D12.37D4 - GroupNames

G = D₁₂.₃₇D₄ order 192 = 2⁶·3

7^th non-split extension by D₁₂ of D₄ acting via D₄/C₂²=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₁₂.₃₇D₄, Dic₆.₃₆D₄, C₄:C₄.₆₆D₆, C₂²:Q₈:₂S₃, C₄.₁₀₂(S₃xD₄), (C₂xQ₈).₅₂D₆, C₆.₄₈C₂²wrC₂, C₆.D₈:₃₈C₂, C₁₂.₁₅₃(C₂xD₄), (C₂xC₁₂).₂₆₆D₄, C₃:₄(D₄.₇D₄), (C₂²xC₆).₉₃D₄, C₆.₁₀₀(C₄oD₈), C₆.SD₁₆:₃₇C₂, (C₂²xC₄).₁₄₄D₆, (C₆xQ₈).₄₆C₂², C₁₂.₅₅D₄:₁₄C₂, C₂.₁₆(C₂³:₂D₆), (C₂xC₁₂).₃₆₆C₂³, C₆.₉₀(C₈.C₂²), C₂³.₃₄(C₃:D₄), C₂.₁₉(Q₈.₁₃D₆), (C₂xD₁₂).₂₄₃C₂², C₂.₁₁(Q₈.₁₁D₆), (C₂²xC₁₂).₁₇₀C₂², (C₂xDic₆).₂₇₀C₂², (C₂xC₃:Q₁₆):₈C₂, (C₃xC₂²:Q₈):₂C₂, (C₂xQ₈:₂S₃):₉C₂, (C₂xC₆).₄₉₇(C₂xD₄), (C₂xC₄oD₁₂).₁₀C₂, (C₂xC₃:C₈).₁₁₅C₂², (C₂xC₄).₁₇₃(C₃:D₄), (C₃xC₄:C₄).₁₁₃C₂², (C₂xC₄).₄₆₆(C₂²xS₃), C₂².₁₇₂(C₂xC₃:D₄), SmallGroup(192,606)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₁₂ — D₁₂.₃₇D₄

Chief series

C₁ — C₃ — C₆ — C₁₂ — C₂xC₁₂ — C₂xD₁₂ — C₂xC₄oD₁₂ — D₁₂.₃₇D₄

Lower central

C₃ — C₆ — C₂xC₁₂ — D₁₂.₃₇D₄

Upper central

C₁ — C₂² — C₂²xC₄ — C₂²:Q₈

Generators and relations for D₁₂.₃₇D₄
G = < a,b,c,d | a¹²=b²=c⁴=1, d²=a⁶, bab=a^-1, cac^-1=a⁷, ad=da, cbc^-1=a³b, bd=db, dcd^-1=c^-1 >

Subgroups: 448 in 152 conjugacy classes, 43 normal (39 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₈, C₂xC₄, C₂xC₄, D₄, Q₈, C₂³, C₂³, Dic₃, C₁₂, C₁₂, D₆, C₂xC₆, C₂xC₆, C₂²:C₄, C₄:C₄, C₄:C₄, C₂xC₈, SD₁₆, Q₁₆, C₂²xC₄, C₂²xC₄, C₂xD₄, C₂xQ₈, C₂xQ₈, C₄oD₄, C₃:C₈, Dic₆, Dic₆, C₄xS₃, D₁₂, D₁₂, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₂xC₁₂, C₃xQ₈, C₂²xS₃, C₂²xC₆, C₂²:C₈, D₄:C₄, Q₈:C₄, C₂²:Q₈, C₂xSD₁₆, C₂xQ₁₆, C₂xC₄oD₄, C₂xC₃:C₈, Q₈:₂S₃, C₃:Q₁₆, C₃xC₂²:C₄, C₃xC₄:C₄, C₃xC₄:C₄, C₂xDic₆, S₃xC₂xC₄, C₂xD₁₂, C₄oD₁₂, C₂xC₃:D₄, C₂²xC₁₂, C₆xQ₈, D₄.₇D₄, C₆.D₈, C₆.SD₁₆, C₁₂.₅₅D₄, C₂xQ₈:₂S₃, C₂xC₃:Q₁₆, C₃xC₂²:Q₈, C₂xC₄oD₁₂, D₁₂.₃₇D₄
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂xD₄, C₃:D₄, C₂²xS₃, C₂²wrC₂, C₄oD₈, C₈.C₂², S₃xD₄, C₂xC₃:D₄, D₄.₇D₄, C₂³:₂D₆, Q₈.₁₁D₆, Q₈.₁₃D₆, D₁₂.₃₇D₄

Smallest permutation representation of D₁₂.₃₇D₄
►On 96 points

Generators in S₉₆

(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 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)

