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## G = (C2×D4).303D4order 128 = 27

### 56th non-split extension by C2×D4 of D4 acting via D4/C22=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — (C2×D4).303D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C42⋊C2 — C23.33C23 — (C2×D4).303D4
 Lower central C1 — C2 — C2×C4 — (C2×D4).303D4
 Upper central C1 — C22 — C2×C4○D4 — (C2×D4).303D4
 Jennings C1 — C2 — C2 — C2×C4 — (C2×D4).303D4

Generators and relations for (C2×D4).303D4
G = < a,b,c,d,e | a2=b4=c2=e2=1, d4=b2, ebe=ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, dbd-1=ab-1, dcd-1=ece=ab2c, ede=ad3 >

Subgroups: 380 in 199 conjugacy classes, 92 normal (38 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×2], C4 [×9], C22, C22 [×2], C22 [×11], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×17], D4 [×12], Q8 [×4], C23, C23 [×2], C23, C42 [×3], C22⋊C4 [×7], C4⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×2], C2×C8 [×2], C2×C8 [×2], M4(2) [×2], C22×C4, C22×C4 [×2], C22×C4 [×5], C2×D4 [×2], C2×D4 [×2], C2×D4 [×3], C2×Q8 [×2], C4○D4 [×8], C22⋊C8 [×4], D4⋊C4 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], Q8⋊C4 [×2], C4.Q8 [×2], C2.D8 [×6], C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2, C42⋊C2 [×2], C4×D4 [×3], C4×Q8, C4⋊D4 [×2], C4⋊D4 [×3], C22⋊Q8 [×2], C22⋊Q8, C22×C8, C2×M4(2), C2×C4○D4 [×2], (C22×C8)⋊C2, C23.24D4, C23.36D4, C2×C2.D8, M4(2)⋊C4, C22.D8 [×2], C23.19D4 [×2], C23.48D4 [×2], C23.20D4 [×2], C23.33C23, C22.31C24, (C2×D4).303D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22.D4 [×4], C22×D4, C2×C4○D4 [×2], C2×C22.D4, D4○D8, Q8○D8, (C2×D4).303D4

Smallest permutation representation of (C2×D4).303D4
On 64 points
Generators in S64
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 36)(10 37)(11 38)(12 39)(13 40)(14 33)(15 34)(16 35)(17 51)(18 52)(19 53)(20 54)(21 55)(22 56)(23 49)(24 50)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 64)
(1 47 5 43)(2 60 6 64)(3 41 7 45)(4 62 8 58)(9 17 13 21)(10 56 14 52)(11 19 15 23)(12 50 16 54)(18 37 22 33)(20 39 24 35)(25 63 29 59)(26 44 30 48)(27 57 31 61)(28 46 32 42)(34 49 38 53)(36 51 40 55)
(1 52)(2 23)(3 54)(4 17)(5 56)(6 19)(7 50)(8 21)(9 62)(10 43)(11 64)(12 45)(13 58)(14 47)(15 60)(16 41)(18 25)(20 27)(22 29)(24 31)(26 49)(28 51)(30 53)(32 55)(33 63)(34 44)(35 57)(36 46)(37 59)(38 48)(39 61)(40 42)
(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)
(2 28)(3 7)(4 26)(6 32)(8 30)(9 38)(10 14)(11 36)(13 34)(15 40)(17 19)(18 56)(20 54)(21 23)(22 52)(24 50)(27 31)(33 37)(41 61)(42 48)(43 59)(44 46)(45 57)(47 63)(49 55)(51 53)(58 64)(60 62)

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C 4D 4E ··· 4N 4O 4P 4Q 8A 8B 8C 8D 8E 8F order 1 2 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 4 4 8 8 8 8 8 8 size 1 1 1 1 2 2 4 4 8 2 2 2 2 4 ··· 4 8 8 8 4 4 4 4 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 4 4 type + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 C4○D4 D4○D8 Q8○D8 kernel (C2×D4).303D4 (C22×C8)⋊C2 C23.24D4 C23.36D4 C2×C2.D8 M4(2)⋊C4 C22.D8 C23.19D4 C23.48D4 C23.20D4 C23.33C23 C22.31C24 C2×D4 C2×Q8 C2×C4 C2 C2 # reps 1 1 1 1 1 1 2 2 2 2 1 1 3 1 8 2 2

Matrix representation of (C2×D4).303D4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 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 1
,
 16 2 0 0 0 0 0 1 0 0 0 0 0 0 11 0 3 0 0 0 0 11 0 3 0 0 16 0 6 0 0 0 0 16 0 6
,
 1 15 0 0 0 0 0 16 0 0 0 0 0 0 0 9 0 15 0 0 8 0 2 0 0 0 0 7 0 8 0 0 10 0 9 0
,
 13 0 0 0 0 0 13 4 0 0 0 0 0 0 3 14 6 11 0 0 3 3 6 6 0 0 14 3 14 3 0 0 14 14 14 14
,
 1 0 0 0 0 0 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,2,1,0,0,0,0,0,0,11,0,16,0,0,0,0,11,0,16,0,0,3,0,6,0,0,0,0,3,0,6],[1,0,0,0,0,0,15,16,0,0,0,0,0,0,0,8,0,10,0,0,9,0,7,0,0,0,0,2,0,9,0,0,15,0,8,0],[13,13,0,0,0,0,0,4,0,0,0,0,0,0,3,3,14,14,0,0,14,3,3,14,0,0,6,6,14,14,0,0,11,6,3,14],[1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;

(C2×D4).303D4 in GAP, Magma, Sage, TeX

(C_2\times D_4)._{303}D_4
% in TeX

G:=Group("(C2xD4).303D4");
// GroupNames label

G:=SmallGroup(128,1830);
// by ID

G=gap.SmallGroup(128,1830);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,100,1018,248,4037,1027,124]);
// Polycyclic

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

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