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## G = C2×C23.36C23order 128 = 27

### Direct product of C2 and C23.36C23

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C2×C23.36C23
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=1, f2=d, g2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ebe=bd=db, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, eg=ge, fg=gf >

Subgroups: 844 in 616 conjugacy classes, 412 normal (32 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×8], C4 [×20], C22, C22 [×10], C22 [×32], C2×C4 [×32], C2×C4 [×60], D4 [×24], Q8 [×8], C23, C23 [×10], C23 [×16], C42 [×24], C22⋊C4 [×40], C4⋊C4 [×40], C22×C4 [×6], C22×C4 [×26], C22×C4 [×20], C2×D4 [×12], C2×D4 [×12], C2×Q8 [×4], C2×Q8 [×4], C24, C24 [×2], C2×C42 [×4], C2×C42 [×10], C2×C22⋊C4 [×10], C2×C4⋊C4 [×2], C2×C4⋊C4 [×8], C42⋊C2 [×16], C4×D4 [×24], C4×Q8 [×8], C4⋊D4 [×8], C22⋊Q8 [×8], C22.D4 [×16], C4.4D4 [×8], C42.C2 [×8], C422C2 [×16], C23×C4 [×3], C23×C4 [×2], C22×D4, C22×D4 [×2], C22×Q8, C22×C42, C2×C42⋊C2 [×2], C2×C4×D4, C2×C4×D4 [×2], C2×C4×Q8, C2×C4⋊D4, C2×C22⋊Q8, C2×C22.D4 [×2], C2×C4.4D4, C2×C42.C2, C2×C422C2 [×2], C23.36C23 [×16], C2×C23.36C23
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], C4○D4 [×12], C24 [×31], C2×C4○D4 [×18], C25, C23.36C23 [×4], C22×C4○D4 [×3], C2×C23.36C23

Smallest permutation representation of C2×C23.36C23
On 64 points
Generators in S64
(1 49)(2 50)(3 51)(4 52)(5 37)(6 38)(7 39)(8 40)(9 29)(10 30)(11 31)(12 32)(13 17)(14 18)(15 19)(16 20)(21 25)(22 26)(23 27)(24 28)(33 63)(34 64)(35 61)(36 62)(41 53)(42 54)(43 55)(44 56)(45 57)(46 58)(47 59)(48 60)
(1 21)(2 22)(3 23)(4 24)(5 43)(6 44)(7 41)(8 42)(9 35)(10 36)(11 33)(12 34)(13 45)(14 46)(15 47)(16 48)(17 57)(18 58)(19 59)(20 60)(25 49)(26 50)(27 51)(28 52)(29 61)(30 62)(31 63)(32 64)(37 55)(38 56)(39 53)(40 54)
(1 19)(2 20)(3 17)(4 18)(5 61)(6 62)(7 63)(8 64)(9 55)(10 56)(11 53)(12 54)(13 51)(14 52)(15 49)(16 50)(21 59)(22 60)(23 57)(24 58)(25 47)(26 48)(27 45)(28 46)(29 43)(30 44)(31 41)(32 42)(33 39)(34 40)(35 37)(36 38)
(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 34)(2 37)(3 36)(4 39)(5 50)(6 13)(7 52)(8 15)(9 58)(10 21)(11 60)(12 23)(14 63)(16 61)(17 38)(18 33)(19 40)(20 35)(22 53)(24 55)(25 30)(26 41)(27 32)(28 43)(29 46)(31 48)(42 45)(44 47)(49 64)(51 62)(54 57)(56 59)
(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 55 19 9)(2 56 20 10)(3 53 17 11)(4 54 18 12)(5 47 61 25)(6 48 62 26)(7 45 63 27)(8 46 64 28)(13 31 51 41)(14 32 52 42)(15 29 49 43)(16 30 50 44)(21 37 59 35)(22 38 60 36)(23 39 57 33)(24 40 58 34)

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 2N 2O 4A ··· 4H 4I ··· 4AB 4AC ··· 4AN order 1 2 ··· 2 2 2 2 2 2 2 2 2 4 ··· 4 4 ··· 4 4 ··· 4 size 1 1 ··· 1 2 2 2 2 4 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C4○D4 C4○D4 kernel C2×C23.36C23 C22×C42 C2×C42⋊C2 C2×C4×D4 C2×C4×Q8 C2×C4⋊D4 C2×C22⋊Q8 C2×C22.D4 C2×C4.4D4 C2×C42.C2 C2×C42⋊2C2 C23.36C23 C2×C4 C23 # reps 1 1 2 3 1 1 1 2 1 1 2 16 16 8

Matrix representation of C2×C23.36C23 in GL6(𝔽5)

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

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,4,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,2,0,0,0,0,3,0,0,0,0,0,0,0,2,3,0,0,0,0,4,3,0,0,0,0,0,0,4,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,2,4,0,0,0,0,0,0,0,2,0,0,0,0,2,0],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3] >;

C2×C23.36C23 in GAP, Magma, Sage, TeX

C_2\times C_2^3._{36}C_2^3
% in TeX

G:=Group("C2xC2^3.36C2^3");
// GroupNames label

G:=SmallGroup(128,2171);
// by ID

G=gap.SmallGroup(128,2171);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,136]);
// Polycyclic

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

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