(1 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 20)(14 19)(15 18)(16 17)(21 24)(22 23)(25 26)(27 36)(28 35)(29 34)(30 33)(31 32)(37 44)(38 43)(39 42)(40 41)(45 48)(46 47)(49 55)(50 54)(51 53)(56 60)(57 59)(61 65)(62 64)(66 72)(67 71)(68 70)(73 77)(74 76)(78 84)(79 83)(80 82)(85 93)(86 92)(87 91)(88 90)(94 96)

(1 85 41 71)(2 92 42 66)(3 87 43 61)(4 94 44 68)(5 89 45 63)(6 96 46 70)(7 91 47 65)(8 86 48 72)(9 93 37 67)(10 88 38 62)(11 95 39 69)(12 90 40 64)(13 79 34 56)(14 74 35 51)(15 81 36 58)(16 76 25 53)(17 83 26 60)(18 78 27 55)(19 73 28 50)(20 80 29 57)(21 75 30 52)(22 82 31 59)(23 77 32 54)(24 84 33 49)

(1 32 7 26)(2 33 8 27)(3 34 9 28)(4 35 10 29)(5 36 11 30)(6 25 12 31)(13 37 19 43)(14 38 20 44)(15 39 21 45)(16 40 22 46)(17 41 23 47)(18 42 24 48)(49 72 55 66)(50 61 56 67)(51 62 57 68)(52 63 58 69)(53 64 59 70)(54 65 60 71)(73 87 79 93)(74 88 80 94)(75 89 81 95)(76 90 82 96)(77 91 83 85)(78 92 84 86)

G:=sub<Sym(96)| (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)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,20)(14,19)(15,18)(16,17)(21,24)(22,23)(25,26)(27,36)(28,35)(29,34)(30,33)(31,32)(37,44)(38,43)(39,42)(40,41)(45,48)(46,47)(49,55)(50,54)(51,53)(56,60)(57,59)(61,65)(62,64)(66,72)(67,71)(68,70)(73,77)(74,76)(78,84)(79,83)(80,82)(85,93)(86,92)(87,91)(88,90)(94,96), (1,85,41,71)(2,92,42,66)(3,87,43,61)(4,94,44,68)(5,89,45,63)(6,96,46,70)(7,91,47,65)(8,86,48,72)(9,93,37,67)(10,88,38,62)(11,95,39,69)(12,90,40,64)(13,79,34,56)(14,74,35,51)(15,81,36,58)(16,76,25,53)(17,83,26,60)(18,78,27,55)(19,73,28,50)(20,80,29,57)(21,75,30,52)(22,82,31,59)(23,77,32,54)(24,84,33,49), (1,32,7,26)(2,33,8,27)(3,34,9,28)(4,35,10,29)(5,36,11,30)(6,25,12,31)(13,37,19,43)(14,38,20,44)(15,39,21,45)(16,40,22,46)(17,41,23,47)(18,42,24,48)(49,72,55,66)(50,61,56,67)(51,62,57,68)(52,63,58,69)(53,64,59,70)(54,65,60,71)(73,87,79,93)(74,88,80,94)(75,89,81,95)(76,90,82,96)(77,91,83,85)(78,92,84,86)>;

G:=Group( (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)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,20)(14,19)(15,18)(16,17)(21,24)(22,23)(25,26)(27,36)(28,35)(29,34)(30,33)(31,32)(37,44)(38,43)(39,42)(40,41)(45,48)(46,47)(49,55)(50,54)(51,53)(56,60)(57,59)(61,65)(62,64)(66,72)(67,71)(68,70)(73,77)(74,76)(78,84)(79,83)(80,82)(85,93)(86,92)(87,91)(88,90)(94,96), (1,85,41,71)(2,92,42,66)(3,87,43,61)(4,94,44,68)(5,89,45,63)(6,96,46,70)(7,91,47,65)(8,86,48,72)(9,93,37,67)(10,88,38,62)(11,95,39,69)(12,90,40,64)(13,79,34,56)(14,74,35,51)(15,81,36,58)(16,76,25,53)(17,83,26,60)(18,78,27,55)(19,73,28,50)(20,80,29,57)(21,75,30,52)(22,82,31,59)(23,77,32,54)(24,84,33,49), (1,32,7,26)(2,33,8,27)(3,34,9,28)(4,35,10,29)(5,36,11,30)(6,25,12,31)(13,37,19,43)(14,38,20,44)(15,39,21,45)(16,40,22,46)(17,41,23,47)(18,42,24,48)(49,72,55,66)(50,61,56,67)(51,62,57,68)(52,63,58,69)(53,64,59,70)(54,65,60,71)(73,87,79,93)(74,88,80,94)(75,89,81,95)(76,90,82,96)(77,91,83,85)(78,92,84,86) );

