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## G = C2×C22.26C24order 128 = 27

### Direct product of C2 and C22.26C24

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×C22.26C24
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C22×C42 — C2×C22.26C24
 Lower central C1 — C22 — C2×C22.26C24
 Upper central C1 — C22×C4 — C2×C22.26C24
 Jennings C1 — C22 — C2×C22.26C24

Generators and relations for C2×C22.26C24
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=1, f2=c, g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede=bd=db, be=eb, bf=fb, bg=gb, fdf-1=cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 1324 in 864 conjugacy classes, 452 normal (14 characteristic)
C1, C2, C2 [×6], C2 [×12], C4 [×16], C4 [×12], C22, C22 [×10], C22 [×52], C2×C4 [×52], C2×C4 [×60], D4 [×80], Q8 [×16], C23, C23 [×14], C23 [×28], C42 [×16], C22⋊C4 [×32], C4⋊C4 [×16], C22×C4 [×2], C22×C4 [×44], C22×C4 [×24], C2×D4 [×40], C2×D4 [×40], C2×Q8 [×8], C2×Q8 [×8], C4○D4 [×64], C24, C24 [×4], C2×C42 [×2], C2×C42 [×10], C2×C22⋊C4 [×8], C2×C4⋊C4 [×4], C4×D4 [×32], C4⋊D4 [×32], C4.4D4 [×16], C41D4 [×8], C4⋊Q8 [×8], C23×C4, C23×C4 [×6], C22×D4 [×10], C22×Q8 [×2], C2×C4○D4 [×16], C2×C4○D4 [×16], C22×C42, C2×C4×D4 [×4], C2×C4⋊D4 [×4], C2×C4.4D4 [×2], C2×C41D4, C2×C4⋊Q8, C22.26C24 [×16], C22×C4○D4 [×2], C2×C22.26C24
Quotients: C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], C2×D4 [×28], C4○D4 [×8], C24 [×31], C22×D4 [×14], C2×C4○D4 [×12], C25, C22.26C24 [×4], D4×C23, C22×C4○D4 [×2], C2×C22.26C24

Smallest permutation representation of C2×C22.26C24
On 64 points
Generators in S64
(1 25)(2 26)(3 27)(4 28)(5 39)(6 40)(7 37)(8 38)(9 13)(10 14)(11 15)(12 16)(17 31)(18 32)(19 29)(20 30)(21 33)(22 34)(23 35)(24 36)(41 45)(42 46)(43 47)(44 48)(49 64)(50 61)(51 62)(52 63)(53 57)(54 58)(55 59)(56 60)
(1 11)(2 12)(3 9)(4 10)(5 29)(6 30)(7 31)(8 32)(13 27)(14 28)(15 25)(16 26)(17 37)(18 38)(19 39)(20 40)(21 50)(22 51)(23 52)(24 49)(33 61)(34 62)(35 63)(36 64)(41 55)(42 56)(43 53)(44 54)(45 59)(46 60)(47 57)(48 58)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)(49 51)(50 52)(53 55)(54 56)(57 59)(58 60)(61 63)(62 64)
(1 50)(2 49)(3 52)(4 51)(5 57)(6 60)(7 59)(8 58)(9 23)(10 22)(11 21)(12 24)(13 35)(14 34)(15 33)(16 36)(17 41)(18 44)(19 43)(20 42)(25 61)(26 64)(27 63)(28 62)(29 47)(30 46)(31 45)(32 48)(37 55)(38 54)(39 53)(40 56)
(1 45)(2 46)(3 47)(4 48)(5 52)(6 49)(7 50)(8 51)(9 57)(10 58)(11 59)(12 60)(13 53)(14 54)(15 55)(16 56)(17 33)(18 34)(19 35)(20 36)(21 31)(22 32)(23 29)(24 30)(25 41)(26 42)(27 43)(28 44)(37 61)(38 62)(39 63)(40 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 31 11 7)(2 32 12 8)(3 29 9 5)(4 30 10 6)(13 39 27 19)(14 40 28 20)(15 37 25 17)(16 38 26 18)(21 59 50 45)(22 60 51 46)(23 57 52 47)(24 58 49 48)(33 55 61 41)(34 56 62 42)(35 53 63 43)(36 54 64 44)

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L ··· 2S 4A ··· 4H 4I ··· 4AB 4AC ··· 4AJ order 1 2 ··· 2 2 2 2 2 2 ··· 2 4 ··· 4 4 ··· 4 4 ··· 4 size 1 1 ··· 1 2 2 2 2 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 C4○D4 kernel C2×C22.26C24 C22×C42 C2×C4×D4 C2×C4⋊D4 C2×C4.4D4 C2×C4⋊1D4 C2×C4⋊Q8 C22.26C24 C22×C4○D4 C22×C4 C2×C4 # reps 1 1 4 4 2 1 1 16 2 8 16

Matrix representation of C2×C22.26C24 in GL5(𝔽5)

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

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

C2×C22.26C24 in GAP, Magma, Sage, TeX

C_2\times C_2^2._{26}C_2^4
% in TeX

G:=Group("C2xC2^2.26C2^4");
// GroupNames label

G:=SmallGroup(128,2174);
// by ID

G=gap.SmallGroup(128,2174);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,352,136]);
// Polycyclic

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

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