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## G = C2×C23.10D4order 128 = 27

### Direct product of C2 and C23.10D4

direct product, p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C2×C23.10D4
 Chief series C1 — C2 — C22 — C23 — C24 — C23×C4 — C22×C22⋊C4 — C2×C23.10D4
 Lower central C1 — C23 — C2×C23.10D4
 Upper central C1 — C24 — C2×C23.10D4
 Jennings C1 — C23 — C2×C23.10D4

Generators and relations for C2×C23.10D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf=bc=cb, bd=db, ebe-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=ce-1 >

Subgroups: 1300 in 626 conjugacy classes, 164 normal (16 characteristic)
C1, C2 [×3], C2 [×12], C2 [×8], C4 [×14], C22 [×3], C22 [×32], C22 [×72], C2×C4 [×8], C2×C4 [×54], D4 [×32], C23, C23 [×22], C23 [×104], C22⋊C4 [×32], C4⋊C4 [×8], C22×C4 [×18], C22×C4 [×26], C2×D4 [×56], C24, C24 [×12], C24 [×24], C2.C42 [×4], C2×C22⋊C4 [×16], C2×C22⋊C4 [×16], C2×C4⋊C4 [×4], C2×C4⋊C4 [×4], C23×C4 [×3], C23×C4 [×2], C22×D4 [×4], C22×D4 [×12], C25 [×2], C2×C2.C42, C23.10D4 [×8], C22×C22⋊C4 [×4], C22×C4⋊C4, D4×C23, C2×C23.10D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×16], C23 [×15], C2×D4 [×24], C4○D4 [×6], C24, C22≀C2 [×4], C4⋊D4 [×12], C22.D4 [×8], C4.4D4 [×4], C22×D4 [×4], C2×C4○D4 [×3], C23.10D4 [×8], C2×C22≀C2, C2×C4⋊D4 [×3], C2×C22.D4 [×2], C2×C4.4D4, C2×C23.10D4

Smallest permutation representation of C2×C23.10D4
On 64 points
Generators in S64
(1 23)(2 24)(3 21)(4 22)(5 57)(6 58)(7 59)(8 60)(9 45)(10 46)(11 47)(12 48)(13 42)(14 43)(15 44)(16 41)(17 51)(18 52)(19 49)(20 50)(25 40)(26 37)(27 38)(28 39)(29 36)(30 33)(31 34)(32 35)(53 62)(54 63)(55 64)(56 61)
(1 63)(2 34)(3 61)(4 36)(5 20)(6 48)(7 18)(8 46)(9 25)(10 60)(11 27)(12 58)(13 30)(14 55)(15 32)(16 53)(17 39)(19 37)(21 56)(22 29)(23 54)(24 31)(26 49)(28 51)(33 42)(35 44)(38 47)(40 45)(41 62)(43 64)(50 57)(52 59)
(1 9)(2 10)(3 11)(4 12)(5 32)(6 29)(7 30)(8 31)(13 18)(14 19)(15 20)(16 17)(21 47)(22 48)(23 45)(24 46)(25 63)(26 64)(27 61)(28 62)(33 59)(34 60)(35 57)(36 58)(37 55)(38 56)(39 53)(40 54)(41 51)(42 52)(43 49)(44 50)
(1 52)(2 49)(3 50)(4 51)(5 56)(6 53)(7 54)(8 55)(9 42)(10 43)(11 44)(12 41)(13 45)(14 46)(15 47)(16 48)(17 22)(18 23)(19 24)(20 21)(25 33)(26 34)(27 35)(28 36)(29 39)(30 40)(31 37)(32 38)(57 61)(58 62)(59 63)(60 64)
(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 12)(4 10)(5 32)(6 8)(7 30)(14 17)(16 19)(22 46)(24 48)(25 63)(26 28)(27 61)(29 31)(33 59)(34 36)(35 57)(37 39)(38 56)(40 54)(41 49)(43 51)(53 55)(58 60)(62 64)

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

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

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

44 conjugacy classes

 class 1 2A ··· 2O 2P ··· 2W 4A ··· 4T order 1 2 ··· 2 2 ··· 2 4 ··· 4 size 1 1 ··· 1 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 C4○D4 kernel C2×C23.10D4 C2×C2.C42 C23.10D4 C22×C22⋊C4 C22×C4⋊C4 D4×C23 C22×C4 C24 C23 # reps 1 1 8 4 1 1 8 8 12

Matrix representation of C2×C23.10D4 in GL7(𝔽5)

 4 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 4
,
 1 0 0 0 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 4
,
 1 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 4
,
 1 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 1 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 1 0
,
 4 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 4

G:=sub<GL(7,GF(5))| [4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0],[4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4] >;

C2×C23.10D4 in GAP, Magma, Sage, TeX

C_2\times C_2^3._{10}D_4
% in TeX

G:=Group("C2xC2^3.10D4");
// GroupNames label

G:=SmallGroup(128,1118);
// by ID

G=gap.SmallGroup(128,1118);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,758,723,100]);
// Polycyclic

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

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