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## G = C2×C23.37C23order 128 = 27

### Direct product of C2 and C23.37C23

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×C23.37C23
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C22×C42 — C2×C23.37C23
 Lower central C1 — C22 — C2×C23.37C23
 Upper central C1 — C22×C4 — C2×C23.37C23
 Jennings C1 — C22 — C2×C23.37C23

Generators and relations for C2×C23.37C23
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=f2=d, g2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, ebe-1=bc=cb, bd=db, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef-1=de=ed, df=fd, dg=gd, eg=ge, fg=gf >

Subgroups: 716 in 584 conjugacy classes, 452 normal (12 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C24, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, C23×C4, C23×C4, C22×Q8, C22×C42, C2×C42⋊C2, C2×C4×Q8, C2×C22⋊Q8, C2×C42.C2, C2×C4⋊Q8, C23.37C23, C2×C23.37C23
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C4○D4, C24, C22×Q8, C2×C4○D4, C25, C23.37C23, Q8×C23, C22×C4○D4, C2×C23.37C23

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

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I ··· 4AB 4AC ··· 4AR order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 4 ··· 4 size 1 1 ··· 1 2 2 2 2 1 ··· 1 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 Q8 C4○D4 kernel C2×C23.37C23 C22×C42 C2×C42⋊C2 C2×C4×Q8 C2×C22⋊Q8 C2×C42.C2 C2×C4⋊Q8 C23.37C23 C22×C4 C2×C4 # reps 1 1 2 4 4 2 2 16 8 16

Matrix representation of C2×C23.37C23 in GL5(𝔽5)

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

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

C2×C23.37C23 in GAP, Magma, Sage, TeX

C_2\times C_2^3._{37}C_2^3
% in TeX

G:=Group("C2xC2^3.37C2^3");
// GroupNames label

G:=SmallGroup(128,2175);
// by ID

G=gap.SmallGroup(128,2175);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,1430,352,136]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=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,e*b*e^-1=b*c=c*b,b*d=d*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,f*e*f^-1=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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