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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

Aliases: C2×Q32⋊C2, C16.1C23, C8.14C24, Q323C22, D8.3C23, C23.54D8, SD322C22, Q16.3C23, M5(2)⋊6C22, (C2×C4).55D8, C4.76(C2×D8), C8.38(C2×D4), (C2×Q32)⋊12C2, (C2×SD32)⋊5C2, (C2×C8).148D4, (C2×M5(2))⋊4C2, C2.29(C22×D8), C22.79(C2×D8), C4.20(C22×D4), (C2×C8).292C23, (C2×C16).34C22, (C2×Q16)⋊56C22, (C22×Q16)⋊21C2, C4○D8.31C22, (C22×C4).533D4, (C2×D8).152C22, (C22×C8).295C22, (C2×C4○D8).20C2, (C2×C4).877(C2×D4), SmallGroup(128,2145)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — C2×Q32⋊C2
C1C2C4C8C2×C8C22×C8C22×Q16 — C2×Q32⋊C2
C1C2C4C8 — C2×Q32⋊C2
C1C22C22×C4C22×C8 — C2×Q32⋊C2
C1C2C2C2C2C4C4C8 — 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 2A2B2C2D2E2F2G4A4B4C4D4E···4J8A8B8C8D8E8F16A···16H
order1222222244444···488888816···16
size1111228822228···82222444···4

32 irreducible representations

dim111111122224
type+++++++++++-
imageC1C2C2C2C2C2C2D4D4D8D8Q32⋊C2
kernelC2×Q32⋊C2C2×M5(2)C2×SD32C2×Q32Q32⋊C2C22×Q16C2×C4○D8C2×C8C22×C4C2×C4C23C2
# reps112281131624

Matrix representation of C2×Q32⋊C2 in GL6(𝔽17)

1600000
0160000
001000
000100
000010
000001
,
14130000
1130000
003795
00215129
0048210
00841514
,
340000
15140000
00103112
00221216
0094153
0013827
,
100000
010000
000120
001002
0000016
0000160

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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