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## G = C2×C23.38C23order 128 = 27

### Direct product of C2 and C23.38C23

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×C23.38C23
 Chief series C1 — C2 — C22 — C23 — C24 — C23×C4 — Q8×C23 — C2×C23.38C23
 Lower central C1 — C22 — C2×C23.38C23
 Upper central C1 — C23 — C2×C23.38C23
 Jennings C1 — C22 — C2×C23.38C23

Generators and relations for C2×C23.38C23
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=1, f2=g2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, ebe=bc=cb, bd=db, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, gfg-1=cf=fc, cg=gc, geg-1=de=ed, df=fd, dg=gd >

Subgroups: 1100 in 756 conjugacy classes, 436 normal (14 characteristic)
C1, C2, C2 [×6], C2 [×8], C4 [×8], C4 [×20], C22, C22 [×10], C22 [×32], C2×C4 [×48], C2×C4 [×52], D4 [×24], Q8 [×40], C23, C23 [×10], C23 [×16], C42 [×8], C22⋊C4 [×40], C4⋊C4 [×40], C22×C4 [×2], C22×C4 [×38], C22×C4 [×16], C2×D4 [×12], C2×D4 [×12], C2×Q8 [×20], C2×Q8 [×52], C4○D4 [×32], C24, C24 [×2], C2×C42 [×2], C2×C22⋊C4 [×10], C2×C4⋊C4 [×10], C42⋊C2 [×8], C22⋊Q8 [×32], C22.D4 [×32], C4.4D4 [×16], C4⋊Q8 [×16], C23×C4, C23×C4 [×4], C22×D4, C22×D4 [×2], C22×Q8, C22×Q8 [×12], C22×Q8 [×8], C2×C4○D4 [×8], C2×C4○D4 [×8], C2×C42⋊C2, C2×C22⋊Q8 [×4], C2×C22.D4 [×4], C2×C4.4D4 [×2], C2×C4⋊Q8 [×2], C23.38C23 [×16], Q8×C23, C22×C4○D4, C2×C23.38C23
Quotients: C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], C2×D4 [×28], C24 [×31], C22×D4 [×14], 2- 1+4 [×4], C25, C23.38C23 [×4], D4×C23, C2×2- 1+4 [×2], C2×C23.38C23

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 4 type + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 2- 1+4 kernel C2×C23.38C23 C2×C42⋊C2 C2×C22⋊Q8 C2×C22.D4 C2×C4.4D4 C2×C4⋊Q8 C23.38C23 Q8×C23 C22×C4○D4 C22×C4 C22 # reps 1 1 4 4 2 2 16 1 1 8 4

Matrix representation of C2×C23.38C23 in GL8(𝔽5)

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

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

C2×C23.38C23 in GAP, Magma, Sage, TeX

C_2\times C_2^3._{38}C_2^3
% in TeX

G:=Group("C2xC2^3.38C2^3");
// GroupNames label

G:=SmallGroup(128,2179);
// by ID

G=gap.SmallGroup(128,2179);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,232,1430,387,184,1123]);
// Polycyclic

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

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