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## G = C2×C23.38D4order 128 = 27

### Direct product of C2 and C23.38D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C2×C23.38D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — Q8×C23 — C2×C23.38D4
 Lower central C1 — C2 — C4 — C2×C23.38D4
 Upper central C1 — C23 — C23×C4 — C2×C23.38D4
 Jennings C1 — C2 — C2 — C2×C4 — C2×C23.38D4

Generators and relations for C2×C23.38D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=fbf-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce3 >

Subgroups: 588 in 376 conjugacy classes, 180 normal (16 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×2], C4 [×6], C4 [×12], C22, C22 [×10], C22 [×12], C8 [×4], C2×C4 [×2], C2×C4 [×26], C2×C4 [×36], Q8 [×8], Q8 [×28], C23, C23 [×6], C23 [×4], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×4], M4(2) [×8], C22×C4 [×2], C22×C4 [×12], C22×C4 [×16], C2×Q8 [×28], C2×Q8 [×42], C24, Q8⋊C4 [×16], C2×C42, C2×C22⋊C4, C2×C4⋊C4 [×2], C42⋊C2 [×4], C42⋊C2 [×2], C22×C8 [×2], C2×M4(2) [×4], C2×M4(2) [×4], C23×C4, C23×C4, C22×Q8 [×14], C22×Q8 [×7], C2×Q8⋊C4 [×4], C23.38D4 [×8], C2×C42⋊C2, C22×M4(2), Q8×C23, C2×C23.38D4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], C23 [×15], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C24, C2×C22⋊C4 [×12], C8.C22 [×4], C23×C4, C22×D4 [×2], C23.38D4 [×4], C22×C22⋊C4, C2×C8.C22 [×2], C2×C23.38D4

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

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I ··· 4X 8A ··· 8H order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 2 2 2 2 2 ··· 2 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 4 type + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 D4 D4 C8.C22 kernel C2×C23.38D4 C2×Q8⋊C4 C23.38D4 C2×C42⋊C2 C22×M4(2) Q8×C23 C22×Q8 C22×C4 C24 C22 # reps 1 4 8 1 1 1 16 7 1 4

Matrix representation of C2×C23.38D4 in GL8(𝔽17)

 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 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
,
 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 4 13 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16
,
 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 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16
,
 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 1 16 4 9 0 0 0 0 0 0 4 0 0 0 0 0 13 0 0 0 0 0 0 0 13 0 0 16
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 13 4 1 15 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 13

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,4,0,0,0,0,0,16,0,13,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,1,0,13,13,0,0,0,0,16,0,0,0,0,0,0,0,4,4,0,0,0,0,0,0,9,0,0,16],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,13,16,0,0,0,0,0,0,4,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,15,0,13] >;

C2×C23.38D4 in GAP, Magma, Sage, TeX

C_2\times C_2^3._{38}D_4
% in TeX

G:=Group("C2xC2^3.38D4");
// GroupNames label

G:=SmallGroup(128,1626);
// by ID

G=gap.SmallGroup(128,1626);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,456,1430,2804,1411,172]);
// Polycyclic

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

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