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## G = C2×C42.28C22order 128 = 27

### Direct product of C2 and C42.28C22

direct product, p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×C42.28C22
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C8 — C2×C8⋊C4 — C2×C42.28C22
 Lower central C1 — C2 — C2×C4 — C2×C42.28C22
 Upper central C1 — C23 — C2×C42 — C2×C42.28C22
 Jennings C1 — C2 — C2 — C2×C4 — C2×C42.28C22

Generators and relations for C2×C42.28C22
G = < a,b,c,d,e | a2=b4=c4=d2=1, e2=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd=b-1c2, ebe-1=bc2, dcd=c-1, ce=ec, ede-1=b2c-1d >

Subgroups: 452 in 218 conjugacy classes, 100 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C8⋊C4, D4⋊C4, Q8⋊C4, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4.4D4, C4.4D4, C4⋊Q8, C4⋊Q8, C22×C8, C22×D4, C22×Q8, C22×Q8, C2×C8⋊C4, C2×D4⋊C4, C2×Q8⋊C4, C42.28C22, C2×C4.4D4, C2×C4⋊Q8, C2×C42.28C22
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4.4D4, C8⋊C22, C8.C22, C22×D4, C2×C4○D4, C42.28C22, C2×C4.4D4, C2×C8⋊C22, C2×C8.C22, C2×C42.28C22

Smallest permutation representation of C2×C42.28C22
On 64 points
Generators in S64
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 63)(10 64)(11 57)(12 58)(13 59)(14 60)(15 61)(16 62)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(33 45)(34 46)(35 47)(36 48)(37 41)(38 42)(39 43)(40 44)
(1 64 17 40)(2 61 18 37)(3 58 19 34)(4 63 20 39)(5 60 21 36)(6 57 22 33)(7 62 23 38)(8 59 24 35)(9 28 43 52)(10 25 44 49)(11 30 45 54)(12 27 46 51)(13 32 47 56)(14 29 48 53)(15 26 41 50)(16 31 42 55)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)
(1 25)(2 52)(3 31)(4 50)(5 29)(6 56)(7 27)(8 54)(9 33)(10 60)(11 39)(12 58)(13 37)(14 64)(15 35)(16 62)(17 49)(18 28)(19 55)(20 26)(21 53)(22 32)(23 51)(24 30)(34 46)(36 44)(38 42)(40 48)(41 59)(43 57)(45 63)(47 61)
(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)

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

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

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

32 conjugacy classes

 class 1 2A ··· 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4N 8A ··· 8H order 1 2 ··· 2 2 2 4 4 4 4 4 4 4 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 8 8 2 2 2 2 4 4 4 4 8 ··· 8 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 4 4 type + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 D4 D4 C4○D4 C8⋊C22 C8.C22 kernel C2×C42.28C22 C2×C8⋊C4 C2×D4⋊C4 C2×Q8⋊C4 C42.28C22 C2×C4.4D4 C2×C4⋊Q8 C42 C22×C4 C2×C4 C22 C22 # reps 1 1 2 2 8 1 1 2 2 8 2 2

Matrix representation of C2×C42.28C22 in GL8(𝔽17)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 8 0 0 0 0 0 0 4 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 2 12 0 10 0 0 0 0 12 2 10 10 0 0 0 0 5 12 10 0 0 0 0 0 14 2 5 3
,
 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 0 1 1 16 15 0 0 0 0 16 0 1 1
,
 16 0 0 0 0 0 0 0 13 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 16 16 1 2 0 0 0 0 0 1 0 16
,
 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 2 2 10 3 0 0 0 0 5 12 10 0 0 0 0 0 15 5 0 7 0 0 0 0 2 14 15 3

G:=sub<GL(8,GF(17))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,4,0,0,0,0,0,0,8,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,12,5,14,0,0,0,0,12,2,12,2,0,0,0,0,0,10,10,5,0,0,0,0,10,10,0,3],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,1,16,0,0,0,0,1,0,1,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,15,1],[16,13,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,16,0,0,0,0,0,0,1,16,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,16],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,2,5,15,2,0,0,0,0,2,12,5,14,0,0,0,0,10,10,0,15,0,0,0,0,3,0,7,3] >;

C2×C42.28C22 in GAP, Magma, Sage, TeX

C_2\times C_4^2._{28}C_2^2
% in TeX

G:=Group("C2xC4^2.28C2^2");
// GroupNames label

G:=SmallGroup(128,1864);
// by ID

G=gap.SmallGroup(128,1864);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,232,758,723,100,2804,172,4037,124]);
// Polycyclic

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

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