallGroup(144,181);
// by ID

G=gap.SmallGroup(144,181);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-2,889,338]);
// Polycyclic

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

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