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## G = C2×Q32⋊C2order 128 = 27

### Direct product of C2 and Q32⋊C2

direct product, p-group, metabelian, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C8 — C2×Q32⋊C2
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C22×C8 — C22×Q16 — C2×Q32⋊C2
 Lower central C1 — C2 — C4 — C8 — C2×Q32⋊C2
 Upper central C1 — C22 — C22×C4 — C22×C8 — C2×Q32⋊C2
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C8 — C2×Q32⋊C2

Generators and relations for C2×Q32⋊C2
G = < a,b,c,d | a2=b16=d2=1, c2=b8, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd=b9, cd=dc >

Subgroups: 372 in 180 conjugacy classes, 92 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C16, C2×C8, C2×C8, D8, D8, SD16, Q16, Q16, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C16, M5(2), SD32, Q32, C22×C8, C2×D8, C2×SD16, C2×Q16, C2×Q16, C2×Q16, C4○D8, C4○D8, C22×Q8, C2×C4○D4, C2×M5(2), C2×SD32, C2×Q32, Q32⋊C2, C22×Q16, C2×C4○D8, C2×Q32⋊C2
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C24, C2×D8, C22×D4, Q32⋊C2, C22×D8, C2×Q32⋊C2

Smallest permutation representation of C2×Q32⋊C2
On 64 points
Generators in S64
(1 40)(2 41)(3 42)(4 43)(5 44)(6 45)(7 46)(8 47)(9 48)(10 33)(11 34)(12 35)(13 36)(14 37)(15 38)(16 39)(17 55)(18 56)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(25 63)(26 64)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)
(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 17 9 25)(2 32 10 24)(3 31 11 23)(4 30 12 22)(5 29 13 21)(6 28 14 20)(7 27 15 19)(8 26 16 18)(33 62 41 54)(34 61 42 53)(35 60 43 52)(36 59 44 51)(37 58 45 50)(38 57 46 49)(39 56 47 64)(40 55 48 63)
(1 24)(2 17)(3 26)(4 19)(5 28)(6 21)(7 30)(8 23)(9 32)(10 25)(11 18)(12 27)(13 20)(14 29)(15 22)(16 31)(33 63)(34 56)(35 49)(36 58)(37 51)(38 60)(39 53)(40 62)(41 55)(42 64)(43 57)(44 50)(45 59)(46 52)(47 61)(48 54)

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4J 8A 8B 8C 8D 8E 8F 16A ··· 16H order 1 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 8 8 8 8 8 8 16 ··· 16 size 1 1 1 1 2 2 8 8 2 2 2 2 8 ··· 8 2 2 2 2 4 4 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 D4 D4 D8 D8 Q32⋊C2 kernel C2×Q32⋊C2 C2×M5(2) C2×SD32 C2×Q32 Q32⋊C2 C22×Q16 C2×C4○D8 C2×C8 C22×C4 C2×C4 C23 C2 # reps 1 1 2 2 8 1 1 3 1 6 2 4

Matrix representation of C2×Q32⋊C2 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
,
 14 13 0 0 0 0 11 3 0 0 0 0 0 0 3 7 9 5 0 0 2 15 12 9 0 0 4 8 2 10 0 0 8 4 15 14
,
 3 4 0 0 0 0 15 14 0 0 0 0 0 0 10 3 1 12 0 0 2 2 12 16 0 0 9 4 15 3 0 0 13 8 2 7
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 2 0 0 0 1 0 0 2 0 0 0 0 0 16 0 0 0 0 16 0

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],[14,11,0,0,0,0,13,3,0,0,0,0,0,0,3,2,4,8,0,0,7,15,8,4,0,0,9,12,2,15,0,0,5,9,10,14],[3,15,0,0,0,0,4,14,0,0,0,0,0,0,10,2,9,13,0,0,3,2,4,8,0,0,1,12,15,2,0,0,12,16,3,7],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,2,0,0,16,0,0,0,2,16,0] >;

C2×Q32⋊C2 in GAP, Magma, Sage, TeX

C_2\times Q_{32}\rtimes C_2
% in TeX

G:=Group("C2xQ32:C2");
// GroupNames label

G:=SmallGroup(128,2145);
// by ID

G=gap.SmallGroup(128,2145);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,-2,-2,253,456,1430,1684,851,242,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^16=d^2=1,c^2=b^8,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d=b^9,c*d=d*c>;
// generators/relations

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