G=PermutationGroup([[(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),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,20),(14,19),(15,18),(16,17),(21,24),(22,23),(25,26),(27,36),(28,35),(29,34),(30,33),(31,32),(37,44),(38,43),(39,42),(40,41),(45,48),(46,47),(49,55),(50,54),(51,53),(56,60),(57,59),(61,65),(62,64),(66,72),(67,71),(68,70),(73,77),(74,76),(78,84),(79,83),(80,82),(85,93),(86,92),(87,91),(88,90),(94,96)], [(1,85,41,71),(2,92,42,66),(3,87,43,61),(4,94,44,68),(5,89,45,63),(6,96,46,70),(7,91,47,65),(8,86,48,72),(9,93,37,67),(10,88,38,62),(11,95,39,69),(12,90,40,64),(13,79,34,56),(14,74,35,51),(15,81,36,58),(16,76,25,53),(17,83,26,60),(18,78,27,55),(19,73,28,50),(20,80,29,57),(21,75,30,52),(22,82,31,59),(23,77,32,54),(24,84,33,49)], [(1,32,7,26),(2,33,8,27),(3,34,9,28),(4,35,10,29),(5,36,11,30),(6,25,12,31),(13,37,19,43),(14,38,20,44),(15,39,21,45),(16,40,22,46),(17,41,23,47),(18,42,24,48),(49,72,55,66),(50,61,56,67),(51,62,57,68),(52,63,58,69),(53,64,59,70),(54,65,60,71),(73,87,79,93),(74,88,80,94),(75,89,81,95),(76,90,82,96),(77,91,83,85),(78,92,84,86)]])

33 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	3	4A	4B	4C	4D	4E	4F	4G	4H	6A	6B	6C	6D	6E	8A	8B	8C	8D	12A	12B	12C	12D	12E	12F	12G	12H
order	1	2	2	2	2	2	2	3	4	4	4	4	4	4	4	4	6	6	6	6	6	8	8	8	8	12	12	12	12	12	12	12	12
size	1	1	1	1	4	12	12	2	2	2	2	2	8	8	12	12	2	2	2	4	4	12	12	12	12	4	4	4	4	8	8	8	8

33 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	4	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+				-	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	S₃	D₄	D₄	D₄	D₄	D₆	D₆	D₆	C₃:D₄	C₃:D₄	C₄oD₈	C₈.C₂²	S₃xD₄	Q₈.₁₁D₆	Q₈.₁₃D₆
kernel	D₁₂.₃₇D₄	C₆.D₈	C₆.SD₁₆	C₁₂.₅₅D₄	C₂xQ₈:₂S₃	C₂xC₃:Q₁₆	C₃xC₂²:Q₈	C₂xC₄oD₁₂	C₂²:Q₈	Dic₆	D₁₂	C₂xC₁₂	C₂²xC₆	C₄:C₄	C₂²xC₄	C₂xQ₈	C₂xC₄	C₂³	C₆	C₆	C₄	C₂	C₂
# reps	1	1	1	1	1	1	1	1	1	2	2	1	1	1	1	1	2	2	4	1	2	2	2

Matrix representation of D₁₂.₃₇D₄ ►in GL₆(F₇₃)

1	1	0	0	0	0
72	0	0	0	0	0
0	0	1	71	0	0
0	0	1	72	0	0
0	0	0	0	72	0
0	0	0	0	0	72

1	1	0	0	0	0
0	72	0	0	0	0
0	0	1	71	0	0
0	0	0	72	0	0
0	0	0	0	72	0
0	0	0	0	56	1

43	13	0	0	0	0
60	30	0	0	0	0
0	0	41	32	0	0
0	0	57	32	0	0
0	0	0	0	17	71
0	0	0	0	72	56

72	0	0	0	0	0
0	72	0	0	0	0
0	0	46	0	0	0
0	0	0	46	0	0
0	0	0	0	72	0
0	0	0	0	56	1

G:=sub<GL(6,GF(73))| [1,72,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,71,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,71,72,0,0,0,0,0,0,72,56,0,0,0,0,0,1],[43,60,0,0,0,0,13,30,0,0,0,0,0,0,41,57,0,0,0,0,32,32,0,0,0,0,0,0,17,72,0,0,0,0,71,56],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,72,56,0,0,0,0,0,1] >;

D₁₂.₃₇D₄ in GAP, Magma, Sage, TeX

D_{12}._{37}D_4

% in TeX

G:=Group("D12.37D4");

// GroupNames label

G:=SmallGroup(192,606);

// by ID

G=gap.SmallGroup(192,606);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,219,184,1123,297,136,6278]);

// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=c^4=1,d^2=a^6,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=c^-1>;

// generators/relations

G = D12.37D4 order 192 = 26·3

7th non-split extension by D12 of D4 acting via D4/C22=C2

G = D₁₂.₃₇D₄ order 192 = 2⁶·3

7^th non-split extension by D₁₂ of D₄ acting via D₄/C₂²=C₂