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## G = C2×C22.36C24order 128 = 27

### Direct product of C2 and C22.36C24

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×C22.36C24
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C42⋊C2 — C2×C22.36C24
 Lower central C1 — C22 — C2×C22.36C24
 Upper central C1 — C23 — C2×C22.36C24
 Jennings C1 — C22 — C2×C22.36C24

Generators and relations for C2×C22.36C24
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=cb=bc, g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, ede-1=gdg-1=bd=db, fef=be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 844 in 566 conjugacy classes, 396 normal (32 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×4], C4 [×22], C22, C22 [×6], C22 [×30], C2×C4 [×28], C2×C4 [×38], D4 [×16], Q8 [×16], C23, C23 [×6], C23 [×18], C42 [×16], C22⋊C4 [×48], C4⋊C4 [×40], C22×C4 [×6], C22×C4 [×18], C22×C4 [×8], C2×D4 [×12], C2×D4 [×8], C2×Q8 [×12], C2×Q8 [×8], C24, C24 [×2], C2×C42 [×2], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×10], C2×C4⋊C4 [×4], C2×C4⋊C4 [×6], C42⋊C2 [×8], C4×D4 [×8], C4×Q8 [×8], C4⋊D4 [×8], C22⋊Q8 [×24], C22.D4 [×16], C4.4D4 [×24], C422C2 [×16], C4⋊Q8 [×8], C23×C4, C23×C4 [×2], C22×D4, C22×D4 [×2], C22×Q8, C22×Q8 [×2], C2×C42⋊C2, C2×C4×D4, C2×C4×Q8, C2×C4⋊D4, C2×C22⋊Q8, C2×C22⋊Q8 [×2], C2×C22.D4 [×2], C2×C4.4D4, C2×C4.4D4 [×2], C2×C422C2 [×2], C2×C4⋊Q8, C22.36C24 [×16], C2×C22.36C24
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], C4○D4 [×4], C24 [×31], C2×C4○D4 [×6], 2+ 1+4 [×2], 2- 1+4 [×2], C25, C22.36C24 [×4], C22×C4○D4, C2×2+ 1+4, C2×2- 1+4, C2×C22.36C24

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

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2M 4A ··· 4L 4M ··· 4AD order 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 size 1 1 ··· 1 4 ··· 4 2 ··· 2 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 4 4 type + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C4○D4 2+ 1+4 2- 1+4 kernel C2×C22.36C24 C2×C42⋊C2 C2×C4×D4 C2×C4×Q8 C2×C4⋊D4 C2×C22⋊Q8 C2×C22.D4 C2×C4.4D4 C2×C42⋊2C2 C2×C4⋊Q8 C22.36C24 C2×C4 C22 C22 # reps 1 1 1 1 1 3 2 3 2 1 16 8 2 2

Matrix representation of C2×C22.36C24 in GL8(𝔽5)

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

G:=sub<GL(8,GF(5))| [4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0],[2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0],[1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0] >;

C2×C22.36C24 in GAP, Magma, Sage, TeX

C_2\times C_2^2._{36}C_2^4
% in TeX

G:=Group("C2xC2^2.36C2^4");
// GroupNames label

G:=SmallGroup(128,2186);
// by ID

G=gap.SmallGroup(128,2186);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,680,1430,387,1123,136]);
// Polycyclic

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

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