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## G = C22×C4.10D4order 128 = 27

### Direct product of C22 and C4.10D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C22×C4.10D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — Q8×C23 — C22×C4.10D4
 Lower central C1 — C2 — C22 — C22×C4.10D4
 Upper central C1 — C23 — C23×C4 — C22×C4.10D4
 Jennings C1 — C2 — C2 — C2×C4 — C22×C4.10D4

Generators and relations for C22×C4.10D4
G = < a,b,c,d,e | a2=b2=c4=1, d4=c2, e2=dcd-1=c-1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ce=ec, ede-1=c-1d3 >

Subgroups: 556 in 368 conjugacy classes, 180 normal (9 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×8], C4 [×8], C22, C22 [×10], C22 [×12], C8 [×8], C2×C4, C2×C4 [×35], C2×C4 [×24], Q8 [×32], C23, C23 [×6], C23 [×4], C2×C8 [×12], M4(2) [×8], M4(2) [×12], C22×C4 [×26], C22×C4 [×8], C2×Q8 [×16], C2×Q8 [×48], C24, C4.10D4 [×16], C22×C8 [×2], C2×M4(2) [×12], C2×M4(2) [×6], C23×C4, C23×C4 [×2], C22×Q8 [×12], C22×Q8 [×8], C2×C4.10D4 [×12], C22×M4(2) [×2], Q8×C23, C22×C4.10D4
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, C4.10D4 [×4], C2×C22⋊C4 [×12], C23×C4, C22×D4 [×2], C2×C4.10D4 [×6], C22×C22⋊C4, C22×C4.10D4

Smallest permutation representation of C22×C4.10D4
On 64 points
Generators in S64
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 57)(24 58)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(31 49)(32 50)(33 46)(34 47)(35 48)(36 41)(37 42)(38 43)(39 44)(40 45)
(1 55)(2 56)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 29)(10 30)(11 31)(12 32)(13 25)(14 26)(15 27)(16 28)(17 38)(18 39)(19 40)(20 33)(21 34)(22 35)(23 36)(24 37)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 64)
(1 15 5 11)(2 12 6 16)(3 9 7 13)(4 14 8 10)(17 57 21 61)(18 62 22 58)(19 59 23 63)(20 64 24 60)(25 49 29 53)(26 54 30 50)(27 51 31 55)(28 56 32 52)(33 48 37 44)(34 45 38 41)(35 42 39 46)(36 47 40 43)
(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 60 11 24 5 64 15 20)(2 23 16 59 6 19 12 63)(3 58 13 22 7 62 9 18)(4 21 10 57 8 17 14 61)(25 35 53 46 29 39 49 42)(26 45 50 34 30 41 54 38)(27 33 55 44 31 37 51 48)(28 43 52 40 32 47 56 36)

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I ··· 4P 8A ··· 8P 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 2 4 type + + + + + - image C1 C2 C2 C2 C4 C4 D4 C4.10D4 kernel C22×C4.10D4 C2×C4.10D4 C22×M4(2) Q8×C23 C23×C4 C22×Q8 C22×C4 C22 # reps 1 12 2 1 4 12 8 4

Matrix representation of C22×C4.10D4 in GL8(𝔽17)

 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 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 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 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 11 1 0 0 0 0 0 0 16 6 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0
,
 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 16 6 0 0 0 0 0 0 11 1 0 0 0 0 0 0 0 0 0 0 5 12 0 0 0 0 0 0 12 12 0 0 0 0 12 12 0 0 0 0 0 0 12 5 0 0

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,11,16,0,0,0,0,0,0,1,6,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,11,0,0,0,0,0,0,6,1,0,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,12,5,0,0,0,0,5,12,0,0,0,0,0,0,12,12,0,0] >;

C22×C4.10D4 in GAP, Magma, Sage, TeX

C_2^2\times C_4._{10}D_4
% in TeX

G:=Group("C2^2xC4.10D4");
// GroupNames label

G:=SmallGroup(128,1618);
// by ID

G=gap.SmallGroup(128,1618);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,456,2804,2028,124]);
// Polycyclic

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

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