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G = C2×C42.78C22order 128 = 27

Direct product of C2 and C42.78C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×C42.78C22
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C8 — C2×C4×C8 — C2×C42.78C22
 Lower central C1 — C2 — C2×C4 — C2×C42.78C22
 Upper central C1 — C23 — C2×C42 — C2×C42.78C22
 Jennings C1 — C2 — C2 — C2×C4 — C2×C42.78C22

Generators and relations for C2×C42.78C22
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, be=eb, dcd=c-1, ce=ec, ede-1=b2cd >

Subgroups: 404 in 206 conjugacy classes, 100 normal (14 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, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C4×C8, D4⋊C4, Q8⋊C4, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4.4D4, C4.4D4, C42.C2, C42.C2, C22×C8, C22×D4, C22×Q8, C2×C4×C8, C2×D4⋊C4, C2×Q8⋊C4, C42.78C22, C2×C4.4D4, C2×C42.C2, C2×C42.78C22
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4.4D4, C4○D8, C22×D4, C2×C4○D4, C42.78C22, C2×C4.4D4, C2×C4○D8, C2×C42.78C22

Smallest permutation representation of C2×C42.78C22
On 64 points
Generators in S64
(1 53)(2 54)(3 55)(4 56)(5 49)(6 50)(7 51)(8 52)(9 46)(10 47)(11 48)(12 41)(13 42)(14 43)(15 44)(16 45)(17 31)(18 32)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(33 63)(34 64)(35 57)(36 58)(37 59)(38 60)(39 61)(40 62)
(1 64 17 10)(2 57 18 11)(3 58 19 12)(4 59 20 13)(5 60 21 14)(6 61 22 15)(7 62 23 16)(8 63 24 9)(25 41 55 36)(26 42 56 37)(27 43 49 38)(28 44 50 39)(29 45 51 40)(30 46 52 33)(31 47 53 34)(32 48 54 35)
(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 27)(2 56)(3 25)(4 54)(5 31)(6 52)(7 29)(8 50)(9 35)(10 47)(11 33)(12 45)(13 39)(14 43)(15 37)(16 41)(17 49)(18 26)(19 55)(20 32)(21 53)(22 30)(23 51)(24 28)(34 64)(36 62)(38 60)(40 58)(42 61)(44 59)(46 57)(48 63)
(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)| (1,53)(2,54)(3,55)(4,56)(5,49)(6,50)(7,51)(8,52)(9,46)(10,47)(11,48)(12,41)(13,42)(14,43)(15,44)(16,45)(17,31)(18,32)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(33,63)(34,64)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62), (1,64,17,10)(2,57,18,11)(3,58,19,12)(4,59,20,13)(5,60,21,14)(6,61,22,15)(7,62,23,16)(8,63,24,9)(25,41,55,36)(26,42,56,37)(27,43,49,38)(28,44,50,39)(29,45,51,40)(30,46,52,33)(31,47,53,34)(32,48,54,35), (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), (1,27)(2,56)(3,25)(4,54)(5,31)(6,52)(7,29)(8,50)(9,35)(10,47)(11,33)(12,45)(13,39)(14,43)(15,37)(16,41)(17,49)(18,26)(19,55)(20,32)(21,53)(22,30)(23,51)(24,28)(34,64)(36,62)(38,60)(40,58)(42,61)(44,59)(46,57)(48,63), (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( (1,53)(2,54)(3,55)(4,56)(5,49)(6,50)(7,51)(8,52)(9,46)(10,47)(11,48)(12,41)(13,42)(14,43)(15,44)(16,45)(17,31)(18,32)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(33,63)(34,64)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62), (1,64,17,10)(2,57,18,11)(3,58,19,12)(4,59,20,13)(5,60,21,14)(6,61,22,15)(7,62,23,16)(8,63,24,9)(25,41,55,36)(26,42,56,37)(27,43,49,38)(28,44,50,39)(29,45,51,40)(30,46,52,33)(31,47,53,34)(32,48,54,35), (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), (1,27)(2,56)(3,25)(4,54)(5,31)(6,52)(7,29)(8,50)(9,35)(10,47)(11,33)(12,45)(13,39)(14,43)(15,37)(16,41)(17,49)(18,26)(19,55)(20,32)(21,53)(22,30)(23,51)(24,28)(34,64)(36,62)(38,60)(40,58)(42,61)(44,59)(46,57)(48,63), (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([[(1,53),(2,54),(3,55),(4,56),(5,49),(6,50),(7,51),(8,52),(9,46),(10,47),(11,48),(12,41),(13,42),(14,43),(15,44),(16,45),(17,31),(18,32),(19,25),(20,26),(21,27),(22,28),(23,29),(24,30),(33,63),(34,64),(35,57),(36,58),(37,59),(38,60),(39,61),(40,62)], [(1,64,17,10),(2,57,18,11),(3,58,19,12),(4,59,20,13),(5,60,21,14),(6,61,22,15),(7,62,23,16),(8,63,24,9),(25,41,55,36),(26,42,56,37),(27,43,49,38),(28,44,50,39),(29,45,51,40),(30,46,52,33),(31,47,53,34),(32,48,54,35)], [(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)], [(1,27),(2,56),(3,25),(4,54),(5,31),(6,52),(7,29),(8,50),(9,35),(10,47),(11,33),(12,45),(13,39),(14,43),(15,37),(16,41),(17,49),(18,26),(19,55),(20,32),(21,53),(22,30),(23,51),(24,28),(34,64),(36,62),(38,60),(40,58),(42,61),(44,59),(46,57),(48,63)], [(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)]])

44 conjugacy classes

 class 1 2A ··· 2G 2H 2I 4A ··· 4L 4M ··· 4R 8A ··· 8P order 1 2 ··· 2 2 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 8 8 2 ··· 2 8 ··· 8 2 ··· 2

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D4 D4 C4○D4 C4○D8 kernel C2×C42.78C22 C2×C4×C8 C2×D4⋊C4 C2×Q8⋊C4 C42.78C22 C2×C4.4D4 C2×C42.C2 C42 C22×C4 C2×C4 C22 # reps 1 1 2 2 8 1 1 2 2 8 16

Matrix representation of C2×C42.78C22 in GL5(𝔽17)

 16 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 16
,
 16 0 0 0 0 0 7 8 0 0 0 15 10 0 0 0 0 0 0 13 0 0 0 4 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 1 0
,
 1 0 0 0 0 0 16 0 0 0 0 6 1 0 0 0 0 0 16 0 0 0 0 0 1
,
 1 0 0 0 0 0 6 2 0 0 0 8 11 0 0 0 0 0 3 14 0 0 0 3 3

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,7,15,0,0,0,8,10,0,0,0,0,0,0,4,0,0,0,13,0],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,16,0],[1,0,0,0,0,0,16,6,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,1],[1,0,0,0,0,0,6,8,0,0,0,2,11,0,0,0,0,0,3,3,0,0,0,14,3] >;

C2×C42.78C22 in GAP, Magma, Sage, TeX

C_2\times C_4^2._{78}C_2^2
% in TeX

G:=Group("C2xC4^2.78C2^2");
// GroupNames label

G:=SmallGroup(128,1862);
// by ID

G=gap.SmallGroup(128,1862);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,680,758,100,2804,172,4037,124]);
// 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,b*e=e*b,d*c*d=c^-1,c*e=e*c,e*d*e^-1=b^2*c*d>;
// generators/relations

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