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## G = (C22×D8).C2order 128 = 27

### 3rd non-split extension by C22×D8 of C2 acting faithfully

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — (C22×D8).C2
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×D4 — C2×C4.4D4 — (C22×D8).C2
 Lower central C1 — C2 — C22×C4 — (C22×D8).C2
 Upper central C1 — C23 — C2×C42 — (C22×D8).C2
 Jennings C1 — C2 — C2 — C22×C4 — (C22×D8).C2

Generators and relations for (C22×D8).C2
G = < a,b,c,d,e | a2=b2=c8=d2=1, e2=c4, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c-1, ece-1=ac3, ede-1=abc2d >

Subgroups: 536 in 213 conjugacy classes, 54 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×5], C22 [×3], C22 [×4], C22 [×20], C8 [×4], C2×C4 [×6], C2×C4 [×13], D4 [×12], Q8 [×6], C23, C23 [×16], C42 [×2], C22⋊C4 [×12], C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×4], D8 [×8], SD16 [×8], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×4], C2×D4 [×10], C2×Q8 [×2], C2×Q8 [×5], C24 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], C2×C42, C2×C22⋊C4 [×4], C2×C4⋊C4, C4.4D4 [×4], C22×C8 [×2], C2×D8 [×6], C2×SD16 [×6], C22×D4 [×2], C22×Q8, C22.7C42, C24.3C22, C2×D4⋊C4, C2×Q8⋊C4, C2×C4.4D4, C22×D8, C22×SD16, (C22×D8).C2
Quotients: C1, C2 [×7], C22 [×7], D4 [×12], C23, C2×D4 [×6], C4○D4, C22≀C2 [×3], C4⋊D4 [×3], C41D4, C4○D8 [×2], C8⋊C22 [×2], C232D4, D4⋊D4 [×2], D4.2D4 [×2], C8.12D4, C83D4, (C22×D8).C2

Smallest permutation representation of (C22×D8).C2
On 64 points
Generators in S64
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(9 52)(10 53)(11 54)(12 55)(13 56)(14 49)(15 50)(16 51)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 57)(24 58)(33 44)(34 45)(35 46)(36 47)(37 48)(38 41)(39 42)(40 43)
(1 55)(2 56)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 32)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 35)(18 36)(19 37)(20 38)(21 39)(22 40)(23 33)(24 34)(41 62)(42 63)(43 64)(44 57)(45 58)(46 59)(47 60)(48 61)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(8 16)(17 62)(18 61)(19 60)(20 59)(21 58)(22 57)(23 64)(24 63)(25 52)(26 51)(27 50)(28 49)(29 56)(30 55)(31 54)(32 53)(33 43)(34 42)(35 41)(36 48)(37 47)(38 46)(39 45)(40 44)
(1 63 5 59)(2 24 6 20)(3 61 7 57)(4 22 8 18)(9 41 13 45)(10 33 14 37)(11 47 15 43)(12 39 16 35)(17 27 21 31)(19 25 23 29)(26 60 30 64)(28 58 32 62)(34 52 38 56)(36 50 40 54)(42 51 46 55)(44 49 48 53)

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

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

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

32 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 8A ··· 8H order 1 2 ··· 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 8 ··· 8 size 1 1 ··· 1 8 8 8 8 2 2 2 2 4 4 4 4 8 8 8 8 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4 C4○D4 C4○D8 C8⋊C22 kernel (C22×D8).C2 C22.7C42 C24.3C22 C2×D4⋊C4 C2×Q8⋊C4 C2×C4.4D4 C22×D8 C22×SD16 C2×C8 C22×C4 C2×D4 C2×Q8 C2×C4 C22 C22 # reps 1 1 1 1 1 1 1 1 4 2 4 2 2 8 2

Matrix representation of (C22×D8).C2 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
,
 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
,
 3 14 0 0 0 0 3 3 0 0 0 0 0 0 7 13 0 0 0 0 4 10 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 3 14 0 0 0 0 14 14 0 0 0 0 0 0 10 4 0 0 0 0 5 7 0 0 0 0 0 0 1 0 0 0 0 0 16 16
,
 4 0 0 0 0 0 0 13 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 15 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],[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],[3,3,0,0,0,0,14,3,0,0,0,0,0,0,7,4,0,0,0,0,13,10,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[3,14,0,0,0,0,14,14,0,0,0,0,0,0,10,5,0,0,0,0,4,7,0,0,0,0,0,0,1,16,0,0,0,0,0,16],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,15,1] >;

(C22×D8).C2 in GAP, Magma, Sage, TeX

(C_2^2\times D_8).C_2
% in TeX

G:=Group("(C2^2xD8).C2");
// GroupNames label

G:=SmallGroup(128,744);
// by ID

G=gap.SmallGroup(128,744);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,456,422,387,2019,1018,521,248,2804,1411,718,172]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^8=d^2=1,e^2=c^4,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,d*c*d=c^-1,e*c*e^-1=a*c^3,e*d*e^-1=a*b*c^2*d>;
// generators/relations

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