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## G = (C2×D4).304D4order 128 = 27

### 57th 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).304D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C42⋊C2 — C23.33C23 — (C2×D4).304D4
 Lower central C1 — C2 — C2×C4 — (C2×D4).304D4
 Upper central C1 — C22 — C2×C4○D4 — (C2×D4).304D4
 Jennings C1 — C2 — C2 — C2×C4 — (C2×D4).304D4

Generators and relations for (C2×D4).304D4
G = < a,b,c,d,e | a2=b4=c2=1, d4=e2=b2, ebe-1=ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, dbd-1=ab-1, dcd-1=ece-1=ab2c, ede-1=ab2d3 >

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 [×6], C2.D8 [×2], 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×C4.Q8, M4(2)⋊C4, C23.46D4 [×2], C23.19D4 [×2], C23.47D4 [×2], C23.20D4 [×2], C23.33C23, C22.31C24, (C2×D4).304D4
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○SD16 [×2], (C2×D4).304D4

Smallest permutation representation of (C2×D4).304D4
On 64 points
Generators in S64
(1 47)(2 48)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)(25 53)(26 54)(27 55)(28 56)(29 49)(30 50)(31 51)(32 52)(33 63)(34 64)(35 57)(36 58)(37 59)(38 60)(39 61)(40 62)
(1 56 5 52)(2 25 6 29)(3 50 7 54)(4 27 8 31)(9 60 13 64)(10 35 14 39)(11 62 15 58)(12 37 16 33)(17 59 21 63)(18 34 22 38)(19 61 23 57)(20 36 24 40)(26 41 30 45)(28 43 32 47)(42 55 46 51)(44 49 48 53)
(1 39)(2 58)(3 33)(4 60)(5 35)(6 62)(7 37)(8 64)(9 27)(10 52)(11 29)(12 54)(13 31)(14 56)(15 25)(16 50)(17 26)(18 51)(19 28)(20 53)(21 30)(22 55)(23 32)(24 49)(34 46)(36 48)(38 42)(40 44)(41 63)(43 57)(45 59)(47 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 12 5 16)(2 24 6 20)(3 10 7 14)(4 22 8 18)(9 46 13 42)(11 44 15 48)(17 43 21 47)(19 41 23 45)(25 62 29 58)(26 39 30 35)(27 60 31 64)(28 37 32 33)(34 55 38 51)(36 53 40 49)(50 57 54 61)(52 63 56 59)

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

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

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

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 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 C4○D4 D4○SD16 kernel (C2×D4).304D4 (C22×C8)⋊C2 C23.24D4 C23.36D4 C2×C4.Q8 M4(2)⋊C4 C23.46D4 C23.19D4 C23.47D4 C23.20D4 C23.33C23 C22.31C24 C2×D4 C2×Q8 C2×C4 C2 # reps 1 1 1 1 1 1 2 2 2 2 1 1 3 1 8 4

Matrix representation of (C2×D4).304D4 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
,
 12 16 0 0 0 0 7 5 0 0 0 0 0 0 1 0 0 15 0 0 16 0 16 1 0 0 0 1 0 1 0 0 1 0 0 16
,
 5 1 0 0 0 0 10 12 0 0 0 0 0 0 4 0 8 0 0 0 0 0 13 13 0 0 13 0 13 0 0 0 4 4 4 0
,
 14 13 0 0 0 0 2 3 0 0 0 0 0 0 6 6 0 0 0 0 14 0 0 0 0 0 11 14 14 3 0 0 3 3 14 14
,
 16 0 0 0 0 0 10 1 0 0 0 0 0 0 4 0 0 0 0 0 13 13 0 0 0 0 13 0 13 0 0 0 0 0 0 4

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],[12,7,0,0,0,0,16,5,0,0,0,0,0,0,1,16,0,1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,15,1,1,16],[5,10,0,0,0,0,1,12,0,0,0,0,0,0,4,0,13,4,0,0,0,0,0,4,0,0,8,13,13,4,0,0,0,13,0,0],[14,2,0,0,0,0,13,3,0,0,0,0,0,0,6,14,11,3,0,0,6,0,14,3,0,0,0,0,14,14,0,0,0,0,3,14],[16,10,0,0,0,0,0,1,0,0,0,0,0,0,4,13,13,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,4] >;

(C2×D4).304D4 in GAP, Magma, Sage, TeX

(C_2\times D_4)._{304}D_4
% in TeX

G:=Group("(C2xD4).304D4");
// GroupNames label

G:=SmallGroup(128,1831);
// by ID

G=gap.SmallGroup(128,1831);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,100,1018,248,4037,1027,124]);
// Polycyclic

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

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