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## G = C4×2- 1+4order 128 = 27

### Direct product of C4 and 2- 1+4

direct product, p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C4×2- 1+4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C2×C4×Q8 — C4×2- 1+4
 Lower central C1 — C2 — C4×2- 1+4
 Upper central C1 — C2×C4 — C4×2- 1+4
 Jennings C1 — C22 — C4×2- 1+4

Generators and relations for C4×2- 1+4
G = < a,b,c,d,e | a4=b4=c2=1, d2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=b2d >

Subgroups: 836 in 746 conjugacy classes, 686 normal (8 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C22×Q8, C2×C4○D4, 2- 1+4, C2×C4×Q8, C4×C4○D4, C23.32C23, C23.33C23, C2×2- 1+4, C4×2- 1+4
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, C23×C4, 2- 1+4, C25, C24×C4, C2×2- 1+4, C2.C25, C4×2- 1+4

Smallest permutation representation of C4×2- 1+4
On 64 points
Generators in S64
(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)
(1 56 11 57)(2 53 12 58)(3 54 9 59)(4 55 10 60)(5 17 62 16)(6 18 63 13)(7 19 64 14)(8 20 61 15)(21 44 26 39)(22 41 27 40)(23 42 28 37)(24 43 25 38)(29 47 34 52)(30 48 35 49)(31 45 36 50)(32 46 33 51)
(1 16)(2 13)(3 14)(4 15)(5 57)(6 58)(7 59)(8 60)(9 19)(10 20)(11 17)(12 18)(21 31)(22 32)(23 29)(24 30)(25 35)(26 36)(27 33)(28 34)(37 47)(38 48)(39 45)(40 46)(41 51)(42 52)(43 49)(44 50)(53 63)(54 64)(55 61)(56 62)
(1 35 11 30)(2 36 12 31)(3 33 9 32)(4 34 10 29)(5 38 62 43)(6 39 63 44)(7 40 64 41)(8 37 61 42)(13 26 18 21)(14 27 19 22)(15 28 20 23)(16 25 17 24)(45 53 50 58)(46 54 51 59)(47 55 52 60)(48 56 49 57)
(1 59 11 54)(2 60 12 55)(3 57 9 56)(4 58 10 53)(5 19 62 14)(6 20 63 15)(7 17 64 16)(8 18 61 13)(21 37 26 42)(22 38 27 43)(23 39 28 44)(24 40 25 41)(29 45 34 50)(30 46 35 51)(31 47 36 52)(32 48 33 49)

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

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

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

68 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2M 4A 4B 4C 4D 4E ··· 4BB order 1 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 size 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2

68 irreducible representations

 dim 1 1 1 1 1 1 1 4 4 type + + + + + + - image C1 C2 C2 C2 C2 C2 C4 2- 1+4 C2.C25 kernel C4×2- 1+4 C2×C4×Q8 C4×C4○D4 C23.32C23 C23.33C23 C2×2- 1+4 2- 1+4 C4 C2 # reps 1 5 10 5 10 1 32 2 2

Matrix representation of C4×2- 1+4 in GL5(𝔽5)

 2 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 4 0 0 0 0 0 0 0 2 0 0 0 0 0 3 0 2 0 0 0 0 0 3 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0
,
 4 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 4 0 0 0 4 0 0 0
,
 4 0 0 0 0 0 0 0 3 0 0 0 0 0 3 0 3 0 0 0 0 0 3 0 0

G:=sub<GL(5,GF(5))| [2,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,2,0,0,0,0,0,3,0,0],[1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0],[4,0,0,0,0,0,0,0,0,4,0,0,0,4,0,0,0,1,0,0,0,1,0,0,0],[4,0,0,0,0,0,0,0,3,0,0,0,0,0,3,0,3,0,0,0,0,0,3,0,0] >;

C4×2- 1+4 in GAP, Magma, Sage, TeX

C_4\times 2_-^{1+4}
% in TeX

G:=Group("C4xES-(2,2)");
// GroupNames label

G:=SmallGroup(128,2162);
// by ID

G=gap.SmallGroup(128,2162);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,232,387,184,1123,172]);
// Polycyclic

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

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