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## G = C22×C4○D8order 128 = 27

### Direct product of C22 and C4○D8

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C22×C4○D8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C22×C4○D4 — C22×C4○D8
 Lower central C1 — C2 — C4 — C22×C4○D8
 Upper central C1 — C22×C4 — C23×C4 — C22×C4○D8
 Jennings C1 — C2 — C2 — C4 — C22×C4○D8

Generators and relations for C22×C4○D8
G = < a,b,c,d,e | a2=b2=c4=e2=1, d4=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d3 >

Subgroups: 1148 in 752 conjugacy classes, 436 normal (14 characteristic)
C1, C2, C2 [×6], C2 [×12], C4, C4 [×7], C4 [×8], C22 [×11], C22 [×44], C8 [×8], C2×C4 [×28], C2×C4 [×44], D4 [×8], D4 [×44], Q8 [×8], Q8 [×12], C23, C23 [×6], C23 [×24], C2×C8 [×28], D8 [×16], SD16 [×32], Q16 [×16], C22×C4 [×2], C22×C4 [×12], C22×C4 [×26], C2×D4 [×12], C2×D4 [×30], C2×Q8 [×12], C2×Q8 [×6], C4○D4 [×32], C4○D4 [×48], C24, C24 [×2], C22×C8 [×2], C22×C8 [×12], C2×D8 [×12], C2×SD16 [×24], C2×Q16 [×12], C4○D8 [×64], C23×C4, C23×C4 [×2], C22×D4 [×2], C22×D4 [×2], C22×Q8 [×2], C2×C4○D4 [×24], C2×C4○D4 [×12], C23×C8, C22×D8, C22×SD16 [×2], C22×Q16, C2×C4○D8 [×24], C22×C4○D4 [×2], C22×C4○D8
Quotients: C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], C2×D4 [×28], C24 [×31], C4○D8 [×4], C22×D4 [×14], C25, C2×C4○D8 [×6], D4×C23, C22×C4○D8

Smallest permutation representation of C22×C4○D8
On 64 points
Generators in S64
(1 28)(2 29)(3 30)(4 31)(5 32)(6 25)(7 26)(8 27)(9 56)(10 49)(11 50)(12 51)(13 52)(14 53)(15 54)(16 55)(17 48)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(33 58)(34 59)(35 60)(36 61)(37 62)(38 63)(39 64)(40 57)
(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)(17 60)(18 61)(19 62)(20 63)(21 64)(22 57)(23 58)(24 59)(25 56)(26 49)(27 50)(28 51)(29 52)(30 53)(31 54)(32 55)(33 46)(34 47)(35 48)(36 41)(37 42)(38 43)(39 44)(40 45)
(1 59 5 63)(2 60 6 64)(3 61 7 57)(4 62 8 58)(9 21 13 17)(10 22 14 18)(11 23 15 19)(12 24 16 20)(25 39 29 35)(26 40 30 36)(27 33 31 37)(28 34 32 38)(41 49 45 53)(42 50 46 54)(43 51 47 55)(44 52 48 56)
(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 11)(2 10)(3 9)(4 16)(5 15)(6 14)(7 13)(8 12)(17 57)(18 64)(19 63)(20 62)(21 61)(22 60)(23 59)(24 58)(25 53)(26 52)(27 51)(28 50)(29 49)(30 56)(31 55)(32 54)(33 47)(34 46)(35 45)(36 44)(37 43)(38 42)(39 41)(40 48)

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L ··· 2S 4A ··· 4H 4I 4J 4K 4L 4M ··· 4T 8A ··· 8P order 1 2 ··· 2 2 2 2 2 2 ··· 2 4 ··· 4 4 4 4 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 2 2 2 2 4 ··· 4 1 ··· 1 2 2 2 2 4 ··· 4 2 ··· 2

56 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D4 D4 C4○D8 kernel C22×C4○D8 C23×C8 C22×D8 C22×SD16 C22×Q16 C2×C4○D8 C22×C4○D4 C22×C4 C24 C22 # reps 1 1 1 2 1 24 2 7 1 16

Matrix representation of C22×C4○D8 in GL4(𝔽17) generated by

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

C22×C4○D8 in GAP, Magma, Sage, TeX

C_2^2\times C_4\circ D_8
% in TeX

G:=Group("C2^2xC4oD8");
// GroupNames label

G:=SmallGroup(128,2309);
// by ID

G=gap.SmallGroup(128,2309);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,-2,477,352,4037,2028,124]);
// Polycyclic

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

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