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## G = C2×C23.19D4order 128 = 27

### Direct product of C2 and C23.19D4

direct product, p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×C23.19D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4⋊C4 — C2×C42⋊C2 — C2×C23.19D4
 Lower central C1 — C2 — C2×C4 — C2×C23.19D4
 Upper central C1 — C23 — C23×C4 — C2×C23.19D4
 Jennings C1 — C2 — C2 — C2×C4 — C2×C23.19D4

Generators and relations for C2×C23.19D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=f2=1, e4=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe-1=bc=cb, bd=db, fbf=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=cde3 >

Subgroups: 492 in 234 conjugacy classes, 100 normal (28 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, C22⋊C8, D4⋊C4, C4.Q8, C2.D8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4⋊D4, C4⋊D4, C22×C8, C23×C4, C22×D4, C22×D4, C2×C22⋊C8, C2×D4⋊C4, C2×C4.Q8, C2×C2.D8, C23.19D4, C2×C42⋊C2, C2×C4⋊D4, C2×C23.19D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22.D4, C4○D8, C8⋊C22, C22×D4, C2×C4○D4, C23.19D4, C2×C22.D4, C2×C4○D8, C2×C8⋊C22, C2×C23.19D4

Smallest permutation representation of C2×C23.19D4
On 64 points
Generators in S64
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 33)(8 34)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 25)(17 46)(18 47)(19 48)(20 41)(21 42)(22 43)(23 44)(24 45)(49 60)(50 61)(51 62)(52 63)(53 64)(54 57)(55 58)(56 59)
(1 56)(2 21)(3 50)(4 23)(5 52)(6 17)(7 54)(8 19)(9 64)(10 47)(11 58)(12 41)(13 60)(14 43)(15 62)(16 45)(18 27)(20 29)(22 31)(24 25)(26 53)(28 55)(30 49)(32 51)(33 57)(34 48)(35 59)(36 42)(37 61)(38 44)(39 63)(40 46)
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 40)(10 33)(11 34)(12 35)(13 36)(14 37)(15 38)(16 39)(17 53)(18 54)(19 55)(20 56)(21 49)(22 50)(23 51)(24 52)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 57)(48 58)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 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 16)(2 38)(3 14)(4 36)(5 12)(6 34)(7 10)(8 40)(9 28)(11 26)(13 32)(15 30)(17 62)(18 43)(19 60)(20 41)(21 58)(22 47)(23 64)(24 45)(25 35)(27 33)(29 39)(31 37)(42 55)(44 53)(46 51)(48 49)(50 57)(52 63)(54 61)(56 59)

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I ··· 4P 4Q 4R 8A ··· 8H order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 4 4 8 ··· 8 size 1 1 ··· 1 4 4 8 8 2 ··· 2 4 ··· 4 8 8 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 C4○D4 C4○D8 C8⋊C22 kernel C2×C23.19D4 C2×C22⋊C8 C2×D4⋊C4 C2×C4.Q8 C2×C2.D8 C23.19D4 C2×C42⋊C2 C2×C4⋊D4 C22×C4 C24 C2×C4 C22 C22 # reps 1 1 2 1 1 8 1 1 3 1 8 8 2

Matrix representation of C2×C23.19D4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 13 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 13 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 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 16 0 0 0 0 0 0 16
,
 0 16 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 16 0 0 0 0 0 0 0 14 3 0 0 0 0 14 14
,
 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,4,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,14,14,0,0,0,0,3,14],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1] >;

C2×C23.19D4 in GAP, Magma, Sage, TeX

C_2\times C_2^3._{19}D_4
% in TeX

G:=Group("C2xC2^3.19D4");
// GroupNames label

G:=SmallGroup(128,1819);
// by ID

G=gap.SmallGroup(128,1819);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,100,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=f^2=1,e^4=d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,e*b*e^-1=b*c=c*b,b*d=d*b,f*b*f=b*c*d,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=c*d*e^3>;
// generators/relations

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