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## G = C22⋊C4×He3order 432 = 24·33

### Direct product of C22⋊C4 and He3

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C22⋊C4×He3
 Chief series C1 — C3 — C6 — C2×C6 — C62 — C22×He3 — C2×C4×He3 — C22⋊C4×He3
 Lower central C1 — C6 — C22⋊C4×He3
 Upper central C1 — C2×C6 — C22⋊C4×He3

Generators and relations for C22⋊C4×He3
G = < a,b,c,d,e,f | a2=b2=c4=d3=e3=f3=1, cac-1=ab=ba, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf-1=de-1, ef=fe >

Subgroups: 437 in 187 conjugacy classes, 77 normal (15 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C23, C32, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C3×C6, C3×C6, C2×C12, C2×C12, C22×C6, C22×C6, He3, C3×C12, C62, C62, C3×C22⋊C4, C3×C22⋊C4, C2×He3, C2×He3, C2×He3, C6×C12, C2×C62, C4×He3, C22×He3, C22×He3, C22×He3, C32×C22⋊C4, C2×C4×He3, C23×He3, C22⋊C4×He3
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C32, C12, C2×C6, C22⋊C4, C3×C6, C2×C12, C3×D4, He3, C3×C12, C62, C3×C22⋊C4, C2×He3, C6×C12, D4×C32, C4×He3, C22×He3, C32×C22⋊C4, C2×C4×He3, D4×He3, C22⋊C4×He3

Smallest permutation representation of C22⋊C4×He3
On 72 points
Generators in S72
(1 3)(2 44)(4 42)(5 7)(6 26)(8 28)(9 11)(10 60)(12 58)(13 15)(14 62)(16 64)(17 19)(18 38)(20 40)(21 23)(22 72)(24 70)(25 27)(29 31)(30 52)(32 50)(33 35)(34 54)(36 56)(37 39)(41 43)(45 47)(46 66)(48 68)(49 51)(53 55)(57 59)(61 63)(65 67)(69 71)
(1 41)(2 42)(3 43)(4 44)(5 27)(6 28)(7 25)(8 26)(9 57)(10 58)(11 59)(12 60)(13 63)(14 64)(15 61)(16 62)(17 39)(18 40)(19 37)(20 38)(21 69)(22 70)(23 71)(24 72)(29 49)(30 50)(31 51)(32 52)(33 55)(34 56)(35 53)(36 54)(45 67)(46 68)(47 65)(48 66)
(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 35 63)(2 36 64)(3 33 61)(4 34 62)(5 17 45)(6 18 46)(7 19 47)(8 20 48)(9 21 49)(10 22 50)(11 23 51)(12 24 52)(13 41 53)(14 42 54)(15 43 55)(16 44 56)(25 37 65)(26 38 66)(27 39 67)(28 40 68)(29 57 69)(30 58 70)(31 59 71)(32 60 72)
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 67 29)(14 68 30)(15 65 31)(16 66 32)(17 21 35)(18 22 36)(19 23 33)(20 24 34)(25 59 43)(26 60 44)(27 57 41)(28 58 42)(37 71 55)(38 72 56)(39 69 53)(40 70 54)(45 49 63)(46 50 64)(47 51 61)(48 52 62)
(13 29 67)(14 30 68)(15 31 65)(16 32 66)(17 21 35)(18 22 36)(19 23 33)(20 24 34)(37 71 55)(38 72 56)(39 69 53)(40 70 54)(45 63 49)(46 64 50)(47 61 51)(48 62 52)

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

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

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

110 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C ··· 3J 4A 4B 4C 4D 6A ··· 6F 6G 6H 6I 6J 6K ··· 6AH 6AI ··· 6AX 12A ··· 12H 12I ··· 12AN order 1 2 2 2 2 2 3 3 3 ··· 3 4 4 4 4 6 ··· 6 6 6 6 6 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 1 1 2 2 1 1 3 ··· 3 2 2 2 2 1 ··· 1 2 2 2 2 3 ··· 3 6 ··· 6 2 ··· 2 6 ··· 6

110 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 3 3 3 3 6 type + + + + image C1 C2 C2 C3 C4 C6 C6 C12 D4 C3×D4 He3 C2×He3 C2×He3 C4×He3 D4×He3 kernel C22⋊C4×He3 C2×C4×He3 C23×He3 C32×C22⋊C4 C22×He3 C6×C12 C2×C62 C62 C2×He3 C3×C6 C22⋊C4 C2×C4 C23 C22 C2 # reps 1 2 1 8 4 16 8 32 2 16 2 4 2 8 4

Matrix representation of C22⋊C4×He3 in GL5(𝔽13)

 1 10 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 12 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 5 12 0 0 0 0 0 5 0 0 0 0 0 5 0 0 0 0 0 5
,
 9 0 0 0 0 0 9 0 0 0 0 0 4 1 7 0 0 6 0 2 0 0 8 0 9
,
 1 0 0 0 0 0 1 0 0 0 0 0 3 0 0 0 0 0 3 0 0 0 0 0 3
,
 9 0 0 0 0 0 9 0 0 0 0 0 1 0 1 0 0 0 9 2 0 0 0 0 3

G:=sub<GL(5,GF(13))| [1,0,0,0,0,10,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,5,0,0,0,0,12,0,0,0,0,0,5,0,0,0,0,0,5,0,0,0,0,0,5],[9,0,0,0,0,0,9,0,0,0,0,0,4,6,8,0,0,1,0,0,0,0,7,2,9],[1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3],[9,0,0,0,0,0,9,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,1,2,3] >;

C22⋊C4×He3 in GAP, Magma, Sage, TeX

C_2^2\rtimes C_4\times {\rm He}_3
% in TeX

G:=Group("C2^2:C4xHe3");
// GroupNames label

G:=SmallGroup(432,204);
// by ID

G=gap.SmallGroup(432,204);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-3,504,533,1109]);
// Polycyclic

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

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